A covariance formula for topological events of smooth Gaussian fields
AA COVARIANCE FORMULA FOR TOPOLOGICAL EVENTSOF SMOOTH GAUSSIAN FIELDS
DMITRY BELIAEV , STEPHEN MUIRHEAD , AND ALEJANDRO RIVERA Abstract.
We derive a covariance formula for the class of ‘topological events’ of smooth Gauss-ian fields on manifolds; these are events that depend only on the topology of the level sets ofthe field, for example (i) crossing events for level or excursion sets, (ii) events measurable withrespect to the number of connected components of level or excursion sets of a given diffeomor-phism class, and (iii) persistence events. As an application of the covariance formula, we derivestrong mixing bounds for topological events, as well as lower concentration inequalities for ad-ditive topological functionals (e.g. the number of connected components) of the level sets thatsatisfy a law of large numbers. The covariance formula also gives an alternate justification of theHarris criterion, which conjecturally describes the boundary of the percolation university classfor level sets of stationary Gaussian fields. Our work is inspired by [44], in which a correlationinequality was derived for certain topological events on the plane, as well as by [40], in which asimilar covariance formula was established for finite-dimensional Gaussian vectors. Introduction
In recent years there has been much progress in the study of the topology of level sets of smoothGaussian fields. Techniques have been developed to estimate their homology (see [37, 38], andalso [10, 14, 21, 31, 47]), and also their large scale connectivity properties (see [1, 5], and also[9, 35, 36, 45]) using ideas from Bernoulli percolation. When studying the topology of level sets,one often has to estimate quantities such asCov( A , A ) := P [ A ∩ A ] − P [ A ] P [ A ] , where A and A are events of topological nature. Since the events A and A in general do notadmit explicit integral representations, the quantity Cov( A , A ) is often estimated indirectly,leading to inequalities of varying precision. In the present work we prove an exact formula forCov( A , A ), where A and A belong to a large class of ‘topological events’.Let us illustrate our formula with a simple example. Let f be an a.s. C centred Gaussianfield on R , with covariance K ( x, y ) := Cov( f ( x ) , f ( y )), such that, for each distinct x, y ∈ R , Mathematical Institute, University of Oxford School of Mathematical Sciences, Queen Mary University of London Institute of Mathematics, EPFL
E-mail addresses : [email protected], [email protected],[email protected] . Date : May 19, 2020.2010
Mathematics Subject Classification.
Key words and phrases.
Gaussian fields, topology, covariance formula.The first author was partially supported by the Engineering and Physical Sciences Research Council (EPSRC)Fellowship EP/M002896/1 “Random Fractals”. The second author was partially supported by the Engineeringand Physical Sciences Research Council (EPSRC) Grant EP/N0094361/1 “The many faces of random character-istic polynomials”. The third author was partially supported by the ERC Starting Grant LIKO, Pr. ID6:76999.The authors would like to thank Hugo Vanneuville for helpful discussions at an early stage of the project, as wellas for pointing out that the covariance formula gives an alternate justification for the Harris criterion (see Sec-tion 2.2.5), and finally, for interesting discussions concerning reformulations of Piterbarg’s formula. The authorswould also like to thank an anonymous referee for pointing out a mistake in the statement of Corollary 1.6 in aprevious version of this article. a r X i v : . [ m a t h . P R ] M a y A COVARIANCE FORMULA FOR TOPOLOGICAL EVENTS OF SMOOTH GAUSSIAN FIELDS ( f ( x ) , ∇ f ( x ) , f ( y ) , ∇ f ( y )) is a non-degenerate Gaussian vector. Let B and B be two boxeson the plane R , not necessarily disjoint, each with two opposite sides distinguished (we callthese ‘left’ and ‘right’, with the remaining sides being ‘top’ and ‘bottom’). For each i ∈ { , } ,consider the event A i that there exists a continuous path in B i ∩ { f ≥ } joining the ‘left’and ‘right’ sides. This is known as a ‘crossing event’ for the excursion set { f ≥ } , and is offundamental importance in the study of the connectivity of the level sets [5]. As a corollary ofour general covariance formula, we establish the following exact formula for Cov( A , A ): Corollary 1.1.
The quantity
Cov( A , A ) is equal to (cid:88) j ,j =0 , , , , (cid:90) F j × F j K ( x , x ) (cid:90) γ t ; x ,x (0) E t ; x ,x (cid:20) (cid:89) i =1 , | det( H j i x i f it ) | Piv t,ixi ( A i ) (cid:21) d t dv F j dv F j , where: • For each i ∈ { , } , F i := ˚ B i denotes the interior of B i , equipped with its two-dimensionalLebesgue measure dv F i , and ( F ij ) j =1 , , , denote the sides of B i , equipped with theirnatural length measure dv F ij ; the F ij are therefore disjoint. • For each t ∈ [0 , , f t = ( f t , f t ) = ( f , tf + √ − t f ) denotes a Gaussian field on R × R that interpolates between ( f , f ) and ( f , f ) , where f and f are independentcopies of f . For each distinct x ∈ B and x ∈ B , γ t ; x ,x (0) denotes the density at of the Gaussian vector (1.1) ( f t ( x ) , ∇ f t | F j ( x ) , f t ( x ) , ∇ f t | F j ( x )) ∈ R × T x F j × R × T x F j , where F ij i denotes the unique face/interior that contains x i ; moreover, E t ; x ,x [ · ] denotesexpectation conditional on the vector (1.1) vanishing, and H j i x i f it = ∇ f it | F ji ( x i ) denotesthe Hessian at the point x i of f it restricted to the face F ij i . • For each i ∈ { , } , t ∈ [0 , and x ∈ B i , Piv t,ix ( A i ) denotes the event that there existsa continuous path in B i ∩ { f it ≥ } joining the ‘left’ and ‘right’ sides, and a continuouspath in B i ∩ { f it ≤ } joining the ‘top’ and ‘bottom’ sides, both of which pass through x (see Figure 1; central panels). This is a natural analogue of a ‘pivotal event’ in Bernoullipercolation (see [12, 24] ). Let us make three observations concerning the formula in Corollary 1.1: • If K is non-negative then so is the integrand in the formula, and we deduce that P [ A ∩ A ] ≥ P [ A ] P [ A ] . This is the analogue of the Fortuyn-Kasteleyn-Ginibre (FKG) inequality (see [12, 24]),originally proven in the Gaussian setting by Pitt [41]. • Assume that f is stationary, let κ ( x ) = K (0 , x ) and denote κ ( r ) = sup | x |≥ r | κ ( x ) | . Since f is Gaussian, the Hessians H j i x i f it have finite moments and so, by stationarity, theconditional expectation in the formula is bounded. Thus, if B and B have sides oflength O ( R ) and are at distance of order at least R , we deduce a ‘strong mixing’ boundfor crossing events, namely that(1.2) | P [ A ∩ A ] − P [ A ] P [ A ] | = O ( R κ ( R )) . In particular, as long as κ ( R ) = o ( R − ), the crossing events A and A are asymptoticallyindependent, recovering the recent result of Rivera and Vanneuville [44]. • Setting B = B (and so A = A ), Corollary 1.1 also yields a formula for the variance of (the indicator function of) the crossing event A i . COVARIANCE FORMULA FOR TOPOLOGICAL EVENTS OF SMOOTH GAUSSIAN FIELDS 3
Figure 1.
An illustration of the crossing events A i and the pivotal eventsPiv x ( A i ) that appear in the covariance formula in Corollary 1.1. Left panels:Two realisations of a field f which exhibit the left-right crossing event for { f > } in the rectangle B (shown in grey). Right panels: After a small perturbation of f (compared to the left panel), the left-right crossing event no longer occurs.Central panels: The ‘pivotal event’ at which the crossing event first fails in thisperturbation; this event can be of two possible types, either involving a level-0critical point x of f in the interior of B (top figure), or involving a level-0 criticalpoint x of f restricted to the top side of B (bottom figure).The main result of this paper (see Theorem 2.14) consists of a vast generalisation of Corol-lary 1.1 to the class of topological events of smooth Gaussian fields on manifolds of any dimension.In particular, this permits a generalisation of the mixing bound (1.2) to arbitrary topologicalevents on manifolds (see Corollary 1.2 for the Euclidean case and Theorem 2.15 for the generalcase). Since the statement of Theorem 2.14 requires several preliminary definitions, in this intro-duction we instead focus on applications of this formula, including (i) the aforementioned strongmixing bounds, and (ii) lower concentration inequalities for additive topological functionals ofthe level sets, such as such as the number of connected components contained in a given domain.Our work was largely inspired by [44] in which the mixing bound (1.2) was first established,improving similar bounds that had previously appeared in [5, 8]. Here we extend the techniquesand results in [44] to arbitrary topological events and to higher dimensions; the key differencein our approach is that we work directly in the continuum , rather than with discretisations ofthe field as in [5, 8, 44].1.1. Topological events.
We begin by describing the class of topological events to which ourresults apply. Broadly speaking, we study events that depend only on the topology of the levelsets { f = (cid:96) } (or excursion sets { f > (cid:96) } ) of a Gaussian field f restricted to reasonable boundeddomains B ⊂ R d . One might think that it would therefore be enough to study homeomorphismclasses of pairs ( { f > (cid:96) } ∩ B, B ), however, this would in fact not identify crossing events, whichdistinguish marked sides of the reference domain B . Moreover, as in the case of a product ofhomeomorphic sets, one might wish to distinguish between factors. For these reasons, we workinstead with equivalence classes induced by isotopies that preserve certain subsets of B , usingthe formalism of stratifications .An affine stratified set in R d is a compact subset B ⊂ R d equipped with a finite partition B = (cid:116) F ∈F F into open connected subsets of affine subspaces of R d , such that for each F, F (cid:48) ∈ F , F ∩ F (cid:48) (cid:54) = ∅ ⇒ F ⊂ F (cid:48) . The partition F is called a stratification of B . When there is no risk of A COVARIANCE FORMULA FOR TOPOLOGICAL EVENTS OF SMOOTH GAUSSIAN FIELDS ambiguity, we will often refer to B itself as an affine stratified set. For example, a closed cubein R d , equipped with the collection of the interiors of its faces of all dimensions, is an affinestratified set.Given an affine stratified set ( B, F ) of R d and a continuous map H : B × [0 , → B , we saythat H is a stratified isotopy if for each t ∈ [0 , H ( · , t ) is a homeomorphism such that for each F ∈ F , H ( F × { t } ) = F . The stratified isotopy class of a subset E ⊂ B , denoted by [ E ] B , is theset of H ( E × { } ) where H : B × [0 , → B ranges over the set of stratified isotopies of B with H ( · ,
0) = id B . We consider the stratified isotopy class [ { f > } ] B of the excursion set { f > } ,which captures what we mean by the ‘topology’ of the level set { f = 0 } restricted to B . Aswe verify in Corollary 5.8, under mild conditions on f the stratified istotopy class [ { f > } ] B ismeasurable with respect to f .A topological event in B is an event measurable with respect to [ { f > } ] B . Importantexamples include: • As in Corollary 1.1, crossing events for level or excursion sets inside a box B , e.g. theevent that a connected component of { f = 0 } ∩ B or { f > } ∩ B intersects opposite( d − B (Corollary 1.1 concerned the case d = 2). • Events that depend on the number of the connected components of a level or excursionset inside a polytope B , or more generally the number of such components of a givendiffeomorphism class (see, e.g., [14, 21, 37, 38, 47]). • The ‘persistence’ event that { f | B > } (see, e.g., [2, 16, 20, 43]).We write σ top ( B ) to denote the σ -algebra of topological events on B .1.2. Strong mixing in the Euclidean setting.
The strong mixing of a random field is definedvia the decay, for domains B and B that are well-separated in space, of the α -mixing coefficient(1.3) α ( B , B ) = sup A ∈ σ ( B ) , A ∈ σ ( B ) | P [ A ∩ A ] − P [ A ] P [ A ] | , where σ ( B ) denotes the sub- σ -algebra generated by the restriction of f to the domain B . Strongmixing is a classical notion in probability theory with important connections to laws of largenumbers, central limit theorems, and extreme value theory (see, e.g., [17, 32, 33, 46]) amongother topics. While for general continuous processes there is a rich literature on strong mixing(see [13] for a review), in the study of smooth random fields the concept of strong mixing is oftenfar too restrictive. For example, if the spectral density of a stationary Gaussian process decaysexponentially (which implies the real analyticity of the covariance kernel and the correspondingsample paths), then by [28] there is no strong mixing regardless of how rapidly correlationsdecay, unless one restricts the class of events that are controlled by the α -mixing coefficient.As a first application of our covariance formula we derive conditions that guarantee the strongmixing of the class of topological events.Let f be an a.s. C stationary Gaussian field on R d with covariance κ ( x ) = Cov( f (0) , f ( x )),and suppose that, for each distinct x, y ∈ R d , ( f ( x ) , ∇ f ( x ) , f ( y ) , ∇ f ( y )) is a non-degenerateGaussian vector. These conditions ensure that κ is C , and that the level set { f = 0 } is a C -smooth hypersurface. For each pair of affine stratified sets B , B ⊂ R d , define the ‘topological’ α -mixing coefficient α top ( B , B ) = sup A ∈ σ top ( B ) , A ∈ σ top ( B ) | P [ A ∩ A ] − P [ A ] P [ A ] | . Corollary 1.2 (Strong mixing for topological events) . There exist c , c > such that, for everypair of affine stratified sets ( B , F ) and ( B , F ) in R d satisfying max α ∈ N d : | α |≤ sup x ∈ B ,x ∈ B | ∂ α κ ( x − x ) | < c , COVARIANCE FORMULA FOR TOPOLOGICAL EVENTS OF SMOOTH GAUSSIAN FIELDS 5 it holds that α top ( B , B ) ≤ c |F ||F | max F ∈F ,F ∈F (cid:90) F × F | κ ( x − x ) | dv F ( x ) dv F ( x ) . In particular, recalling that ¯ κ ( s ) = sup | x |≥ s | κ ( x ) | , if (1.4) lim | x |→∞ | ∂ α κ ( x ) | = 0 , for all α ∈ N d such that | α | ≤ , then for every pair of disjoint affine stratified sets B , B ⊂ R d there exist c , c > such that (1.5) α top ( sB , sB ) ≤ c s d ¯ κ ( c s ) for all s ≥ . Corollary 1.2 demonstrates that topological events on well-separated boxes B , B ⊂ R d areindependent up to an additive error that depends (up to a constant) solely on the double integralof the absolute value of the covariance kernel on the boxes; we expect this result to have manyapplications. Later we present a generalisation of Corollary 1.2 to Gaussian fields on generalmanifolds (see Theorem 2.15). The proof of Corollary 1.2 is given in Section 6. Remark 1.3.
The constant c in Corollary 1.2 can be chosen in a way that depends only onthe dimension d , on κ (0), and on the Hessian of κ at 0, whereas the constant c can be chosenin a way that depends, in addition to these, also on max j ( ∂ κ (0) /∂x j ). Remark 1.4.
We do not assume that the field f is centred. Since adding a constant does notchange the covariance kernel, Corollary 1.2 also bounds the strong mixing of topological eventsthat are defined in terms of non-zero levels. Notably, neither c nor c depends on the meanvalue of the field. Remark 1.5.
As explained above, the mixing bound in Corollary 1.2 was already known intwo dimensions, at least in the case of crossing events [44] (see also (1.2)); our results extendsthis mixing bound to arbitrary dimensions and arbitrary topological events. Note also thatan analogue of (1.5) was recently established [36] for a version of the α -mixing coefficient thatcontrols all events (not necessarily topological) that depend monotonically on f (this includes,for instance, crossing events for { f > } ); in this case the factor s d can be improved to s d .1.3. Application to lower concentration for topological counts.
We next present a simpleapplication of Corollary 1.2 to give a taste of the utility of mixing bounds. A topological count is a set of integer-valued random variables N = N ( B ), indexed by affine stratified sets B ⊂ R d ,each of which is measurable with respect to the corresponding σ -algebra σ top ( B ). We calla topological count super-additive if, for every affine stratified set B and every collection ofdisjoint affine stratified sets ( B i ) i ≤ k contained in B ,(1.6) N ( B ) ≥ (cid:88) i ≤ k N ( B i ) . Examples of super-additive topological counts include the number of connected components oflevel or excursion sets that are fully contained in a set [38], or more generally the number ofconnected components of these sets that have a certain diffeomorphism class [14, 21, 47]. Inone dimension, topological counts reduce to the number of solutions to { f = 0 } in intervals, aquantity studied extensively since the works of Kac and Rice in the 1940s [26, 42]. We say thata topological count N satisfies a law of large numbers if there exists a c N > B ⊂ R d , as s → ∞ (1.7) N ( sB ) s d Vol( B ) → c N in probability . Nazarov–Sodin have shown [37, 38] (see also [7, 31]) that if f is ergodic (and under certainmild extra conditions) the number of connected components of level or excursion sets satisfies A COVARIANCE FORMULA FOR TOPOLOGICAL EVENTS OF SMOOTH GAUSSIAN FIELDS a law of large numbers, and in fact, (1.7) converges a.s. and in mean; the same result was latershown to be true also for the number of connected components of a given diffeomorphism type[10, 14, 47] (in the one dimensional case this follows immediately from the ergodic theorem). Aswas shown in [44], quantitative mixing bounds can be used to deduce the lower concentrationof super-additive topological counts:
Corollary 1.6 (Lower concentration for topological counts) . Let N denote a super-additivetopological count that satisfies a law of large numbers (1.7) with limiting constant c N > .Assume that (1.4) holds. Then for every affine stratified set B ⊂ R d and constants ε, C > ,there exist c , c B > such that, for every s ≥ , (1.8) P (cid:20) N ( sB ) s d Vol( B ) ≤ c N − ε (cid:21) ≤ c inf r ∈ [1 ,s ] (cid:16) e − C ( s/r ) d + e c B ( s/r ) d ( rs ) d ¯ κ ( r ) (cid:17) , where the constant c B > depends only on the stratified set B . In particular, if there exist c , α > such that κ ( x ) ≤ c | x | − α for every | x | ≥ , then for every ε, δ > we can set r = c s/ (log s ) /d for a sufficiently large choice of c > (depending on c B , α and δ ) and apply (1.8) for C > sufficiently large (depending on c and δ ) to deduce the existence of a c > such that, for every s ≥ , P (cid:20) N ( sB ) s d Vol( B ) ≤ c N − ε (cid:21) ≤ c s d − α + δ . Similarly, if there exist c , α, β > such that κ ( x ) ≤ c e − β | x | α for every | x | ≥ , then setting r = c s d/ ( d + α ) for a sufficiently large choice of c > and then choosing C sufficiently large wededuce that for every γ > there is c > such that, for every s ≥ , P (cid:20) N ( sB ) s d Vol( B ) ≤ c N − ε (cid:21) ≤ c exp (cid:16) − γs dα/ ( d + α ) (cid:17) . Remark 1.7.
As for Corollary 1.2, Corollary 1.6 was also already known in two dimensions(at least in the case of the number of connected components of level sets [44]) but not inhigher dimensions. A stronger version of Corollary 1.6 was also recently established in theone dimensional case (i.e. for the number of zeros of a one-dimensional stationary Gaussianprocess [4]), and also for the number of connected components of the zero level set of randomspherical harmonics (RSHs) [37]; the results in [4, 37] are proven using very different techniquesto ours, and in the latter case relies heavily on the specific structure of the RSHs.2.
A covariance formula for topological events
In this section we present our covariance formula in the general setting of smooth Gaussianfields on smooth manifolds. We also discuss further applications of the formula beyond thosewe gave in Section 1, and give a sketch of its proof.2.1.
The covariance formula.
We begin by fixing definitions, starting with the ‘stratified sets’on which we work; our main reference is [23]. Let (
M, g ) be a smooth Riemannian manifold ofdimension d . Definition 2.1 (Stratified set) . Let B ⊂ M be a compact subset. Assume there is a partitionof B into a finite collection F of smooth locally closed submanifolds, called strata , satisfying thefollowing additional properties: • The strata cover B , i.e. B = (cid:96) F ∈F F . • Any two strata F and F satisfy F ∩ F (cid:54) = ∅ ⇔ F ⊂ F . This allows us to equip F with the partial order < defined such that, for any two strata F and F , F ∩ F (cid:54) = ∅ ⇔ F = F or F < F . COVARIANCE FORMULA FOR TOPOLOGICAL EVENTS OF SMOOTH GAUSSIAN FIELDS 7 • For each F < F the following is true. Consider any embedding of M in Euclideanspace, and let ( x k ) k ∈ N and ( y k ) k ∈ N be sequences of points satisfying (i) for each k ∈ N , x k ∈ F and y k ∈ F , (ii) x k and y k converge to a common point y ∈ F , (iii) the tangentplanes T x k F converge to a limit τ , and (iv) the lines λ k generated by the vectors x k − y k converge to a limit λ . Then it holds that λ ⊂ τ . Equivalently, it is enough that thiscondition be fulfilled for one fixed embedding of M in Euclidean space. Limits τ of thiskind are called generalised tangent spaces at y . • For each F , F ∈ F such that F < F , there exists a smooth sub-bundle T F | F of T M | F , whose rank is the dimension of F , that contains T F as a sub-bundle, and suchthat (i) the map y (cid:55)→ T y F , with values in the adequate Grassmannian bundle definedon F , extends by continuity to F together with all of its derivatives, and (ii) for eachsequence of points x k ∈ F converging to a limit x ∈ F , lim k → + ∞ T x F = T x F | F . Wecall T F | F the generalised tangent bundle of F over F (see Figure 2).The collection F is called a tame stratification of B . A stratified set of M is a pair ( B, F )consisting of a compact subset B ⊂ M and a tame stratification F of B . When there is no riskof ambiguity, we will often write that B ⊂ M is a stratified set without explicit mention of itstame stratification F . Figure 2.
Left: An example of a tame stratification F = { F , F } of a compactset B . Here the generalised tangent bundle T x F | F is well-defined since, asthe points x k converge to x ∈ F , the respective tangent planes also converge.Right: A rough depiction of the ‘rapid spiral sheet’, which is an example of a setthat cannot be tamely stratified (see Example 2.6); here tangent planes do notconverge, and so the generalised tangent bundle is not well-defined. Remark 2.2.
A partition F of a compact subset B satisfying the first three properties requiredin Definition 2.1 is called a Whitney stratification (see for instance Part I, Section 1.2 of [23]);indeed, the third property is known as ‘Whitney’s condition (b)’. While Whitney stratificationshave many interesting properties, sometimes the structure of a stratification can force functionson it to have degenerate stratified critical points (see Example 2.6). To avoid such pathologies,we add the additional fourth condition which is satisfied in most natural examples. In fact,this additional ‘tameness’ property is only used at a single place in the proof of the covarianceformula, namely, to prove Claim 4.6.Let us present several important examples (and one non-example) of stratified sets, beginningwith the trivial stratification: Example 2.3 (Trivial stratification) . Let M be a compact manifold without boundary. Then F = { M } is a tame stratification of M . Moreover, let Ω ⊂ M be a compact subset with smoothboundary ∂ Ω. Then F = { ˚Ω , ∂ Ω } is a tame stratification of Ω. A COVARIANCE FORMULA FOR TOPOLOGICAL EVENTS OF SMOOTH GAUSSIAN FIELDS
In the case that M = R d , by gluing boxes and other polytopes together one obtains setsequipped with a natural stratification that will, in most case, be tame. Our definition of ‘affinestratified set’, introduced in Section 1, covers all such examples: Example 2.4 (Affine stratified sets) . The affine stratified sets introduced in Section 1 arestratified sets of M = R d .One can also consider individual ‘polytopes’, such as the boxes in Corollary 1.1, to be stratifiedsets of M = R d : Example 2.5 (Polytopes) . A polytope in R d is naturally equipped with a stratification whosestrata are the faces of the polytope of all dimensions. Though to our knowledge there is noconsensus on the definition of a polytope in R d , it is easy to check whether or not a specificexample satisfies Definition 2.1.We also present one non-example, in the form of the ‘rapid spiral’: Example 2.6 (Rapid spiral) . The rapid spiral B = { r = e − θ } (see Figure 2) admits a naturalpartition that satisfies all the conditions of a tame stratification except the last; in particular,this partition is a Whitney stratification. The rapid spiral B exhibits certain pathologies thatresult from the lack of tameness, for instance, there are no stratified Morse functions on B (see[23, Part I, Example 2.2.2]).We next extend the definition of topological events given in Section 1 to the general settingof stratified sets. Let f be a continuous Gaussian field on M , defined on a probability space Ω.Let µ : M → R and K : M × M → R denote respectively the mean and covariance kernel of f .Assume that f satisfies the following condition (generalising the conditions in Section 1): Condition 2.7.
The field f is a.s. C . Moreover, for each distinct x, y ∈ M , the Gaussianvector ( f ( x ) , d x f, f ( y ) , d y f ) ∈ R × T ∗ x M × R × T ∗ y M is non-degenerate.This condition ensures that µ is C and that K is of class C , . Let us now define the classof topological events on a stratified set B . Definition 2.8 (Topological events) . Let ( B, F ) be a stratified set of M . A stratified homeo-morphism of B is a homeomorphism h : B → B such that for each F ∈ F , h ( F ) = F . A stratifiedisotopy of B is a continuous map H : B × [0 , → B such that for each t ∈ [0 , H ( · , t ) : B → B is a stratified homeomorphism of B . We say that two stratified homeomorphisms h , h : B → B are F - isotopic if there exists a stratified isotopy H such that H ( · ,
0) = h and H ( · ,
1) = h .Let D denote the excursion set { f > } . The stratified isotopy class of D in B , denoted[ D ] B , is the set of h ( D ∩ B ) where h ranges over all stratified homeomorphisms of B that are F -isotopic to the identity. As we establish in Corollary 5.8, under Condition 2.7 there are acountable number of stratified isotopy classes, and we equip the set of classes with its maximal σ -algebra. We will also verify in Corollary 5.8 that the map [ D ] B from the probability spaceΩ into the set of stratified isotopy classes is measurable. A topological event on B is an event A ⊂ Ω measurable with respect to the random variable [ D ] B .Henceforth we fix two stratified sets ( B , F ) and ( B , F ) of M (not necessarily disjoint).Our main formula expresses the covariance between topological events on B and B in termsof an integral over the ‘pivotal measure’ of the events. This measure is defined in terms of (i)‘pivotal points’, and (ii) a certain interpolation between f and an independent copy of itself;we introduce these concepts now. Our definition of ‘pivotal points’ is related to the notion of‘pivotal sites’ in percolation theory (see [24, Section 2.4]), whereas the interpolation is based onthe classical interpolation argument of Piterbarg [40]. COVARIANCE FORMULA FOR TOPOLOGICAL EVENTS OF SMOOTH GAUSSIAN FIELDS 9
Definition 2.9 (Pivotal points) . Fix ˆ A ⊂ C ( M ). For every u ∈ C ( M ), we say that x ∈ M is pivotal for u ( with respect to ˆ A ) if, for any open neighbourhood W of x in M , there exists afunction h ∈ C c ( W ) such that for every sufficiently small δ > u + δh ∈ ˆ A and u − δh / ∈ ˆ A . Sucha function u is described as having a pivotal point at x ∈ M , and we denote by Piv x ( ˆ A ) ⊂ C ( M )the set of all such u ’s. If h can be chosen so that h ≥
0, we say that x is positively pivotal for u , and we denote by Piv + x ( ˆ A ) ⊂ C ( M ) the set of such u ’s. Similarly, x is negatively pivotal for u if h can be chosen so that h ≤
0, and we denote Piv − x ( ˆ A ) ⊂ C ( M ) the set of such u ’s. Definition 2.10 (Interpolation) . Let ˜ f be an independent copy of f . For each t ∈ [0 , M × M (2.1) f t ( x ) = ( f t ( x ) , f t ( x )) := ( f ( x ) , t ( f ( x ) − µ ( x )) + (cid:112) − t ( ˜ f ( x ) − µ ( x )) + µ ( x )) . Observe that f t and f t have the same law as f , and Cov( f t ( x ) , f t ( x )) = tK ( x , x ); inparticular, f and f are independent, while f = f . Also, observe that f t and f t both satisfyCondition 2.7. For each x ∈ F ∈ F and x ∈ F ∈ F , denote by γ t ; x ,x (0) the density atzero of the Gaussian vector(2.2) ( f t ( x ) , d x f t | F , f t ( x ) , d x f t | F )in orthonormal coordinates of R × T ∗ x F × R × T ∗ x F , and denote by E t ; x ,x [ · ] expectationconditional on the vector (2.2) vanishing; this conditional expectation is well defined and de-scribed by the usual Gaussian regression formula ([3, Proposition 1.2]) since the vector (2.2)is non-degenerate. Note that, since x and x correspond to unique strata F and F , to easenotation we have dropped the explicit dependence of γ t ; x ,x (0) and E t ; x ,x [ · ] on F and F .We are now ready to define the pivotal measure, or more precisely, two ‘signed’ pivotalmeasures. Fix topological events A and A on B and B respectively. Denote by (cid:101) A , (cid:101) A the measurable sets of stratified isotopy classes in B and B respectively that define thesetopological events, and let ˆ A (resp. ˆ A ) be the set of functions u ∈ C ( M ) such that [ { u > } ] B ∈ (cid:101) A (resp. [ { u > } ] B ∈ (cid:101) A ).Denote by dv g the Riemannian volume measure on M . Similarly, for each stratum F ∈ F ∪F ,denote by dv F the Riemannian volume measure induced by g F , the restriction of g to F . If u ∈ C ( M ) and x is a critical point of u , we denote by H x u the Hessian of u at x (which iswell defined since x is a critical point of u ; see for instance [39, Chapter 1]). More generally, if F ⊂ M is a smooth submanifold of M and d x u | F = 0, then let H Fx u be the Hessian of u | F at x . Definition 2.11 (Pivotal measures) . For each t ∈ [0 ,
1] and σ ∈ {− , + } , define the signedpivotal intensity function I σt ( x , x ) on B × B to be(2.3) (cid:88) σ ,σ ∈{− , + } ,σ σ = σ γ t ; x ,x (0) E t ; x ,x (cid:104) | det( H F x f t ) det( H F x f t ) | ; f t ∈ Piv σ x ( ˆ A ) , f t ∈ Piv σ x ( ˆ A ) (cid:105) where F and F denote the (unique) strata in F and F that contain x and x respectively,and the determinants are taken with respect to orthonormal bases of T x i F i . The signed pivotalmeasures d π σ ( x , x ) on B × B are defined, for σ ∈ {− , + } , asd π σ ( x , x ) = (cid:16) (cid:90) I σt ( x , x ) d t (cid:17) dv F ( x )dv F ( x ) . We emphasise that, although the ‘pivotal measures’ depend on both (i) the stratified sets B i , and (ii) the topological events A i , to ease notation we have left these dependencies implicit.Observe also that d π σ is a sum of measures of different dimensions that are supported on pairs ofstrata ( F , F ) ∈ F ×F . On each such pair, the measures d π ± are singular with respect to eachother and mutually continuous with respect to the product of Riemannian volume measures. Remark 2.12.
By definition, the Hessian on 0-dimensional strata is always equal to zero. Thisimplies that, when at least one of x i belongs to a 0-dimensional stratum, the corresponding termis zero independently of how we interpret dv F for 0-dimensional F . Hence in (2.3), as well asin all subsequent formulae of similar type, we can discard the contribution from 0-dimensionalstrata. Remark 2.13. If A and A are both increasing events (meaning that, for i ∈ { , } , if u ∈ ˆ A i and h is a non-negative function, then u + h ∈ ˆ A i ), then the negative pivotal measure d π − is identically zero since Piv − x i ( ˆ A i ) is empty by definition. The same is true if A and A areboth decreasing events, since then Piv + x i ( ˆ A i ) is empty. Similarly, if A is increasing and A isdecreasing, then d π + is identically zero.We are now ready to present our covariance formula in full generality: Theorem 2.14 (Covariance formula for topological events) . Let ( B , F ) and ( B , F ) be strat-ified sets of M . Let f be a Gaussian field on M satisfying Condition 2.7. Then the covarianceof topological events A and A on B and B respectively can be expressed as P [ A ∩ A ] − P [ A ] P [ A ] = (cid:90) B × B K ( x, y ) (cid:0) d π + ( x, y ) − d π − ( x, y ) (cid:1) , where dπ + and dπ − denote the pivotal measures introduced in Definition 2.11. Let us offer some intuition behind the covariance formula in Theorem 2.14. The starting pointof our analysis is the observation that P [ A ∩ A ] = P [ f ∈ ˆ A × ˆ A ] and P [ A ] P [ A ] = P [ f ∈ ˆ A × ˆ A ] , and hence P [ A ∩ A ] − P [ A ] P [ A ] = (cid:90) ddt P [ f t ∈ ˆ A × ˆ A ] d t. As we explain in Section 2.3, the structure of the Gaussian measure allows us to express ddt P [ f t ∈ ˆ A × ˆ A ]as an integral, over pairs of strata ( F , F ) ∈ F × F , of the (signed) two-point intensityfunctions I σt of critical points that are ‘pivotal’ for the events A and A respectively, weightedby a term that is the inner product of the outward normal vectors at the boundary of the events A and A ; by the properties of the Gaussian measure (in particular, the reproducing propertyof the covariance kernel), this inner product is just (a normalisation of) the covariance kernel K .To understand the form of the intensity functions I σt , notice that pivotal points are necessarilycritical points at the zero level. Hence we can understand I σt as a restriction to pivotal pointsof the standard two-point intensity function for critical points of f t on ( F , F ) at the zero level,which by the well-known Kac-Rice formula (see [3, Chapter 6]) is given by γ t ; x ,x (0) E t ; x ,x (cid:2) | det( H F x f t ) det( H F x f t ) | (cid:3) . Note that our intensity functions are signed; this is because we must distinguish pairs of pivotalpoints that are pivotal ‘in the same direction’, in the sense that a local increase in f causes theevents A and A to both occur or to both not occur, from those that are pivotal ‘in oppositedirections’.It is possible that some variant of Theorem 2.14 remains true for a wider class of smoothrandom fields. The Kac-Rice formula applies far beyond the Gaussian setting, and in principleone can also express the intensity of pivotal points for non-Gaussian fields. As for the initialinterpolation step, by formulating it using the Ornstein-Uhlenbeck semigroup (as in, say, [15] oras suggested in [48]) the setting could perhaps be extended to measures related to other Markovsemigroups. We leave this for future investigation. COVARIANCE FORMULA FOR TOPOLOGICAL EVENTS OF SMOOTH GAUSSIAN FIELDS 11
Applications.
We next present applications of the covariance formula in Theorem 2.14;some of these have already been discussed (see Corollaries 1.2 and 1.6), but here we give ex-tensions to more general settings. The proofs will be deferred to Section 6. Throughout thissection we assume that f satisfies Condition 2.7.2.2.1. Strong mixing for topological events.
Our first application generalises the strong mixingstatement in Corollary 1.2 to the set-up in Section 2.1. For a stratified set B ⊂ M , let σ top ( B )denote the σ -algebra consisting of topological events in B , and for a pair of stratified sets B , B ⊂ M , define the corresponding ‘topological’ α -mixing coefficient(2.4) α top ( B , B ) = sup A ∈ σ top ( B ) , A ∈ σ top ( B ) | P [ A ∩ A ] − P [ A ] P [ A ] | . Theorem 2.15 (Strong mixing for topological events) . There exists a constant c d > , dependingonly on the dimension of the manifold M , such that for every pair of stratified sets ( B , F ) and ( B , F ) of M , α top ( B , B ) ≤ c d (cid:88) F ∈F ,F ∈F c F ,F (cid:90) F × F | K ( x , x ) | dv F ( x ) dv F ( x ) , where c F ,F is equal to the maximum, over i, j, k ∈ { , } , of sup x ∈ F ,x ∈ F (cid:0) E (cid:2) (cid:107) H F i x i f (cid:107) op | d x i f | F i = 0 (cid:3)(cid:1) d i (cid:112) det(∆( x , x )) max (cid:40) , (cid:16) K ( x j , x j ) det( d x k ⊗ d x k K | F i × F i ) (cid:112) det(∆( x , x )) (cid:17) d i (cid:41) , and where (cid:107) · (cid:107) op denotes the ( L -)operator norm, d i = dim( F i ) , and ∆( x , x ) is the covariancematrix, in orthonormal coordinates, of the (non-degenerate) Gaussian vector ( f ( x ) , d x f | F , f ( x ) , d x f | F ) . Remark 2.16.
All the terms in the definition of c F ,F can be written as a quotient of powersof polynomials of partial derivatives of K of order at most (2 , c F ,F depends continuously on the C , norm of K , and (ii) c F ,F is homogeneous in K (the degreeof homogeneity is easily seen to be −
1, which compensates the presence of K ( x , x ) in theintegral).2.2.2. Sequences of fields: The Kostlan ensemble.
In Corollary 1.2 we stated a quantitativemixing bound for rescaled (affine) stratified sets sB and sB as s → ∞ . In the setting ofcompact manifolds M , it is often more appropriate to work with a sequence of Gaussian fieldson M that converge to a local limit, and consider the topological mixing between fixed disjointstratified sets B , B ⊂ M (in fact, this includes the setting in Corollary 1.2 as a special case,by rescaling the field rather than the sets).Rather than work in full generality, here we work only with the Kostlan ensemble , which isthe sequence ( f n ) n ∈ N of smooth centred isotropic Gaussian fields on S d with covariance kernels K ( x, y ) = cos n ( d S d ( x, y )) = (cid:104) x, y (cid:105) n , where d S d ( · , · ) denotes the spherical distance; it is easy to check that each f n satisfies Condi-tion 2.7. The sequence f n converges to a local limit on the scale s n = 1 / √ n , in the sense thatfor any x ∈ S d the rescaled field(2.5) f (exp x ( x/ √ n )) , x ∈ R d converges on compact sets to the smooth stationary Gaussian field on R d with covariance κ ( x ) = e −(cid:107) x − y (cid:107) / ; here exp x : R d → S d denotes the exponential map based at x . The Kostlanensemble is a natural model for random homogeneous polynomials (see [29, 30]), and its levelsets have been the focus of recent study [9]. Its local limit is known as the Bargmann-Fock field.
Corollary 2.17 (Strong mixing for the Kostlan ensemble) . For each pair of disjoint stratifiedsets B , B ⊂ S d that are contained in an open hemisphere, there exist c , c > such that, foreach n ≥ , α n ; top ( B , B ) ≤ c e − c n , where α n ; top denotes the ‘topological’ mixing coefficient (2.4) for the field f n . Remark 2.18.
Since f n are homogeneous polynomials, they are naturally defined on the realprojective space rather than the sphere, which makes it natural to restrict B and B to becontained in an open hemisphere. Indeed, f n is degenerate at antipodal points.The lower concentration result in Corollary 1.6 can also be generalised to the setting ofsequences of Gaussian fields on manifolds; again we focus just on the Kostlan ensemble ( f n ) n ∈ N on S d . We define a topological count N = N n ( B ) for f n analogously to in Section 1, aftersubstituting affine stratified sets B ⊂ R d with general stratified sets B ⊂ S d ; these counts arenow indexed by B ⊂ S d and n ∈ N . A topological count N is called super-additive if (1.6) holdsfor each N n . We say that a topological count N satisfies a law of large numbers if there existsa c N > B ⊂ S d , as n → ∞ ,(2.6) N n ( B ) n d/ Vol( B ) → c N in probability;the scale n d/ can be understood as the natural volume scaling induced by the rate s n = 1 / √ n at which the Kostlan ensemble converges to a local limit in (2.5). Corollary 2.19 (Lower concentration for topological counts of the Kostlan ensemble) . Let N denote a super-additive topological count that satisfies a law of large numbers (2.6) with limitingconstant c N > . Then for every stratified set B ⊂ S d and every ε > , there exist c , c > such that, for every n ≥ , (2.7) P (cid:20) N n ( B ) n d/ Vol( B ) ≤ c N − ε (cid:21) ≤ c e − c n d/ ( d +2) . In particular, taking B = S d with its trivial stratification F = { S d } , the conclusion of Corol-lary 2.19 is true for N n the number of connected components of { f n > } or { f n = 0 } on thesphere S d (see [38] for a proof of the law of large numbers for N n ).2.2.3. Decorrelation for topological counts.
In the classical theory of strong mixing, a majorapplication of mixing bounds is to prove central limit theorems (CLTs) (see, e.g., [17, 32, 46]).Although establishing CLTs for topological counts is beyond the scope of this work, we illustratehere how mixing bounds can be used to deduce the ‘decorrelation’ of topological counts, a keyintermediate step in proving a CLT.For simplicity we return to the Euclidean setting of Section 1. We say that a topological count N has a finite two-plus-delta moment on an affine stratified set B ⊂ R d if there exist δ, c > E [ N ( B ) δ ] < c < ∞ . Although the finiteness of two-plus-delta moments is not known for the topological counts dis-cussed in Section 1 (except in the one-dimensional case), in principle one can bound (2.8) bythe purely local quantity E [( f in B ) δ ] , which we suspect is finite in great generality. COVARIANCE FORMULA FOR TOPOLOGICAL EVENTS OF SMOOTH GAUSSIAN FIELDS 13
Corollary 2.20 (Decorrelation for topological counts) . Fix affine stratified sets B , B ⊂ R d and suppose that N and N are topological counts that have finite two-plus-delta moments (2.8) on B and B with constants δ, c > . Then (2.9) Cov (cid:0) N ( B ) , N ( B ) (cid:1) ≤ c / (2+ δ ) α top ( B , B ) δ/ (2+ δ ) . We expect that standard methods (i.e. [19, 46]) should allow one to deduce, from Corol-lary 2.20, a CLT for rescaled topological counts that satisfy a law of large numbers wheneverstrong enough two-plus-delta moment bounds can be established, at least as long as κ ( x ) decaysat a high enough polynomial rate (with the polynomial exponent depending on δ ).2.2.4. Positive association for increasing topological events.
Recall that a random vector is saidto be ‘positively associated’ if increasing events (or equivalently decreasing events) are positivelycorrelated. To state an analogous property for continuous random fields some care must betaken to specify an appropriate class of increasing events, and here we restrict the discussion totopological events. An important example of topological events that are increasing are crossingevents for the excursion set { f > } (but not crossing events for the level set { f = 0 } ), and thefact that crossing events are positively correlated is crucial in the analysis of level set percolation[5, 9, 36, 45].In the setting of Gaussian fields, it is known that the class of increasing topological eventson a stratified set are positively correlated if and only if the covariance kernel K is positive.The standard approach is to invoke a classical result that (finite-dimensional) Gaussian vectorsare positively associated if and only if they are positively correlated [41], and then to apply anapproximation argument (see [44]). Here we deduce, directly from our exact formula, a quanti-tative version of this result, whose proof is immediate from Theorem 2.14 and the observationin Remark 2.13. Corollary 2.21 (Positive associations) . Let A and A be topological events on stratified sets B and B , and suppose that A and A are both increasing. Then (2.10) P [ A ∩ A ] − P [ A ] P [ A ] = (cid:90) B × B K ( x, y ) d π + ( x, y ) , where d π + is the measure defined in Definition 2.11. In particular, A and A are positivelycorrelated if K | B × B ≥ . The fact that positive associations fails in general if a Gaussian field is not positively cor-related is a serious limitation to many applications; for example, the current theory of levelset percolation for Gaussian fields fails more or less completely unless K ≥ quantified , which gives hope that the errors that arise might be controllable.2.2.5. The Harris criterion.
Lastly, we present an informal discussion of the ‘Harris crite-rion’ (HC), demonstrating in particular that Theorem 2.14 can be used to give an alternativederivation of this criterion.In its original formulation (see, e.g., [49]), the HC was a heuristic to determine whether long-range correlations influence the large-scale connectivity of discrete critical percolation models.Translated to the setting of Gaussian fields on R d (see [11]), the HC claims that the connectivityof the level set of smooth centred Gaussian fields will, at the critical level (cid:96) c ≤ d = 2, but believed to be strictly negative if d ≥ s /ν − d (cid:90) B s × B s κ ( x − y ) dxdy → s → ∞ , where B s denotes the ball of radius s centred at the origin, and ν is the correlation lengthexponent of critical percolation, widely believed to be universal and satisfy ν = / , d = 2 , ∈ (1 / , , d = 3 , , , / , d ≥ . In the positively-correlated case κ ≥
0, (2.11) is roughly equivalent to demanding that κ haspolynomial decay with exponent at least 2 /ν . The original argument of Harris (as translated toour setting in [11]) goes as follows. Define m s = 1 | B s | (cid:90) B s f ( x ) dx to be the average value of f on the ball B s . The fluctuations of m s are of order(2.12) (cid:112) E [ m s ] = 1 | B s | (cid:18)(cid:90) B s × B s κ ( x − y ) dxdy (cid:19) / . Recall now that the behaviour of critical (and near-critical) percolation follows a set of power-laws with certain universal exponents , one of which is the correlation length exponent ν . Roughlyspeaking, this claims that the connectivity of percolation with probability p ∈ [0 ,
1] closelyapproximates the connectivity of critical percolation on the ball B s as long as | p − p c | (cid:28) s − /ν ,where p c is the critical probability. Under the assumption that f can be replaced by m s + f on B s , the ‘percolation hypothesis’ therefore generates a contradiction unless m s (cid:28) s − /ν ,and combining with (2.12) gives (2.11). Note that the HC should really be understood as a necessary condition for the ‘percolation hypothesis’, since the argument assumes the ‘percolationhypothesis’ and derives a contradiction.We now demonstrate that Theorem 2.14 yields an alternative criterion, more or less equivalentto (2.11), that we claim is also a necessary condition for the ‘percolation hypothesis’. Fix a pairof disjoint boxes B , B ⊂ R d and, for each s ≥ i ∈ { , } , let A si denote the crossing eventsfor the critical level set in sB i . Note that pivotal points for crossing events roughly correspondto four-arm saddles at distance s , i.e. saddle points x such that all four arms of the level set { f = f ( x ) } hit the ball of radius s around x . Putting this approximation into Theorem 2.14,we deduce that P [ A s ∩ A s ] − P [ A s ] P [ A s ] ≈ c κ (cid:90) sB × sB κ ( x − y ) I s ( x, y ) dxdy ≈ c κ I s (0) (cid:90) sB × sB κ ( x − y ) dxdy. where I s denotes the intensity of four-arm saddles at distance s , and where in the last step weused stationarity and an (unjustified) factorisation of this intensity. Consider now the universalexponent ζ that is believed to describe the decay of the probability of critical ‘four-arm’ eventsfor all percolation models. If the ‘percolation hypothesis’ is true, then I s (0) ≈ s − ζ , and sinceunder the ‘percolation hypothesis’ the events A s and A s decorrelate, we end up with the followingcriterion for this hypothesis:(2.13) s − ζ (cid:90) sB × sB κ ( x − y ) dxdy → s → ∞ . To compare to (2.11), recall that by the ‘Kesten scaling relations’ [27] ζ = d − /ν , and so theexponents 2 /ν − d and − ζ in (2.11) and (2.13) match. The only difference is the domain ofintegration, but as s → ∞ this difference is negligible under mild assumptions on the decay ofcovariance. COVARIANCE FORMULA FOR TOPOLOGICAL EVENTS OF SMOOTH GAUSSIAN FIELDS 15
Proof sketch.
Theorem 2.14 can be considered as a generalisation to topological eventsof a simple formula, essentially due to Piterbarg [40], that gives a covariance formula for finite-dimensional Gaussian vectors. This lemma is both the inspiration for Theorem 2.14, and alsoone of the key ingredients in the proof. We state Piterbarg’s formula in the simplest case ofstandard Gaussian vectors, since this is all that we need, but a similar statement exists forgeneral non-degenerate Gaussian vectors; for completeness, we give the proof in Appendix B.
Lemma 2.22 (Piterbarg’s formula; see [40, Theorem 1.4]) . For each t ∈ [0 , , let X t and Y t bejointly Gaussian vectors in R m , not necessarily centred, whose covariance matrix is (cid:18) I tItI I (cid:19) ; that is, Cov( X t,i , X t,j ) = Cov( Y t,i , Y t,j ) = δ i,j and Cov( X t,i , Y t,j ) = tδ i,j . Let γ t ( x, y ) denotethe density of Z t = ( X t , Y t ) ∈ R m . Let A and B be domains in R m whose boundaries arepiecewise smooth, and which have surface areas, inside the ball of radius R , that grow at mostpolynomially in R . Denote by ν A and ν B the outward unit normal vectors on the boundaries of A and B respectively. Then P [ Z t ∈ A × B ] is differentiable in t ∈ (0 , , and ddt P [ Z t ∈ A × B ] = (cid:90) ∂A × ∂B (cid:104) ν A ( x ) , ν B ( y ) (cid:105) γ t ( x, y ) d x d y, where by (cid:82) d x d y we understand integration with respect to the natural m − dimensional mea-sures on ∂A and ∂B respectively.In particular, if X denotes an arbitrary translation of a standard Gaussian vector in R m , then P [ X ∈ A ∩ B ] − P [ X ∈ A ] P [ X ∈ B ] = (cid:90) (cid:90) ∂A × ∂B (cid:104) ν A ( x ) , ν B ( y ) (cid:105) γ t ( x, y ) d x d y d t ; the integral converges since the integral (cid:82) s dt on the right-hand side exists for all s < , andconverges as s → to the left-hand side. Let us now give a brief sketch of the proof of Theorem 2.14, showing how Piterbarg’s formulaplays an essential role. We begin by considering the case of finite-dimensional Gaussian fields, i.e.the case in which f is a Gaussian vector in a finite-dimensional space of continuous functions V (see Proposition 3.9). More precisely, we fix (cid:104)· , ·(cid:105) a scalar product on V and take f to be atranslation of the standard Gaussian vector in V . The scalar product also induces a volumemeasure du on V , and allows us to identify V with R dim( V ) up to isometries. Hence, we canapply Piterbarg’s formula in V and deduce that(2.14) ddt P (cid:104) f t ∈ ˆ A × ˆ A (cid:105) = (cid:90) ∂ ˆ A × ∂ ˆ A (cid:104) ν A ( u ) , ν A ( u ) (cid:105) γ t ( u , u ) d u d u , where f t = ( f t , f t ) has covariance (cid:18) I tItI I (cid:19) in orthonormal coordinates of V × V equipped withthe product scalar product (this coincides with the definition of f t in (2.1)).The next step is to analyse the boundaries of ˆ A i . The path ( f it ) t ∈ [0 , is a generic deformationof f . By standard arguments in Morse theory, along this deformation the topology of the set { f it ≥ } changes only when f it passes through a non-degenerate critical point at level 0 (whichcan cause f it to either enter or exit ˆ A i ); if such a change in topology occurs we say that thiscritical point is pivotal for the event ˆ A i and the function f it . We will see (in Lemma 3.10) that,if we exclude a subset E ⊂ ∂ ˆ A i of positive codimension containing the functions with multiplestratified critical points at level 0, we can define a surjectionΞ : ∂ ˆ A i \ E (cid:16) B i , that induces submersions on each stratum of B i , by associating to each u i ∈ ∂ ˆ A i \ E its uniquecritical point at level 0. The fibre Ξ − ( x i ) is an open subset of the subspace of functions forwhich x i is a stratified critical point at level 0. We will see that it is equal to Piv x i ( ˆ A i ) up to anegligible set.Using the map Ξ, the coarea formula allows us to switch from an integral over ∂ ˆ A × ∂ ˆ A ⊂ V to a sum of integrals over pairs of faces of B and B . We obtain that (2.14) is equal to (cid:88) F ∈F ,F ∈F (cid:90) F × F (cid:18) (cid:90) Ξ − ( x ) × Ξ − ( x ) (cid:104) ν ∂ ˆ A ( u ) , ν ∂ ˆ A ( u ) (cid:105) Jac ⊥ F ( u )Jac ⊥ F ( u ) γ t ( u , u ) du du (cid:19) dv F ( x )dv F ( x )where, in the inner integral, the measures du i are the natural volume measures on the fibres ofΞ − ( x i ) and the terms Jac ⊥ F i ( u i ) are the normal Jacobians of Ξ at u i .We then turn our attention to the unit normal vectors in the integrand (see Lemma 3.12).Consider u i ∈ ∂ ˆ A i \ E such that Ξ( u i ) = x i . Since x i is the only place at which the topology of { u i ≥ } can change by infinitesimal perturbations, T u i ∂ ˆ A i is the subspace of functions v ∈ V such that v ( x i ) = 0. Since K is the reproducing kernel of V , K ( x i , · ) is orthogonal to T u i ∂ ˆ A i .Thus, (cid:104) ν ∂ ˆ A ( u ) , ν ∂ ˆ A ( u ) (cid:105) = ± K ( x , x ) (cid:112) K ( x , x ) K ( x , x ) , where the sign depends on whether a small positive perturbation of u i at x i makes u i enter orexit ˆ A i .Finally, in Lemmas 3.11 and 3.14 we (i) compute the Jacobian of Ξ at u i ∈ Ξ − ( x i ) and (ii)reinterpret the integral over Ξ( x ) − × Ξ − ( x ) as an expectation in f t = ( f t , f t ) conditionedon the fact that for i = 1 , x i is a critical point of f it at level 0, containing the indicators thatthe x i are pivotal for f it . This process involves some standard computations of Jacobians ofevaluation maps and a careful study of the relations between the different metrics on the spaces V × V and F × F . As a result, we get exactly the term which appears in the definition of thepivotal intensity functions (see (2.3)), namely (cid:104) ν ∂ ˆ A ( u ) , ν ∂ ˆ A ( u ) (cid:105) Jac ⊥ F ( u )Jac ⊥ F ( u ) = ± K ( x , x ) (cid:89) i =1 , (cid:12)(cid:12) det (cid:0) H F i x i u i (cid:1)(cid:12)(cid:12) , which completes the proof in the finite-dimensional case.To extend Theorem 2.14 to the general case, it remains only to argue that f can always beapproximated by finite-dimensional fields and that we can successfully pass to the limit in thecovariance formula. This latter step is mainly technical, and requires us to show, among otherthings, that the boundary of Ξ − ( x i ) = Piv x i ( ˆ A i ) is a null set for the field f conditioned on theexistence of critical points at x and x .3. Heart of the proof: the finite-dimensional case
In this section we state and prove a reinterpretation of our covariance formula in the case wherethe space V is finite-dimensional (see Proposition 3.9). As discussed in the proof sketch above,we prove this proposition by applying Piterbarg’s formula (Lemma 2.22) and then obtaining arather explicit description of boundaries of topological events (see Lemma 3.10).Throughout this section, and indeed for the remainder of the paper, ( B, F ) denotes an arbi-trary stratified set of M . COVARIANCE FORMULA FOR TOPOLOGICAL EVENTS OF SMOOTH GAUSSIAN FIELDS 17
Restating the formula in terms of the discriminant.
In this subsection we state thefinite-dimensional version of the formula (Proposition 3.9). For this we introduce an alternativenotion of ‘pivotal sets’ defined in terms of the ‘discriminant’. Definition 3.1 (Critical points) . Let u ∈ C ( M ). A stratified critical point of u in B is a point x ∈ B such that d x u | F = 0, where F ∈ F is the (unique) stratum containing x . When there isno ambiguity we will refer to stratified critical points as critical points for brevity. The level ofa critical point refers to its critical value.Assume now that u ∈ C ( M ). A stratified critical point x of u is said to be a non-degenerate if (i) H Fx u is non-degenerate, and (ii) for each F (cid:48) ∈ F such that F (cid:48) > F , d x u does not vanishon T x F (cid:48) | F (see Definition 2.1). Roughly speaking (ii) means that d x u vanishes on T x F but noton tangent spaces to higher dimensional strata. Note that we define non-degeneracy in terms ofthe generalised tangent bundle T x F (cid:48) | F ; this is since all strata are open and disjoint, so T x F (cid:48) isnot defined.In the following definitions V ⊂ C ( M ) denotes an arbitrary linear subspace (i.e. not nec-essarily finite-dimensional). To define the discriminant, it will be convenient to introduce thefollowing subsets of V : Notation 3.2.
For each x ∈ M , V x ⊂ V denotes the linear subspace of u ∈ V such that u ( x ) = 0. Moreover, V (cid:48) x denotes the linear subspace of V x such that also d x u | F = 0, where F is the (unique) stratum containing x ; in other words, V (cid:48) x contains the functions that possess astratified critical point at x ∈ B at level 0. Similarly, for each F ∈ F , V (cid:48) F = ∪ x ∈ F V (cid:48) x denotes thefunctions that possess a stratified critical point on F at level 0. Definition 3.3 (Discriminant) . The discriminant associated to B in V is the set D B ( V ) = ∪ F ∈F V (cid:48) F , that is, the set of functions that possess a stratified critical point in B at level 0. Foreach u ∈ V \ D B ( V ), the B -discriminant class of u (in V ), written as [ u ] ( B,V ) is the connectedcomponent of V \ D B ( V ) containing u . By Lemma C.1 the discriminant is closed; since C ( M ) isseparable, the number of classes is therefore at most countable. We will denote by (cid:101) σ discr ( B, V )the complete σ -algebra of all collections of B -discriminant classes.Before defining the alternate notion of ‘pivotal sets’ in terms of the discriminant, we introducefurther subsets of V (cid:48) x and V (cid:48) F defined above; as we verify later (see Proposition 4.1), these subsetsare of full measure: Notation 3.4.
For each x ∈ M , (cid:101) V (cid:48) x ⊂ V (cid:48) x denotes the set of u ∈ V such that x is a non-degenerate stratified critical point at level 0 and there are no other stratified critical points in B at this level. Similarly, for each F ∈ F , (cid:101) V (cid:48) F = ∪ x ∈ F (cid:101) V (cid:48) x denotes the subset of V (cid:48) F consisting offunctions that have a non-degenerate stratified critical point on stratum F at level 0, and noother stratified critical points in B at this level. Definition 3.5 (‘Pivotal sets’ in terms of the discriminant) . Let ˜ A be an element of (cid:101) σ discr ( B, V ).We denote by ˆ A ⊂ V the set of functions whose B -discriminant class is in ˜ A , and by ˆ σ discr ( B, V )the σ -algebra of all possible sets ˆ A of this type. For ˆ A ∈ ˆ σ discr ( B, V ), we define the ‘pivotalsets’ (cid:103)
Piv x ( ˆ A ) = ∂ ˆ A ∩ (cid:101) V (cid:48) x and (cid:103) Piv F ( ˆ A ) = ∂ ˆ A ∩ (cid:101) V (cid:48) F ; note that these are subsets of the discrimi-nant D B ( V ). For each σ ∈ { + , −} , let (cid:103) Piv σx ( ˆ A ) be the set of u ∈ (cid:103) Piv x ( ˆ A ) such that there exists h ∈ V with h ( x ) > η > u + σηh ∈ ˆ A .Finally, we introduce the key conditions on the space V : In fact, in the cases that matter to us, this alternative notion of ‘pivotal sets’ coincides with that of Definition2.9 up to null sets. See Remark 5.3.
Condition 3.6.
For each distinct x, y ∈ M , let V (cid:48) x,y ⊂ V denotes the set of u ∈ V such that( u ( x ) , d x u, u ( y ) , d y u ) vanishes. Then the following map is surjective: V (cid:48) x,y → Sym (cid:0) T ∗ y M (cid:1) u (cid:55)→ H y u . Condition 3.7.
For each distinct x, y ∈ M , the following map is surjective: V → R × T ∗ x M × R × T ∗ y Mu (cid:55)→ ( u ( x ) , d x u, u ( y ) , d y u ) . Remark 3.8.
For every smooth M there exists a finite-dimensional subspace V ⊂ C ∞ ( M )satisfying Conditions 3.6 and 3.7. Indeed, given a smooth mapping G : M → R N for some N ∈ N , the coordinates of G generate an N -dimensional subspace of C ∞ ( M ) which we denote by V G . For any distinct x, y ∈ M , the set of G such that V G does not satisfy Conditions 3.6 and 3.7at x and y has codimension arbitrarily large as N → ∞ . Therefore, by the multijet transversalitytheorem (see Theorem 4.13, Chapter II of [22]), applied to the multijet ( x, y ) (cid:55)→ ( j G ( x ) , j G ( y )),the set of G such that V G satisfies Conditions 3.6 and 3.7 is a residual subset of C ∞ ( M, R N )for sufficiently large N . In particular, such spaces exist.We are now ready to present our finite-dimensional restatement of the covariance formula: Proposition 3.9.
Recall the notation introduced in Section 2.1. Let V be a finite-dimensionalsubspace of C ( M ) that satisfies Conditions 3.6 and 3.7, and assume that the support of f isexactly V , so that f is a non-degenerate Gaussian vector in V . Let ( B , F ) and ( B , F ) bestratified sets of M . For each i ∈ { , } , let ˆ A i ∈ ˆ σ discr ( B i , V ) and let A i be the event { f ∈ ˆ A i } .Then, for each t ∈ [0 , , ddt P (cid:104) f t ∈ ˆ A × ˆ A (cid:105) = (cid:88) σ ,σ ∈{− , + } (cid:88) F ∈F , F ∈F (cid:90) F × F K ( x , x ) × γ t ; x ,x (0) × σ σ E t ; x ,x (cid:104) (cid:103) Piv σ x ( ˆ A ) × (cid:103) Piv σ x ( ˆ A ) ( f t , f t ) (cid:12)(cid:12) det (cid:0) H F x f t (cid:1)(cid:12)(cid:12)(cid:12)(cid:12) det (cid:0) H F x f t (cid:1)(cid:12)(cid:12)(cid:105) dv F ( x )dv F ( x ) . Proof of Proposition 3.9.
Throughout this section we assume that V , f and ˆ A i are as inthe statement of Proposition 3.9, in particular V is finite-dimensional and satisfies Conditions 3.6and 3.7 (although all the notation that is introduced applies equally to arbitrary linear subspaces V of C ( M )). We continue to use ( B, F ) to denote an arbitrary stratified set of M , and wealso define an arbitrary ˆ A ∈ ˆ σ discr ( B, V ). We rely on four technical lemmas (namely Lemmas3.10–3.12 and 3.14), whose proofs are deferred to Section 4.The starting point of the proof is to apply Piterbarg’s formula to the events ˆ A and ˆ A ; forthis we need to study the regularity of their boundaries. The structure of ∂ ˆ A i is described bythe following lemma: Lemma 3.10.
For each F ∈ F , the set (cid:103) Piv F ( ˆ A ) (from Definition 3.5) is a smooth (immersed)conical hypersurface of V . If x is the unique level- stratified critical point of some u ∈ (cid:103) Piv F ( ˆ A ) ,then T u (cid:103) Piv F ( ˆ A ) = V x (see Notation 3.4). Moreover, there exists a subset E ⊂ ∂ ˆ A of zero N − dimensional Hausdorff measure such that ∂ ˆ A = E ∪ (cid:71) F ∈F (cid:103) Piv F ( ˆ A ) . By Lemma 3.10, the boundaries of the sets ˆ A and ˆ A are smooth up to null sets, whichimplies that their N − R is of order R N − , ensuring thatPiterbarg’s formula applies to these sets. COVARIANCE FORMULA FOR TOPOLOGICAL EVENTS OF SMOOTH GAUSSIAN FIELDS 19
Figure 3.
Outside of a null set, the boundary of ˆ A is a hypersurface (cid:101) V (cid:48) F whichis covered by the disjoint union over x ∈ F of the (cid:101) V (cid:48) x . Left (functional view): Asmall neighbourhood U of u in V is split by (cid:101) V (cid:48) F into two parts, C and C , whichare inside two different topological classes (one of them belongs to ˆ A and onedoes not). Right (spatial view): When u changes continuously within U ∩ (cid:101) V (cid:48) F ,the corresponding level-0 stratified critical point x changes continuously within F . Central panels shows three functions in (cid:101) V (cid:48) F and their critical points. Smallperturbations of these functions all belong to the same topological class, forperturbations positive near the critical point they belong to C (right panels)and for negative perturbations to C (left panels).Now, consider a coordinate system orthonormal with respect to the scalar product (cid:104)· , ·(cid:105) in-duced by f . We typically denote u = ( u , . . . , u N ) to be the set of coordinates of an elementof V . For each t ∈ [0 , γ t : R N × R N → R be the density of the Gaussian vector withcovariance (cid:18) I N tI N tI N I N (cid:19) and mean ( µ, µ ), where µ ∈ R N is such that E [ f ] = (cid:80) µ i u i . This density gives the distributionof f t as defined in (2.1). Piterbarg’s formula (Lemma 2.22) implies that(3.1) ddt P (cid:104) f t ∈ ˆ A × ˆ A (cid:105) = (cid:90) ∂ ˆ A × ∂ ˆ A (cid:104) ν ˆ A ( u ) , ν ˆ A ( u ) (cid:105) γ t ( v , v ) d H N − ( u ) d H N − ( u ) , where the integral is taken on the product of the smooth part of the boundaries of ˆ A and ˆ A ,which are seen as subsets of R N through the coordinate system fixed above, and where ν ˆ A (resp. ν ˆ A ) is the outward unit normal vector to ˆ A (resp. ˆ A ) defined on the smooth part ofits boundary. Applying the expression for the smooth part of the boundary of ˆ A and ˆ A inLemma 3.10, we have(3.2) ddt P (cid:104) f t ∈ ˆ A × ˆ A (cid:105) = (cid:88) F ∈F , F ∈F (cid:90) (cid:103) Piv F ( ˆ A ) × (cid:103) Piv F ( ˆ A ) (cid:104) ν ˆ A ( u ) , ν ˆ A ( u ) (cid:105) γ t ( u , u ) d H N − ( u ) H N − ( u ) . The next step is to to apply the coarea formula to the integrals in (3.2). For each F ∈ F ,let Ξ F denote the function (cid:101) V (cid:48) F → F which maps u ∈ (cid:101) V (cid:48) F to the unique x ∈ F such that u has astratified critical point on F at level 0. We note that (cid:103) Piv x ( ˆ A ) = (Ξ F ) − ( x ) ∩ (cid:103) Piv F ( ˆ A ) , which means that we can parametrise (cid:101) V (cid:48) F by pairs ( x, u ) where x ∈ F and u ∈ (cid:103) Piv x ( ˆ A ). Thenext lemma shows that Ξ F is a submersion and gives an expression for its normal Jacobian: Lemma 3.11.
For each F ∈ F , the map Ξ F is a submersion. Moreover, for each u ∈ (cid:103) Piv F ( ˆ A ) ,if x := Ξ F ( u ) then the normal Jacobian of Ξ F at u is J F ( u ) := Jac ⊥ [Ξ F ] ( u ) = Jac ⊥ ( L x ) | det ( H Fx u ) | , where L x : V x → T ∗ x F denotes the linear operator u (cid:55)→ d x u | F , and where the determinant istaken in orthonormal coordinates of T x F . Using Lemma 3.11 we can apply the coarea formula to the integrals in (3.2), converting themfrom integrals over part of the boundary of the events to integrals over the faces of the stratifiedsets. As a result, each integral in (3.2) can be written as(3.3) (cid:90) F × F Γ( t ; x , x ) dv F ( x )dv F ( x ) , where, for each x ∈ F , x ∈ F and t ∈ [0 , t ; x , x ) = (cid:90) (cid:103) Piv x ( ˆ A ) × (cid:103) Piv x ( ˆ A ) (cid:104) ν ˆ A ( u ) , ν ˆ A ( u ) (cid:105) γ t ( u , u ) J F ( u ) J F ( u ) dv V (cid:48) x ( u )dv V (cid:48) x ( u ) , and where (cid:103) Piv x ( ˆ A ) × (cid:103) Piv x ( ˆ A ) is viewed as an open subset of V (cid:48) x × V (cid:48) x . Here we haveidentified the spaces V (cid:48) x i with their images in R N in the coordinate system fixed previously. Themeasures dv V (cid:48) xi are defined as the canonical N − dim( F i ) − V (cid:48) x i of R N .We next interpret the normal vectors in (3.4) in more tractable terms (using the sets fromDefinition 3.5): Lemma 3.12.
The fibre (cid:103)
Piv x ( ˆ A ) is the disjoint union of the two subsets (cid:103) Piv + x ( ˆ A ) and (cid:103) Piv − x ( ˆ A ) .Moreover, for each σ ∈ { + , −} and each u ∈ (cid:103) Piv σx ( ˆ A ) , the outward unit normal vector of ˆ A at u is ν ˆ A ( u ) = − σ K ( x, · ) (cid:107) K ( x, · ) (cid:107) = − σ K ( x, · ) (cid:112) K ( x, x ) . Remark 3.13.
Since K is the reproducing kernel in V , the evaluation map Ev x defined by v (cid:55)→ v ( x ) is equal to the map v (cid:55)→ (cid:104) v, K ( x, · ) (cid:105) . Hence K ( x, · ) is orthogonal to V x , and so (cid:107) K ( x, · ) (cid:107) can also be interpreted as Jac ⊥ (Ev x ), the normal Jacobian of the evaluation operator.Since K is the reproducing kernel in V , it satisfies (cid:104) K ( x , · ) , K ( x , · ) (cid:105) = K ( x , x ). Hence(3.5) (cid:104) ν ˆ A ( u ) , ν ˆ A ( u ) (cid:105) = σ ( u , u ) K ( x , x ) (cid:107) K ( x , · ) (cid:107) (cid:107) K ( x , · ) (cid:107) = σ ( u , u ) K ( x , x ) (cid:112) K ( x , x ) K ( x , x ) , where σ ( u , u ) = + if either ( u , u ) ∈ (cid:103) Piv + x ( ˆ A ) × (cid:103) Piv + x ( ˆ A ) or ( u , u ) ∈ (cid:103) Piv − x ( ˆ A ) × (cid:103) Piv − x ( ˆ A ), and σ ( u , u ) = − otherwise. Thus, by Lemma 3.11 and (3.5),(3.6) Γ( t ; x , x ) = (cid:90) (cid:103) Piv x ( ˆ A ) × (cid:103) Piv x ( ˆ A ) Υ x ,x ( u , u ) γ t ( u , u ) dv V (cid:48) x ( u )dv V (cid:48) x ( u ) , COVARIANCE FORMULA FOR TOPOLOGICAL EVENTS OF SMOOTH GAUSSIAN FIELDS 21 where(3.7) Υ x ,x ( u , u ) = σ ( u , u ) K ( x , x ) (cid:112) K ( x , x ) K ( x , x ) × (cid:12)(cid:12) det (cid:0) H F x u (cid:1)(cid:12)(cid:12) (cid:12)(cid:12) det (cid:0) H F x u (cid:1)(cid:12)(cid:12) Jac ⊥ ( L x )Jac ⊥ ( L x ) , and where L x i : V x → T ∗ x i F i denotes the linear operator u (cid:55)→ d x i u | F i . The integral in thedefinition of Γ can be interpreted as a conditional expectation: Lemma 3.14.
For each t ∈ [0 , and each distinct x ∈ F and x ∈ F , Γ( t ; x , x ) = K ( x , x ) γ t ; x ,x (0) × E t ; x ,x (cid:104) σ ( f t , f t ) (cid:103) Piv x ( ˆ A ) × (cid:103) Piv x ( ˆ A ) ( f t , f t ) (cid:12)(cid:12) det (cid:0) H F x f t (cid:1)(cid:12)(cid:12) (cid:12)(cid:12) det (cid:0) H F x f t (cid:1)(cid:12)(cid:12)(cid:105) . Combining (3.2), (3.3) and Lemma 3.14 yields the formula in Proposition 3.9.4.
Proof of the auxiliary lemmas
In this subsection we prove the auxiliary lemmas from Section 3, namely Lemmas 3.10–3.12,and Lemma 3.14. While we make use of the notation from Section 3, we do not rely on resultsfrom that section.4.1.
Differential topology in the space of functions: Proof of Lemmas 3.10–3.12.
Throughout this section V denotes a linear subspace of C ( M ); moreover, with the exceptionof the statement of Proposition 4.1, we will assume that V is finite-dimensional and satisfiesConditions 3.6 and 3.7. Again we fix an arbitrary stratified set ( B, F ) in M and ˆ A ∈ ˆ σ discr ( B, V ).We begin with a couple of definitions; for the time being we work independently of the choiceof ˆ A . Let F ∈ F , and recall from Section 3 the subsets (cid:101) V (cid:48) F ⊂ V (cid:48) F ⊂ V and the map Ξ F ( u ) whichsends u ∈ (cid:101) V (cid:48) F to its unique non-degenerate stratified critical point at level 0. Let I F be the set ofpairs ( u, x ) ∈ V × F such that x is a stratified critical point of u at level 0 (so that in fact u ∈ V (cid:48) F ),and let (cid:101) I F be the set of pairs ( u, x ) ∈ I F such that x is the unique non-degenerate stratifiedcritical point of u at level 0 (so that u ∈ (cid:101) V (cid:48) F ). By Condition 3.7, the map ( u, x ) (cid:55)→ ( u ( x ) , d x u )is a submersion on V × F , and so I F is a smooth submanifold of V × F whose codimension isone plus the dimension of F . Moreover, for each ( u, x ) ∈ I F ,(4.1) T ( u,x ) I F = (cid:8) ( v, τ ) ∈ V × T x F : v ( x ) = 0 , d x v | F + H Fx u ( τ, · ) = 0 (cid:9) . Let pr F : I F → V and pr F : I F → F be the projections onto the first and second coordinates.Note that V (cid:48) F = pr F ( I F ) and (cid:101) V (cid:48) F = pr F ( (cid:101) I F ), and observe also that the map Ξ F ( u ) completesthe following commutative diagram:(4.2) (cid:101) I F (cid:101) V (cid:48) F F pr F pr F Ξ F Lemmas 3.10 and 3.11 both pertain to elements of this diagram: for Lemma 3.11 this is explicitlyso, whereas for Lemma 3.10 it is since, as we shall see, (cid:103)
Piv F ( ˆ A ) is an open subset of (cid:101) V (cid:48) F . Inthe proof of Lemmas 3.10 and 3.11, we use the following proposition (whose proof is postponeduntil the very end of the subsection): Proposition 4.1.
Let F ∈ F . Then the set (cid:101) I F is open in I F and the set (cid:101) V (cid:48) F is open in D B .Moreover, if V has finite dimension N ∈ N and satisfies Conditions 3.6 and 3.7, then H N − (cid:0) V (cid:48) F \ (cid:101) V (cid:48) F ) = 0 . Remark 4.2.
Although we only apply Proposition 4.1 to finite-dimensional V , we state it infull generality so as to clarify which tools are used to prove each point. Remark 4.3.
Roughly speaking, Proposition 4.1 ensures that if the field f is conditioned tohave a stratified critical point at level 0, then a.s. this critical point is non-degenerate, and thereare no other stratified critical points at level 0. Proof of Lemma 3.10.
To show that (cid:103)
Piv F ( ˆ A ) is a smooth (immersed) conical hypersurface of V ,we first show that (cid:101) V (cid:48) F is a smooth immersed (although maybe not embedded) hypersurface of V .By Proposition 4.1, (cid:101) I F is a smooth submanifold of V × F with the same tangent space as I F ateach point. The mapping pr F : (cid:101) I F → V is one-to-one, and we claim that it has constant rank.To see this, let us take ( u, x ) ∈ (cid:101) I F and check that d ( u,x ) pr F (cid:0) T ( u,x ) I F (cid:1) = V x . The inclusion ⊂ is clear by (4.1). For the reverse inclusion, let v ∈ V x and define λ = − d x v | F .Since ( u, x ) ∈ (cid:101) I F , H Fx u is non-degenerate, and so there exists τ ∈ T x F such that H Fx u ( τ, · ) = λ .Therefore, ( v, τ ) ∈ T ( u,x ) I F and d ( u,x ) pr F ( v, τ ) = v , which proves the reverse inclusion. To sumup, pr F is a mapping of corank one on (cid:101) I F , and so its image (cid:101) V (cid:48) F is a smooth immersed (althoughmaybe not embedded) hypersurface of V with the tangent space(4.3) T u (cid:101) V (cid:48) F = d ( u,x ) pr F (cid:0) T ( u,x ) I F (cid:1) = V x . Next, we show that (cid:103)
Piv F ( ˆ A ) is open in (cid:101) V (cid:48) F . Indeed, by Proposition 4.1, (cid:101) V (cid:48) F is open in D B .Moreover, (cid:101) V (cid:48) F is a smooth submanifold of V , which implies that, for each u ∈ (cid:101) V (cid:48) F , there exists U ⊂ V containing u such that ( u, U ∩ (cid:101) V (cid:48) F , U ) (cid:39) (0 , R N − × { } , R N ) and such that U ∩ D B = U ∩ (cid:101) V (cid:48) F . Hence there exist exactly two B -discriminant classes C , C that intersect U and(4.4) C ∩ (cid:101) V (cid:48) F ∩ U = C ∩ (cid:101) V (cid:48) F ∩ U = (cid:101) V (cid:48) F ∩ U as illustrated in Figure 3. In particular, if u ∈ (cid:103) Piv F ( ˆ A ) then (cid:101) V (cid:48) F ∩ U ⊂ (cid:103) Piv F ( ˆ A ), and so (cid:103) Piv F ( ˆ A )is an open subset of (cid:101) V (cid:48) F .To sum up, since (cid:103) Piv F ( ˆ A ) is open in (cid:101) V (cid:48) F and since (cid:101) V (cid:48) F is a smooth (immersed) hypersurfaceof V , (cid:103) Piv F ( ˆ A ) is also a smooth (immersed) hypersurface of V . Noting also that ˆ A is conicalhence so is ∂ ˆ A , and observing moreover that, by (4.3), T u (cid:103) Piv F ( ˆ A ) = V x for every ( u, x ) ∈ (cid:101) I F ,we complete the proof of the first two statements of the lemma.For the third statement of the lemma, we define E = ∂ ˆ A \ (cid:16) (cid:71) F ∈F (cid:101) V (cid:48) F (cid:17) . By the definition of (cid:103)
Piv F ( ˆ A ) = ∂ ˆ A ∩ (cid:101) V (cid:48) F , we have(4.5) ∂ ˆ A = E ∪ (cid:71) F ∈F (cid:103) Piv F ( ˆ A ) . Moreover, we claim that H N − ( E ) = 0. To see this, observe that ∂ ˆ A ⊂ D B := (cid:83) F ∈F V (cid:48) F .Indeed, since the discriminant D B is closed (see Lemma C.1), the B -discriminant class of any u ∈ V \ D B forms a neighbourhood of u ; in particular, u / ∈ ∂ ˆ A . Hence we have an alternateexpression for E : E = (cid:71) F ∈F ∂ ˆ A ∩ (cid:16) V (cid:48) F \ (cid:101) V (cid:48) F (cid:17) . Since by Proposition 4.1 the N − H N − ( E ) = 0. (cid:3) COVARIANCE FORMULA FOR TOPOLOGICAL EVENTS OF SMOOTH GAUSSIAN FIELDS 23
Proof of Lemma 3.11.
We first show that Ξ F is a submersion. Let ( u, v ) ∈ T (cid:101) V (cid:48) F , so that thereexist x ∈ F and τ ∈ T x F such that (( u, x ) , ( v, τ )) ∈ T (cid:101) I F . In particular, by (4.1) we have d x v | F + H Fx u ( τ, · ) = 0. Since H Fx u is non-degenerate, τ is uniquely determined by v . More precisely,let ˇ H Fx u be the image of H Fx u by the canonical isomorphism ( T ∗ F ) ⊗ (cid:39) Hom( T ∗ F, T F ). Then τ = − (cid:0) ˇ H Fx u (cid:1) − ( d x v | F ). Since the diagram (4.2) commutes, we have proven that d u Ξ F ( v ) = − (cid:0) ˇ H Fx u (cid:1) − ( d x v ) . By Condition 3.7, the map v (cid:55)→ d x v is surjective when restricted to V x . Hence Ξ F is a submersion,which proves the first statement of the lemma.Let us now show that the Jacobian of Ξ F is as claimed in the lemma. Let g − F be the metricinduced on T ∗ F by the metric g F on T F . Since ( ˇ H Fx u ) − is an isomorphism ( T ∗ x F, g − F,x ) → ( T x F, g
F,x ), the normal Jacobian of Ξ F is the product of the Jacobian of ( ˇ H Fx u ) − and of thenormal Jacobian of the map L x : ( V x , (cid:104)· , ·(cid:105) ) → ( T ∗ x F, g − F,x ), defined in the statement of the lemmato be L x ( v ) = d x v . Since the first Jacobian is the absolute value of the inverse of det( H Fx u ), i.e.the determinant of the matrix of the bilinear form H Fx u in a g − F,x -orthonormal basis of T x F , theproof is complete. (cid:3) Remark 4.4.
Although for our purposes we do not need to compute Jac ⊥ ( L x ) explicitly (sinceit eventually cancels out in the main formula), for completeness we haveJac ⊥ ( L x ) = (cid:112) det ( L x L ∗ x ) = (cid:113) det ( d x ⊗ d x K x | F,F ) , where K x ( y , y ) = K ( y , y ) − K ( x, y ) K ( y , x ) /K ( x, x ) is the covariance kernel of f conditionedon f ( x ) = 0 or, equivalently, of the orthogonal projection of f onto V x ; this follows from thesame routine computation as in Remark 3.13. More generally, if L : V → R k is a linear operator,the orthogonal Jacobian of Lf is the square root of the determinant of the covariance of Lf .Let us now complete the proof of Lemma 3.12; for this we rely on elements from the proof ofLemma (3.10): Proof of Lemma 3.12.
Let u ∈ (cid:103) Piv F ( ˆ A ), x = Ξ F ( u ), and take U , C and C as in (4.4). ByLemma 3.10, we have T u (cid:103) Piv F ( ˆ A ) = V x . In particular, for any such v , (cid:104) K ( x, · ) , v (cid:105) = v ( x ) = 0,so K ( x, · ) is orthogonal to T x (cid:103) Piv F ( ˆ A ). Moreover, (cid:104) K ( x, · ) , K ( x, · ) (cid:105) = K ( x, x ), which must bepositive (otherwise all functions in V vanish at x which contradicts Condition 3.7). Therefore,the outward unit normal vector ν ˆ A ( u ) to A at u is plus or minus(4.6) v x := K ( x, · ) (cid:112) K ( x, x ) . The sign of this vector depends on which of the C i belongs to ˆ A . More precisely, a perturbation u + ηh (with η (cid:28)
1) enters ˆ A whenever (cid:104) v x , h (cid:105) = h ( x ) has the right sign. In particular, this showsthat the sets (cid:103) Piv + x ( ˆ A ) and (cid:103) Piv − x ( ˆ A ) form a partition of (cid:103) Piv x ( ˆ A ) and that, for each σ ∈ { + , −} and each u ∈ (cid:103) Piv σx ( ˆ A ), ν ˆ A ( u ) = − σ K ( x, · ) √ K ( x,x ) . (cid:3) Finally, we prove Proposition 4.1. For this we use the following standard fact which we statewithout proof:
Lemma 4.5.
Let h : M → M (cid:48) be a Lipschitz map and let S ⊂ M be a k -dimensional submanifoldof M . Then the Hausdorff dimension of h ( S ) is at most k . In particular, H d ( h ( S )) = 0 forevery d > k . Proof of Proposition 4.1.
Let us first give some intuition. The set D B \ (cid:101) V (cid:48) F consists of functionswhich, in addition to having a level-0 critical point on F , are degenerate in some way. Weexpress the five different cases of degeneracy as the vanishing of five explicit smooth functionalsof pairs ( u, x ) ∈ V × F or triplets ( u, x, y ) ∈ V × F × F for some F ∈ F . From this we deduceboth that (cid:101) I F is open in I F and that its complement has positive codimension. We then concludeby projecting the vanishing loci onto V .Recall that V ⊂ C ( M ) is a linear space. Let d denote the dimension of F , and let F , F ∈ F be strata of dimensions d and d respectively. We consider the following five subsets:(1) If F < F , let I F,F be the set of pairs ( u, x ) ∈ I F such that d x u ∈ T ∗ F M | x .(2) Let I F be the set of pairs ( u, x ) ∈ I F such that H x u is singular.(3) If F < F , let I F,F ,F be the set of triplets ( u, x, y ) ∈ I F × F such that x and y aredistinct, y is also a stratified critical point of u and d y u ∈ T ∗ F M | y .(4) Let I F,F be the set of triplets ( u, x, y ) ∈ I F × F such that x and y are distinct, y isalso a stratified critical point of u and H F y u is singular.(5) Let I F,F be the set of triplets ( u, x, y ) ∈ I F × F such that x and y are distinct and y is also a stratified critical point of u with critical value 0. Claim 4.6.
Each of the five subsets defined above is a closed subset of V × F (resp. V × F × F ,as appropriate). Moreover, if we assume in addition that V has finite dimension N ∈ N andsatisfies Conditions 3.6 and 3.7, then each of these subsets is a finite union of submanifolds ofcodimension at least N + d + 1 (resp. N + d + d + 1 ). Remark 4.7.
The proof of Claim 4.6 is the only place in the paper where we use the fact that F is a tame stratification of B , rather than merely a Whitney stratification. Proof.
We begin with a couple of definitions. For each F ∈ F , let T ∗ F M be the conormal bundle to F , that is, for each x ∈ F , T ∗ F M | x is the set of ξ ∈ T ∗ x M such that ξ | T x F = 0. This is asmooth vector bundle whose rank is exactly the codimension of F in M . In particular, T ∗ F M hascodimension d in T ∗ M . Recall from the definition of a tame stratification that, given F , F ∈ F such that F < F , the set of limit points of T F with basepoints on F defines a vector bundleover F , denoted by T F | F , which we call the generalised tangent bundle of F over F . Thisallows us to extend the definition of conormal bundle as follows: the conormal bundle to F over F , denoted T ∗ F M | F , is the set of ( x, ξ ) ∈ T ∗ M | F such that ξ vanishes on T ∗ F | x . Thisdefines a smooth vector bundle over F whose rank is the codimension of F in M . Thus, apoint x ∈ F is a non-degenerate stratified critical point of some u ∈ C ( M ) if and only if it is anon-degenerate critical point of u | F and for each F ∈ F such that F < F , ( x, d x u ) / ∈ T ∗ F M | F .Now, assume first that V has finite dimension N ∈ N and satisfies Conditions 3.6 and 3.7.Since the proofs all follow the same structure, we cover in detail only the case of I F,F , and thenindicate what changes need to be made in the other cases.Consider the map Φ : I F → T ∗ M | F defined by ( u, x ) (cid:55)→ d x u , and recall the expression of thetangent spaces of I F given in (4.1). By Condition 3.7, the map Φ is a submersion. Moreover,the set T ∗ F M | F is a smooth submanifold of T ∗ M | F of codimension 1 + d that is also a closedsubset, and therefore I F,F = Φ − (cid:0) T ∗ F M | F (cid:1) is a smooth submanifold of I F of codimension d as well as a closed subset of this space. We have thus covered the case of I F,F .For I F we consider the map Φ : I F → Sym ( T ∗ F ) defined by ( u, x ) (cid:55)→ H Fx u , which is asubmersion by Condition 3.6. Instead of T ∗ F M | F , we consider the zero set of the determinantmap det : Sym ( T ∗ F ) → R induced by some auxiliary metric. Its zero set W F is closed and canbe partitioned into the spaces of matrices of fixed rank in { , . . . , d − } so it is a finite unionof smooth submanifolds of positive codimension. Since I F = Φ − ( W F ), we are done. COVARIANCE FORMULA FOR TOPOLOGICAL EVENTS OF SMOOTH GAUSSIAN FIELDS 25
The cases I F,F ,F and I F,F are analogous to the first two cases. The maps Φ and Φ shouldbe replaced by maps Φ and Φ defined on I F,F = { ( u, x, y ) ∈ V × F × F : u ( x ) = 0 , d x u | F =0 , d y u | F = 0 } which is a smooth submanifold of I F × F of codimension d and whose tangentspace at ( u, x, y ) is { ( v, τ , τ ) ∈ V × T x F × T y F : v ( x ) = 0 , d x v | F + H x u ( τ , · ) = 0 , d y v | F + H y u ( τ , · ) = 0 } . They should be defined as follows: Φ : ( u, x, y ) (cid:55)→ ( u ( x ) , d x u, d y u ) and Φ ( u, x, y ) (cid:55)→ H y u . Asfor I F , Condition 3.7 should be replaced by Condition 3.6 in the case of I F,F .Finally, for I F,F we can consider the map Φ : I F,F → R that maps each triple ( u, x, y ) to u ( y ). This map is a submersion by Condition 3.7. The conclusion follows accordingly.This ends the proof of the finite-dimensional part of the claim. Consider now the general case.Observe that we still have I F,F = Φ − (cid:0) T ∗ F M | F (cid:1) , which is the preimage of a closed subset bya continuous map; in particular it is also closed. Since the same argument works with the fourother cases, we also deduce the infinite-dimensional case of the claim. (cid:3) Let us now use Claim 4.6 to prove that (cid:101) I F is open in I F . Consider ( u k , x k ) ∈ ( I F \ (cid:101) I F ) N thatconverges in I F ; we claim its limit ( u, x ) belongs to I F \ (cid:101) I F . Observe that I F \ (cid:101) I F is the unionof the following sets:(1) The union over the { F ∈ F : F < F } of the sets I F,F .(2) The set I F .(3) The union over { F , F ∈ F : F < F } of the images of the projections I F,F ,F → I F .(4) The union over F ∈ F of the images of the projections I F,F → I F .(5) The union over F ∈ F of the images of the projections I F,F → I F .Since the above union is over a finite set, one of them contains an infinite number of terms ofthe sequence ( u k , x k ) k ∈ N . We can and will thus assume, up to extraction, that the sequence( u k , x k ) belongs to one of the sets just described. We now describe what happens in each case:(1) By Claim 4.6, I F,F is closed in I F , so ( u, x ) ∈ I F,F ⊂ I F \ (cid:101) I F .(2) We reason likewise.(3) By construction, for each k ∈ N , ( u k , x k ) is the projection of a triplet in I F,F ,F . Bycompactness of B , we can extract a subsequence for which the third coordinate of thetriplet converges in F . Since the subsequence must have the same limit in the projectionas the full sequence, we just denote it by ( u k , x k , y k ) k ∈ N ∈ ( I F,F ,F ) N so that the thirdcoordinate converges to some y ∈ F . If y ∈ F then ( u, x, y ) ∈ I F × F . Then byClaim 4.6, I F,F ,F is closed in I F × F so ( u, x, y ) ∈ I F ,F , which implies that ( u, x )belongs to its projection onto I F . If, on the other hand, y / ∈ F , (by Definition 2.1), y must belong to some F such that F ∈ F such that F < F . Then d y u ∈ T ∗ F M | y (actually we even have d y u ∈ T ∗ F M | y ). If y (cid:54) = x , we must then have ( u, x, y ) ∈ I F,F ,F so( u, x ) belongs to its projection onto I F . Otherwise, if y = x , then, ( u, x ) = ( u, y ) ∈ I F,F .(4) We reason as in the third case. As before, up to extraction, we can find ( y k ) k ∈ N ∈ F converging to some y ∈ F such that for each k ∈ N , ( u k , x k , y k ) ∈ I F,F . Again, asbefore, if y belongs to some face F < F , we have d y u ∈ T ∗ F M | y so ( u, x, y ) ∈ I F,F .Otherwise, if y ∈ F , using Claim 4.6 we deduce that ( u, x, y ) ∈ I F,F .(5) We reason as in the fourth case.We have therefore proven that I F \ (cid:101) I F is closed in I F .Next, we show that (cid:101) V (cid:48) F is open in D B . By construction, D B is the union of the projectionsonto the first coordinates of the sets I F for F (cid:54) = F and of the sets I F , I F , I F,F ,F , I F,F and I F,F defined above, taken over all the adequate F and F . As before we take ( u k ) k ∈ N ∈ ( D B \ (cid:101) V (cid:48) F ) N converging to some u ∈ V (cid:48) F and, up to extraction, there exist two strata F ≤ F and a sequence ( x k ) k ∈ N ∈ F N and x ∈ F such that for each k ∈ N , ( u k , x k ) ∈ I F \ (cid:101) I F andlim k →∞ ( u k , x k ) = ( u, x ). By Lemma C.1, x is a stratified critical point of u . Let us prove that u ∈ D B \ (cid:101) V (cid:48) F . From now on, the reasoning is analogous to that used for I F \ (cid:101) I F .(1) If F (cid:54) = F , then, ( u, x ) ∈ I F so u / ∈ (cid:101) V (cid:48) F .(2) If F > F = F , then, as before d x u ∈ T ∗ | F M | x and so ( u, x ) ∈ I F,F and u / ∈ (cid:101) V (cid:48) F .(3) If F = F = F then for each k ∈ N , ( u k , x k ) belongs to I F \ (cid:101) I F which is closed in I F sothat ( u, x ) / ∈ (cid:101) I F and so u / ∈ (cid:101) V (cid:48) F .This proves that D B \ (cid:101) V (cid:48) F is closed in D B as announced.To finish, assume that V has finite dimension N ∈ N and satisfies Conditions 3.6 and 3.7. By(the finite-dimensional case of) Claim 4.6, V (cid:48) F \ (cid:101) V (cid:48) F is a finite union of projections of submanifoldsof V × F and V × F × F for F ∈ F of codimensions at least dim( F )+2 and dim( F )+dim( F )+2respectively. By Lemma 4.5, we must therefore have H N − ( V (cid:48) F \ (cid:101) V (cid:48) F ) = 0. (cid:3) Conditional expectation computation: Proof of Lemma 3.14.
In this section weprove Lemma 3.14, that is, we rewrite the function Γ defined by (3.4) (see also (3.6)) in termsof a conditional expectation.Fix t ∈ [0 ,
1) and distinct x ∈ F and x ∈ F . In the first part of the proof the exactexpression of Υ x ,x , defined by (3.7), will not play any role except through the fact that it isbounded by a polynomial in u , u . Let P x ,x be the orthogonal projector in V × V (equippedwith the product metric) onto the subspace V (cid:48) x × V (cid:48) x , and let P ⊥ x ,x = I − P x ,x be thecomplementary orthogonal operator onto the orthogonal complement, which we denote by ( V (cid:48) x × V (cid:48) x ) ⊥ . We write ( u , u ) = w + w ⊥ where w = P x ,x ( u , u ) and w ⊥ = P ⊥ x ,x ( u , u ). Let usdefine j x ,x : V × V → R × T ∗ x F × R × T ∗ x F by j x ,x ( u , u ) = ( u ( x ) , d x u | F , u ( x ) , d x u | F ) . Note that the space V (cid:48) x × V (cid:48) x is exactly the kernel of j x ,x , hence j x ,x is a linear isomorphismfrom ( V (cid:48) x × V (cid:48) x ) ⊥ onto R × T ∗ x F × R × T ∗ x F . With this notation we can rewrite the integralin (3.6) as(4.7) Γ( t ; x , x ) = (cid:90) V (cid:48) x × V (cid:48) x (cid:103) Piv x ( ˆ A ) × (cid:103) Piv x ( ˆ A ) ( w )Υ x ,x ( w ) γ t ( w ) dw, where dw = dv V (cid:48) x dv V (cid:48) x . In the same spirit we write g t = P x ,x f t and g ⊥ t = P ⊥ x ,x f t so that f t = g t + g ⊥ t . The density of f t , conditioned on g ⊥ t = 0, at w = ( u , u ) ∈ V (cid:48) x × V (cid:48) x is given by γ f t | g ⊥ t =0 ( w ) = γ t ( u , u ) γ g ⊥ t (0) , where γ g ⊥ t (0) is the density of g ⊥ t evaluated at 0. Notice that for f t , conditioning on g ⊥ t = 0 is thesame as conditioning on ( f t ( x ) , d x f t | F , f t ( x ) , d x f t | F ) = 0. Since by definition ( u , u ) = w on V (cid:48) x × V (cid:48) x , (4.7) becomesΓ( t ; x , x ) = γ g ⊥ t (0) (cid:90) V (cid:48) x × V (cid:48) x (cid:103) Piv x ( ˆ A ) × (cid:103) Piv x ( ˆ A ) ( w )Υ x ,x ( w ) γ f t | g ⊥ t =0 ( w ) dw (4.8) = γ g ⊥ t (0) E t ; x ,x (cid:104) (cid:103) Piv x ( ˆ A ) × (cid:103) Piv x ( ˆ A ) ( f t )Υ x ,x ( f t , f t ) (cid:105) . COVARIANCE FORMULA FOR TOPOLOGICAL EVENTS OF SMOOTH GAUSSIAN FIELDS 27
In the above expression, the density γ g ⊥ t (0) is with respect to the orthogonal coordinates in( V (cid:48) x × V (cid:48) x ) ⊥ , and we need to express it in terms of K . Let (cid:101) Q t ; x ,x be the covariance matrix of g ⊥ t in some orthonormal system of coordinates in ( V (cid:48) x × V (cid:48) x ) ⊥ . Let Q t ; x ,x be the covariance of( f t ( x ) , d x f t | F , f t ( x ) , d x f t | F ) = j x ,x ( f t ) = j x ,x ( g t )in any orthonormal coordinate system of R × T ∗ x F × R × T ∗ x F equipped with the productmetric. Treating j x ,x as an isomorphism from ( V (cid:48) x × V (cid:48) x ) ⊥ onto R × T ∗ x F × R × T ∗ x F we seethat the covariances (cid:101) Q t ; x ,x and Q t ; x ,x are linked by the following relation (cid:101) Q t ; x ,x = (cid:0) j ∗ x ,x (cid:1) − Q t ; x ,x j − x ,x . In particular, det( (cid:101) Q t ; x ,x ) = det( Q t ; x ,x ) / det( j x ,x j ∗ x ,x ) − . Recalling that γ t ; x ,x (0) is thedensity of j x ,x ( f t ) at 0, we have γ g ⊥ t (0) = γ t ; x ,x (0) (cid:113) det (cid:0) j x ,x j ∗ x ,x (cid:1) . It remains to compute (cid:113) det (cid:0) j x ,x j ∗ x ,x (cid:1) . Notice first that j x ,x factors as the direct productof the two linear maps j x i : V (cid:48)⊥ x i → R × T ∗ x i F i for i ∈ { , } defined as j x i ( u ) = ( u ( x i ) , d x i u | F i ),(4.9) det (cid:0) j x ,x j ∗ x ,x (cid:1) = det (cid:0) j x j ∗ x (cid:1) det (cid:0) j x j ∗ x (cid:1) . To compute det (cid:0) j x i j ∗ x i (cid:1) note that, since j x i is 0 on V (cid:48) x i , this determinant does not depend onwhether j x i acts on V (cid:48)⊥ x i or the entire V ; we treat it as an operator on V . Next, we write V asorthogonal sum of V x i which is the space of functions such that v ( x i ) = 0 and its orthogonalcomplement which is spanned by K ( x i , · ) (see the discussion preceding (4.6)). Let us chooseorthonormal coordinates in V that are adopted to this decomposition, that is K ( x i , · ) / (cid:107) K ( x i , · ) (cid:107) must be one of the basis vectors. In this coordinates j x i factors as u (cid:55)→ u ( x i ) acting on the spanof K ( x i , · ) (this is the operator Ev x i from Remark 3.13) and u (cid:55)→ d x i u | F i on V x (which is theoperator L x i ). The factorisation implies that (cid:113) det (cid:0) j x i j ∗ x i (cid:1) = (cid:113) det (cid:0) Ev x i Ev ∗ x i (cid:1) det (cid:0) L x i L ∗ x i (cid:1) = Jac ⊥ (Ev x i )Jac ⊥ ( L x i ) . Plugging this computation into (4.8) we see that Γ is equal to γ t ( x , x ) (cid:112) K ( x , x ) K ( x , x )Jac ⊥ ( L x )Jac ⊥ ( L x ) × E t ; x ,x (cid:104) (cid:103) Piv x ( ˆ A ) × (cid:103) Piv x ( ˆ A ) ( w )Υ x ,x ( f t , f t ) (cid:105) . Recalling the definition of Υ x ,x ( u , u ), and in particular pulling the terms (cid:112) K ( x i , x i ) andJac ⊥ ( L x i ) from this definition out of the expectation (since they do not depend on u i ) so thatthey cancel with those already present, we deduce the result. Remark 4.8.
The cancellations in the above derivation are not so mysterious, since the relevantterms are Jacobians of evaluations of f and its differential and they appear, first, when we switchfrom space coordinates to functional coordinates, and then once again when we move back.5. Proof of the main theorem: from the finite to the infinite-dimensional case
In this section we complete the proof of the covariance formula in Theorem 2.14. The basicidea is to (i) reinterpret topological events in terms of the discriminant, (ii) approximate thefield f by a sequence of fields f k taking values in a finite-dimensional spaces V k , and then (iii)pass to the limit in the formula of Proposition 3.9.In Section 5.1 we show that the boundary of pivotal events is well behaved, which will allowus to take limits of the expectations in the right-hand side of Proposition 3.9. In Section 5.2 weverify that topological events are encoded by the discriminant. Next, in Section 5.3 we construct the finite-dimensional approximation and state an abstract continuity lemma for expectationsthat we use in the proof. Finally in Section 5.4 we assemble these elements into a proof ofTheorem 2.14.At the end of the section we also verify that Corollary 1.1 is indeed a special case of Theo-rem 2.14, as claimed in Section 1.5.1. On the boundary of pivotal events.
In this section we compare pivotal events in dif-ferent subspaces of C ( M ), link the two distinct notions of pivotal events we have introduced,and study the boundary of pivotal events.Recall that ( B, F ) denotes an arbitrary stratified set of M . Fix a linear subspace V ⊂ C ( M ),not necessarily finite-dimensional. Also fix (cid:101) A ∈ (cid:101) σ discr ( B, C ( M )), and let ˆ A ∈ ˆ σ discr ( B, C ( M ))be the set of u ∈ C ( M ) whose discriminant class (in C ( M )) belongs to (cid:101) A . Observe that theset ˆ A V = ˆ A ∩ V belongs to ˆ σ discr ( B, V ), i.e. it is encoded by the V -discriminant. Indeed, it isthe set of functions u ∈ V whose discriminant class in C ( M ) belongs to (cid:101) A . Recall also thedefinition, for x ∈ B and ˆ A ∈ ˆ σ discr ( B, V ), of the sets (cid:103)
Piv x ( ˆ A ) and (cid:103) Piv σx ( ˆ A ) from Definition 3.1.The main result of this section is the following: Lemma 5.1 (On pivotal events) . Suppose that V contains the constant functions on M . Then (1) (cid:103) Piv x ( ˆ A V ) = (cid:103) Piv x ( ˆ A ) ∩ V and, for each σ ∈ { + , −} , (cid:103) Piv σx ( ˆ A V ) = (cid:103) Piv σx ( ˆ A ) ∩ V .Moreover, let f be a Gaussian field on M satisfying Condition 2.7. Then, conditionally on x being a stratified critical point of f with f ( x ) = 0 , a.s. (2) f ∈ (cid:103) Piv x ( ˆ A ) if and only if (i) f ∈ Piv x ( ˆ A ) and (ii) H Fx f is a non-degenerate bilinearform. Moreover, for each σ ∈ { + , −} , the same is true if we replace (cid:103) Piv x ( ˆ A ) by (cid:103) Piv σx ( ˆ A ) and Piv x ( ˆ A ) by Piv σx ( ˆ A ) . (3) If x is a non-degenerate critical point then f / ∈ ∂ (cid:103) Piv x ( ˆ A ) , where (cid:103) Piv x ( ˆ A ) is seen as asubset of the space V (cid:48) x . Remark 5.2.
Note that we only apply Lemma 5.1 to approximations of the field f (as opposedto f itself), so it is irrelevant that the constant functions will not belong to the Cameron-Martinspace of f in general. Remark 5.3.
If we had been willing to impose a non-degeneracy condition on the Hessian of f ,we could have concluded from Lemma 5.1 that, conditionally on x being a level-0 stratifiedcritical point of f , a.s. f ∈ (cid:103) Piv x ( ˆ A ) if and only if f ∈ Piv x ( ˆ A ), and this is the sense in whichwe think of (cid:103) Piv x ( ˆ A ) and Piv x ( ˆ A ) as equal up to null sets. Since non-degeneracy of the Hessianis unnecessary for the result to hold, we do not do this.In order to prove Lemma 5.1, we use the following result: Lemma 5.4.
Let u ∈ C ( M ) be such that u has a unique non-degenerate stratified critical point x at level (c.f. the set ∪ F ∈F (cid:101) V (cid:48) F ). Then (1) For ε > small enough, neither u + ε nor u − ε have a stratified critical point atlevel . Moreover, let C = C ( u ) and C = C ( u ) be the connected components in C ( M ) \ D B ( C ( M )) of u + ε and u − ε respectively. Then, C ∪ C (resp. C , C ) isa neighbourhood of u in C ( M ) (resp. in the set of functions u (cid:48) ∈ C ( M ) such that u ( x ) ≥ , in the set of functions u (cid:48) ∈ C ( M ) such that u ( x ) ≤ ). (2) Let U ⊂ M be a neighbourhood of x . Then there is a neighbourhood U of u in thediscriminant D B ( C ( M )) such that, for each u (cid:48) ∈ U , u (cid:48) has exactly one stratified criticalpoint at level , which is non-degenerate and belongs to U . Moreover, we have C ( u (cid:48) ) = C ( u ) and C ( u (cid:48) ) = C ( u ) (defined as in (1) ). COVARIANCE FORMULA FOR TOPOLOGICAL EVENTS OF SMOOTH GAUSSIAN FIELDS 29
Remark 5.5.
If we were working in a finite-dimensional space, we could think of u as belongingto the smooth part of the discriminant. Since this discriminant is a hypersurface, this would meanthat in a small neighbourhood of u the discriminant would be diffeomorphic to a hyperplane,separating the ambient space into two connected components C and C , and moreover smallperturbations of u would yield the same C and C . Lemma 5.4 encodes (part of) this intuition. Remark 5.6.
The first point of Lemma 5.4 implies that the definition of (cid:103)
Piv σx ( ˆ A ) does notchange if one takes, in the definition of this set, h ∈ C ( M ) instead of merely h ∈ V . Similarly,the definition does not change if one requires h to be a positive constant (which assists in showinga function does not belong to (cid:103) Piv σx ( ˆ A )). Proof of Lemma 5.4.
We start by showing that the property that u has a stratified critical pointnear x is stable under C perturbations; this involves isolating x as a critical point in a uniformway. Fix a neighbourhood U of x . Since x is a non-degenerate critical point, it is isolated inthe set of critical points of u (see Lemma C.2), which is compact by Lemma C.1. Thus, thecritical value u ( x ) is isolated in the set of critical values of u . In particular, for each ε >
0, both u + ε and u − ε belong to C ( M ) \ D B ( C ( M )), which justifies the existence of C and C . Letus show that C ∪ C is a neighbourhood of u in C ( M ). Let F ∈ F be the stratum containing x . For each r >
0, let B r be the Riemmanian ball of radius r > F centred at x . Since x is a non-degenerate critical point at x , the section du , which is C , vanishes transversally at x on the stratum F and stays bounded from below on the higher strata near x . Therefore, thereexist r = r ( u ) > η = η ( u ) > w ∈ C ( M ) such that (cid:107) w (cid:107) C ( B ) ≤ η , thefollowing holds: • The ball B r is included in U ; • The section d ( u + w ) | F vanishes exactly once on B r ; • For any F (cid:48) (cid:54) = F , d ( u + w ) | F (cid:48) does not vanish on B r ; • u + w has no stratified critical points with critical value in [ − η, η ] outside of B r ; • If moreover (cid:107) w (cid:107) C ( B ) ≤ η/ | u + w | ≤ η/ B r .In particular, for each s ∈ (0 , η ], u ± s does not belong to the discriminant. Let w ∈ C ( M ) besuch that (cid:107) w (cid:107) C ( B ) ≤ η/
8. Let us show that u + w ∈ C ∪ C , and that if u + w ∈ D B ( C ( M ))then u + w has a unique stratified critical point at level 0 which belongs to B r . To this end, wewill first consider a path ( v t ) t from u to u + w where w is a small perturbation. Along this pathwe will find further perturbations v t,s = v t + s of v t , for suitable choices of s , that do not belongto the discriminant and that belong to the two connected components C and C .More precisely, for each t ∈ [0 ,
1] and each s ∈ [ − η/ , η/ v t,s = u + tw + s . Then, foreach t ∈ [0 ,
1] and each s ∈ [ − η/ , η/ (cid:107) v t,s − u (cid:107) C ( B ) ≤ η and (cid:107) v t, − u (cid:107) C ( M ) ≤ η/
8. Inparticular, v t,s has a unique stratified critical point in B r , which we call y t (since it does notdepend on s ) and no other stratified critical points with critical value in [ − η, η ]. Moreover,sup B r | v t, | ≤ η/ B r v t,η/ ≥ η/ B r v t, − η/ ≤ − η/
4. In particular, foreach t ∈ [0 , v t, ± η/ / ∈ D B ( C ( M )). Thus, v , ± η/ ∈ C ∪ C . Now, for each s ∈ [ − η/ , η/ v ,s = u + w + s . In particular, if u ( y ) + w ( y ) ≥ v ,s does not belong to the discriminantfor s ∈ (0 , η/
2] and converges to u + w as s →
0. If on the other hand, u ( y ) + w ( y ) ≤
0, thesame approximation holds by taking s ∈ [ − η/ ,
0) and s →
0. In any case, by construction ofthis approximation v ,s ∈ C ∪ C as long as s (cid:54) = 0 so that u + w ∈ C ∪ C = C ∪ C . This showsthat C ∪ C is a neighbourhood of u in C ( M ) and that for each v ∈ ( C ∪ C ) ∩ D B ( C ( M )), v has a unique stratified critical point at level 0, which is in B r ⊂ U .Next notice that u + w is in the same connected component of the complement of the dis-criminant as u + s for s (cid:28) w ( x ) <
0, and is in the same connected component as u − s for s (cid:28) w ( x ) >
0, which proves that C (resp. C ) is a neighbourhood of u in the set offunctions taking non-negative (resp. non-positive) values at x . Finally, if u ( y t ) + tw ( y t ) = 0, by construction u + w + ε (resp. u + w − ε ) is in the same connected component as u + ε (resp. u − w ) for ε > C i ( u ) = C i ( u + w ) for i ∈ { , } . This ends theproof of the lemma. (cid:3) Proof of Lemma 5.1.
We prove the three statements in the lemma sequentially:(1). In order to prove that (cid:103)
Piv x ( ˆ A V ) = (cid:103) Piv x ( ˆ A ) ∩ V and (cid:103) Piv σx ( ˆ A V ) = (cid:103) Piv σx ( ˆ A ) ∩ V , it is enoughthat ∂ ˆ A V = ∂ ˆ A ∩ V , where ∂ ˆ A V is the boundary of ˆ A V in V . Clearly, ∂ ˆ A V ⊂ ∂ ˆ A ∩ V . Onthe other hand, let u ∈ ∂ ˆ A ∩ V . Then, by Lemma 5.4, there exist C and C two connectedcomponents of C ( M ) \ D B ( C ( M )) such that for ε > u + ε ∈ C , u − ε ∈ C and C ∪ C is a neighbourhood of u in C ( M ). Let us assume that C ⊂ ˆ A and C ⊂ C ( M ) \ ˆ A since u ∈ ∂ ˆ A , and the case C ⊂ ˆ A and C ⊂ C ( M ) \ ˆ A follows by exchanging ˆ A and its complement.Since u ∈ V and the constant functions belong to V , u ± ε ∈ V . In particular, letting ε →
0, wededuce that u ∈ ˆ A ∩ V ∩ V \ ˆ A = ∂ ˆ A V , from which it follows that ∂ ˆ A V = ∂ ˆ A ∩ V as announced.(2). Let f x denote the field f conditioned on f ( x ) = 0 and on x being a stratified critical pointof f . Then, f x is a.s. C . Assume now that f x ∈ Piv + x ( ˆ A ). Then, there exists a (random) h ∈ C ( M ) satisfying h ≥ δ > f x + δh ∈ ˆ A and f x − δh / ∈ ˆ A . In particular, f x ∈ ∂ ˆ A . Moreover, since f satisfies Condition 2.7, by the regressionformula ( f x ( y ) , d y f x ) is non-degenerate for y (cid:54) = x , and so by Bulinskaya’s lemma ([3, Proposition1.20]) a.s. f x has no other stratified critical points at level 0. If we also assume that H Fx f x isnon-degenerate, then by Remark 5.6 f x ∈ (cid:103) Piv + x ( ˆ A ). Thus, we have shown that if f x ∈ Piv + x ( ˆ A )and H Fx f x is non-degenerate then a.s. f x ∈ (cid:103) Piv + x ( ˆ A ).Conversely, assume that f x ∈ (cid:103) Piv + x ( ˆ A ) and let U ⊂ M be a neighbourhood of x in M . Then, H Fx f x is non-degenerate. Since f satisfies Condition 2.7, as before by Bulinskaya’s lemma a.s. f x has no other critical points at level 0. We may thus apply Lemma 5.4 to f x , which impliesthat there exists a neighbourhood U of f x in C ( M ), two connected components C and C of C ( M ) \ D B ( C ( M )), and a geodesic ball B r ⊂ U of radius r > x , such that thefollowing holds: • For all small enough δ > f x + δ ∈ C and f x − δ ∈ C . • The union C ∪ C covers U . • Each v ∈ U ∩ D B ( C ( M )) has a unique stratified critical point in B r and no stratifiedcritical points at level 0 outside of B r .Since f x ∈ (cid:103) Piv + x ( ˆ A ), we have C ⊂ ˆ A and C ⊂ ˆ A c . Let h ∈ C c ( W ) be equal to 1 on B r . Then,for all small enough δ > f x ± h ∈ U , so f x has no stratified critical points at level 0 outside of B r . Inside B r it coincides with f x up to a constant ± δ . In particular, if δ (cid:54) = 0, f x ± h ∈ C ∪ C .Moreover, by considering the path ( f x ± ( δh + s (1 − h ))) s ∈ [0 ,δ ] , we conclude that f x + h ∈ C ⊂ ˆ A and f x − h ∈ C ⊂ ˆ A c . But h is supported arbitrarily close to x . Thus, f x ∈ Piv + x ( ˆ A ). Reasoningsymmetrically, we get the same statement with the + exponent replaced by − , and combiningthe two results we get the same property for Piv σx ( ˆ A ) replaced by Piv x ( ˆ A ).(3). Assume that x is a non-degenerate critical point of f x . Then, Lemma 5.4 applies so thatthere are two discriminant classes C and C such that C ∪ C is a neighbourhood of f x in C ( M )and there is a neighbourhood W of f x in the discriminant such that for each v ∈ W , and eachsmall enough ε > v + ε ∈ C and v − ε ∈ C . If f x ∈ (cid:103) Piv x ( ˆ A ) then exactly one of the two classesbelongs to ˆ A , and hence the elements of W will all belong to the boundary of ˆ A . Therefore, f x belongs to the interior of (cid:103) Piv x ( ˆ A ) in the space V (cid:48) x of functions in C ( M ) with a stratified criticalpoint at x at level 0. Similarly, if f x / ∈ (cid:103) Piv x ( ˆ A ) then either both C and C are subsets of ˆ A orneither of them are. So then, as before, the elements of W cannot belong to the boundary of COVARIANCE FORMULA FOR TOPOLOGICAL EVENTS OF SMOOTH GAUSSIAN FIELDS 31 ˆ A so that f x is in the interior of V (cid:48) x \ (cid:103) Piv x ( ˆ A ). In both cases, f x / ∈ ∂ (cid:103) Piv x ( ˆ A ), which proves thelast part of the proposition. (cid:3) The topological class is encoded by the discriminant.
In this subsection we verifythat topological events are encoded by the discriminant, making explicit the link between theevents that appear in Theorem 2.14 and the events that appear in Proposition 3.9; in passing,we also prove the measurability of the stratified isotopy classes.In this section ( B, F ) again denotes an arbitrary stratified set of M ; nevertheless, here weprefer to view F as a general Whitney stratification (see Remark 2.2) since we make use of thestandard theory of Whitney stratifications. Recall the definition of B -discriminant classes fromDefinition 3.3, as well as the definition of the stratified isotopy class from Definition 2.8. Lemma 5.7 (Topological class is encoded by the discriminant) . Suppose that u, v ∈ C ( M ) have the same B -discriminant class in C ( M ) . Then their excursion sets { u > } and { v > } have the same stratified isotopy class, i.e., [ { u > } ] B = [ { v > } ] B . Since the discriminant classes are C -open (see Lemma C.1), there are at most countablymany of them. This immediately implies the following: Corollary 5.8.
There are at most countably many stratified isotopy classes of subsets of B .Moreover, the map [ D ] B from the probability space Ω into the set of stratified isotopy classes ismeasurable. Before proving Lemma 5.7, let us recall some standard facts about Whitney stratifications;they can all be easily checked from the definitions of the objects they involve: • If I ⊂ R is an open interval, then the collection F I = ( F × I ) F ∈F is a Whitney stratifi-cation of B × I . • For each open subset W ⊂ M , F W = ( F ∩ W ) F ∈F is a Whitney stratification of B ∩ W . • Consider f : M → N a smooth map between two Riemannian manifolds. Assume that f | B is proper and that for each F ∈ F , f | F : F → N is a submersion. Then, for each y ∈ N , the preimage f − ( y ) ∩ B is naturally equipped with a Whitney stratification F y whose strata are the intersections F ∩ f − ( y ) where F ∈ F (see Definition 1.3.1 of PartI of [23]).The proof of Lemma 5.7 is a standard application of Thom’s first isotopy lemma (see (8.1) of[34]) and the isotopy extension theorem (see [18]). In fact, the only place we use C regularityin this proof is when we apply Thom’s first isotopy lemma. Proof of Lemma 5.7.
Let u ∈ C ( M ) \ D B ( C ( M )), i.e. u has no stratified critical points in B at level 0. Recall that, by Lemma C.1 and since B is compact, the set of critical points of u iscompact. In particular, this set is at positive distance from the zero set of u and there exists abounded open neighbourhood W ⊂ M of u − (0) in M and a convex neighbourhood U of u in C ( M ) such that for each v ∈ U and each face F ∈ F , dv | F (cid:54) = 0 in W and v (cid:54) = 0 on B \ W .We will prove that for each v ∈ U ∩ C ( M ), [ { v > } ] B = [ { u > } ] B . To do so, notice thatsince U is open and convex, there exists I an open interval containing [0 ,
1] such that for each t ∈ I , u t = tv + (1 − t ) u ∈ U ∩ C ( M ). The family F W,I = (( F ∩ W ) × I ) F ∈F defines a Whitneystratification of ( B ∩ W ) × I in W × I . Moreover, since for each t ∈ I , u t has no critical pointson any face of F inside W , the map( W ∩ B ) × I → R × I , U : ( x, t ) (cid:55)→ ( u t ( x ) , t )is a submersion when restricted to any face of F W,I . It is proper since B is compact and id I : I → I is proper. In particular, by Thom’s first isotopy lemma, since U is C there exists astratified homeomorphism h : W × I → ( W × I ) ∩ U − (0) × R × I (where U − (0) ∩ B is equippedwith the preimage Whitney stratification that exists since F is transverse to { } in R × I ) such that U ◦ h − is the projection on the last two factors. Note that U − (0) = { ( x, t ) ∈ M × I :( u t ( x ) , t ) = (0 , } = u − (0) × { } . In particular, the map( B ∩ u − (0)) × I → B × I , ( z, t ) (cid:55)→ ( f t ( x ) , t ) := h − (( x, , , t )defines an isotopy of u − (0) in B such that for each t ∈ I , f t ( u − (0) ∩ B ) = u − t (0) ∩ B . Sinceit is constructed from h it extends to an isotopy of a tubular neighbourhood of u − (0) in B that preserves strata of F . By Corollary 1.4 of [18] (and its extension provided in Section 7of the same article), there exists a continuous isotopy B × I → B × I ( x, t ) → (Φ t ( x ) , t ) suchthat for each t ∈ I , Φ t is a stratified homeomorphism of B and Φ t ◦ f = f t . In particular,Φ t ( u − (0) ∩ B ) = u − t (0) ∩ B for each t ∈ I . Since Φ = id , and Φ t is continuous in t , we alsohave Φ ( { u > } ∩ B ) = { u > } ∩ B and so [ { u > } ] B = [ { v > } ] B . Given that this istrue for all v ∈ U ∩ C ( M ), we have shown that equivalence classes for the equivalence relationgenerated by the map u (cid:55)→ [ { u > } ] B are C ( M )-open. In particular, since D B ( C ( M )) is C -closed (by Lemma C.1 or just C -closed by the present argument) each topological class in C ( M ) \ D B ( C ( M )) must be a union of connected components of C ( M ) \ D B ( C ( M )) andthe proof is over. (cid:3) Approximation results.
To deduce Theorem 2.14 from Proposition 3.9, we approximatethe field f by a sequence of fields ( f k ) k ∈ N taking values in finite-dimensional subspaces ( V k ) k ∈ N of C ( M ). Then, we integrate the result of Proposition 3.9 and pass to the limit.In this subsection, we first show the existence of an approximating sequence in a generalsetting (see Lemma 5.9), and then state the abstract continuity lemma for expectations (seeLemma 5.10) which we use to show the convergence of the terms in Proposition 3.9. Lemma 5.9 (Existence of finite-dimensional approximations) . Fix l ∈ N and let f be an a.s. C l Gaussian field on a smooth manifold M of dimension d . Let V ⊂ C l ( M ) be a linear subspaceof C l ( M ) such that f belongs a.s. to V . Then the following holds: (1) There exists a sequence ( V k ) k ∈ N of finite-dimensional linear subspaces of V and a se-quence of Gaussian fields ( f k ) k ∈ N , all defined in the same probability space as f , thatconverges in probability to f in the topology of uniform C l convergence on compact sub-sets of M , and such that for each k ∈ N , f k ∈ V k a.s. and f k defines a non-degenerateGaussian vector in V k . If f is centred then the f k can also be chosen to be centred. (2) Moreover, let W ⊂ V be a finite-dimensional subspace. Then, we may find sequences ( V k ) k ∈ N and ( f k ) k ∈ N as in (1) such that W ⊂ V k .Proof. Consider a countable atlas ( U j , φ j ) j ∈ N of M . Let J ⊂ N , let η > I ⊂ M be a locally finite set such that, for each j ∈ J and z ∈ φ j ( U j ), thereexists x ∈ φ j ( I ∩ U j ) for which | z − x | ≤ η . Let ε >
0, fix j ∈ J and let B ⊂ U j be a compactsubset. Let us prove that there exists η = η ( j, B, ε ) > η ≤ η , the field f I := E [ f | f I ] satisfies(5.1) P (cid:104) (cid:107) f ◦ φ − j − f I ◦ φ − j (cid:107) C l ( B ) > ε (cid:105) < ε . Since the seminorms (cid:107) · (cid:107) C l ( B ) , where j ∈ N and B ⊂ U j ranges over the compact subsets of U j ,generate the topology of C l ( M ), repeating the above construction for a sequence ( ε k ) k ∈ N → f k ) k ∈ N of fields satisfying (5.1) for ε = ε k which proves the first point of thelemma.To prove (5.1), fix α ∈ N d and let g = ∂ α ( f ◦ φ − j ) and g I = ∂ α ( f I ◦ φ − j ). Also, let N = N ( d, l ) ∈ N be the number of multi-indices α ∈ N d such that | α | ≤ l . Observe that, foreach z ∈ φ j ( B ), there exists z ∈ φ j ( U j ∩ I ) such that | x − z | < η so that | g ( z ) − g I ( z ) | ≤ | g ( z ) − g ( x ) | + E [ | g ( x ) − E [ g ( z ) | f | I ] | ] ≤ | g ( z ) − g ( x ) | + E [ | g ( z ) − g ( x ) | ] . COVARIANCE FORMULA FOR TOPOLOGICAL EVENTS OF SMOOTH GAUSSIAN FIELDS 33
In particular, E (cid:2) sup z ∈ B | g ( z ) − g I ( z ) | (cid:3) ≤ E (cid:104) sup x,y ∈ B, | x − y |≤ η | g ( x ) − g ( y ) | (cid:105) =: 2 E [ X η ] . Now, since g is a continuous Gaussian field on B (which is compact), it is a.s. bounded andthe family ( X η ) η> is uniformly L (see for instance [3, Theorem 2.9]). Moreover, since g iscontinuous, X η converges a.s. to 0 as η →
0. In particular, lim η → E [ X η ] = 0. Thus, there exists η α = η α ( ε, d, l ) > η ≤ η α , P [sup z ∈ B | g ( z ) − g I ( z ) | > ε ] ≤ ε/C . The estimate(5.1) follows by taking a union bound of the probability of the events sup z ∈ B | g ( z ) − g I ( z ) | where α ∈ N d ranges over all the multiindices such that | α | ≤ l , and will be valid for η ≤ η =min | α |≤ l η α .Consider now a sequence ( η k ) k ∈ N of positive real numbers converging to 0 and ( I k ) k ∈ N anincreasing sequence of finite subsets of M , such that for each k ∈ N and j ≤ k , φ j ( I k ∩ U j ) isan η k -net of φ j ( U j ). For each k ∈ N , let f k = f I k defined as above (with J = [0 , k ]). Then, foreach j ∈ N and each compact subset B ⊂ , by (5.1), lim k →∞ (cid:107) f ◦ φ − j − f k ◦ φ − j (cid:107) C l ( B ) = 0 inprobability so ( f k ) k ∈ N converges to f in probability.We claim that each f k belongs to a finite-dimensional subspace V k . Indeed, let K be thecovariance of f . By the regression formula (Proposition 1.2 of [3]), for each k ∈ N , f k is arandom linear combination of the functions K ( · , x ) for x ∈ I k and of E [ f ] the mean of f . Henceit belongs to the finite-dimensional subspace V k generated by these functions. Moreover, since f k is the mean of a random variable with values in V , we have V k ⊂ V and if f is centred, byconstruction, f k is centred. This concludes the proof of the first statement.For the second statement, take ( f k ) k ∈ N as above, let ( h , . . . , h m ) be a basis of W , and let ξ , . . . , ξ m be independent standard normals. Then, clearly, k ( ξ h + · · · + ξ m h m ) convergesto 0 in probability in C l ( M ), so that replacing f k by f k + 1 k ( ξ h + · · · + ξ m h m )yields the required result. (cid:3) Next we state without proof an abstract continuity lemma for expectations; this can beconsidered a simple variant of the standard Portemanteau lemma.
Lemma 5.10.
Let ( X, Y ) and ( X k , Y k ) k ∈ N be random variables with values in R × E , where E isa Polish space. Assume that the sequence ( X k , Y k ) k ∈ N converges in law towards ( X, Y ) , and thatthe sequence ( X k ) k ∈ N is uniformly integrable. Let A ⊂ E and assume P [ X (cid:54) = 0 , Y ∈ ∂A ] = 0 .Then lim k →∞ E (cid:2) X k [ Y k ∈ A ] (cid:3) = E (cid:2) X [ Y ∈ A ] (cid:3) . Completing the proof of Theorem 2.14.
To complete the proof of Theorem 2.14 weassemble the previous elements together, namely we: • Approximate f by a sequence of finite-dimensional fields ( f k ) k ∈ N with nice regularityand non-degeneracy properties constructed using Lemma 5.9. • Use Lemma 5.7 to encode the topological events A and A via the discriminant. • Apply Proposition 3.9 to the fields f k and the events encoded by the discriminant. • Pass to the limit in each term of the formula given by Proposition 3.9, using Lemmas5.1 and 5.10. • Show that the two definitions of pivotal events coincide using Lemma 5.1.
Proof of Theorem 2.14.
Recall that ( B , F ) and ( B , F ) are stratified sets of M , and A and A are topological events on B and B respectively. By Lemma 5.7, for each i ∈ { , } thereexists ˆ A i ∈ ˆ σ discr ( B i , C ( M )) such that P [ f ∈ ˆ A i (cid:52) A i ] = 0, and so it will be sufficient to workwith the events ˆ A i ∈ ˆ σ discr ( B i , C ( M )). Let us first define the approximating sequence of fields. By Remark 3.8, there exists a finite-dimensional subspace W ⊂ C ( M ) satisfying Conditions 3.6 and 3.7 that contains the constantfunctions. Hence we may define a sequence of Gaussian fields ( f k ) k ∈ N , taking values in a se-quence of finite-dimensional linear subspaces W ⊂ V k ⊂ C ( M ), that satisfy all the propertiesguaranteed by Lemma 5.9 (setting (cid:96) = 2, and so in particular the f k converge in probability inthe topology of uniform C on compact sets). Since W satisfies Conditions 3.6 and 3.7 so doeseach V k , and so Proposition 3.9 applies to the sets ˆ A i,k = ˆ A i ∩ V for i ∈ { , } and the field f k .Next, recall that f and f denote independent copies of f , and let ( f k ) k ∈ N and ( f k ) k ∈ N be independent copies of ( f k ) k ∈ N , with f k converging to f and f k converging to f (i.e. in C ). Similarly, recall that f t denotes the interpolation ( f t , f t ) = ( f , t ( f − µ ) + √ − t ( f − µ ) + µ ), and define for each k ∈ N the interpolation f t,k = ( f t,k , f t,k ) analogously. ApplyingProposition 3.9 we have, for each k ∈ N and t ∈ [0 , P (cid:104) f t,k ∈ ˆ A ,k × ˆ A ,k (cid:105) − P (cid:104) f ,k ∈ ˆ A ,k × ˆ A ,k (cid:105) (5.2) = (cid:88) F ∈F , F ∈F (cid:90) t (cid:90) F × F K k ( x , x ) × Λ k ( s ; x , x ) γ s,k ; x ,x (0) dv F ( x )dv F ( x )d s, where Λ k ( s ; x , x ) equals(5.3) E s ; x ,x (cid:104) σ ( f s,k , f s,k ) (cid:103) Piv x ( ˆ A ,k ) × (cid:103) Piv x ( ˆ A ,k ) ( f s,k , f s,k ) | det (cid:0) H x f s,k | F (cid:1) || det (cid:0) H x f s,k | F (cid:1) | (cid:105) , and where K k is the covariance of f k , and γ s,k ; x ,x is the density of( f s,k ( x ) , d x f s,k | F , f s,k ( x ) , d x f s,k | F )in orthonormal coordinates (recall that the subscript s ; x , x in the expectation denotes condi-tioning on this vector vanishing).Let us compute the limits of both sides of (5.2) as k → ∞ , beginning with the left-hand side.Notice that for each t ∈ [0 ,
1) and k ∈ N , f tk ∈ ˆ A ,k × ˆ A ,k if and only if f tk ∈ ˆ A × ˆ A . Since ∂ ( ˆ A × ˆ A ) ⊂ D B ( C ( M )) × C ( M ) ∪ C ( M ) × D B ( C ( M )), and since f satisfies Condition 2.7,by Bulinskaya’s lemma (Proposition 1.20 of [3]) P [( f t , f t ) ∈ ∂ ( ˆ A × ˆ A )] = 0 . Thus, by Lemma 5.10 (setting X = 1, Y = ( f t , f t ), E = C ( M ) × C ( M ) and A = ˆ A × ˆ A ),we have(5.4) lim k →∞ P [ f t,k ∈ ˆ A ,k × ˆ A ,k ] = P [ f t ∈ ˆ A × ˆ A ] . We turn now to the right-hand side of (5.2); we begin by computing the pointwise limitof the integrand, and then apply the dominated convergence theorem. Fix F ∈ F , F ∈F , x ∈ F , x ∈ F and s ∈ [0 , t ] (so that s < F ) , dim( F ) > x (cid:54) = x by removing a set of measure zero from the integralin ( x , x ). Now, since f sk converges in probability to f s as k → ∞ , it also converges in law.In particular K k ( x , x ) converges to K ( x , x ) and γ s,k ; x ,x (0) converges to γ s ; x ,x (0). Todeal with Λ k ( s ; x , x ), we note that f s,k converges in law to f s in C ( M ) as k → ∞ so that K k converges to K in C ( M × M ). Now, since the vector ( f s ( x ) , d x f s | F , f s ( x ) , d x f s | F )is non-degenerate, by the regression formula (Proposition 1.2 of [3]), the law of f conditionedon this vector vanishing is well-defined and depends continuously in K . Since the covarianceof a field determines its law, we deduce that the sequence of fields ( f s,k ) k ∈ N conditioned on COVARIANCE FORMULA FOR TOPOLOGICAL EVENTS OF SMOOTH GAUSSIAN FIELDS 35 ( f s,k ( x ) , d x f s,k | F , f s,k ( x ) , d x f s,k | F ) = 0 converges in law to f with the above conditioning.We denote the conditional law of these fields by P s ; x ,x [ . . . ]. By the first statement of Lemma 5.1 P s ; x ,x (cid:104) f t,k ∈ (cid:103) Piv x ( ˆ A ,k ) × (cid:103) Piv x ( ˆ A ,k ) (cid:52) (cid:103) Piv x ( ˆ A ) × (cid:103) Piv x ( ˆ A ) (cid:105) = 0so, if we temporarily set A = (cid:103) Piv x ( ˆ A ) × (cid:103) Piv x ( ˆ A ) , X s = X s ( x , x ) = σ ( f s ) | det (cid:0) H F x f s (cid:1) || det (cid:0) H F x f s (cid:1) | and X s,k = X s,k ( x , x ) = σ ( f s,k ) | det (cid:0) H F x f s,k (cid:1) || det (cid:0) H F x f s,k (cid:1) | , we have Λ k ( s ; x , x ) = E s ; x ,x [ A ( f s,k ) X s,k ( x , x )] . Since under the conditioning f k converges in law to f , and since these are Gaussian fields, thesequence ( X s,k ( x , x )) k ∈ N is uniformly integrable (though the bound may depend on s , x and x ). On the other hand, the random variables f s,k and f s take values in the Polish space C ( M ).By the third point of Lemma 5.1, a.s. either the Hessian of one of the f is ’s is degenerate, whichimplies that either X s = 0 or f t / ∈ ∂A . Moreover, the pair ( X s,k , f s,k ) converges in probabilityto the pair ( X s , f s ). By Lemma 5.10, we have lim k →∞ Λ k ( s ; x , x ) = Λ( s ; x , x ) which is equalto(5.5) E s ; x ,x (cid:104) σ ( f s , f s ) (cid:103) Piv x ( ˆ A ) × (cid:103) Piv x ( ˆ A ) ( f s , f s ) | det (cid:0) H F x f s (cid:1) || det (cid:0) H F x f s (cid:1) | (cid:105) . In summary, the integrand of the right hand side of (5.2) converges pointwise to the samequantity with f s,k replaced by f s everywhere.To apply the dominated convergence theorem to the right hand side of (5.2), we must find auniform L bound on the integrand. To bound Λ k ( s ; x , x ) × γ s,k ; x ,x (0) we use Lemma A.4with X is,k = H x i f is,k , Y s,k = Y s,k ( x , x ) = ( f s,k ( x ) , d x f s,k | T x F , f s,k ( x ) , d x f s,k | T x F ) for each i ∈ { , } and k ∈ N . For any finite-dimensional Gaussian vector X in a space equipped with ascalar product, let DC( X ) be the determinant of the covariance of X in orthonormal coordinates.The covariances of the coordinates of the X is,k are bounded in terms of derivatives up to ordertwo in each variable of the covariances K k ; since these are uniformly bounded, there exists aconstant C < ∞ for which, for each k ∈ N , | Λ k ( s ; x , x ) γ s,k ; x ,x (0) | ≤ C (cid:112) DC( Y s,k ) . Next, for i, j ∈ { , } let Y ijk ( x j ) = ( f ik ( x j ) , d x j f ik | T xj F j ) so that, for any j , j ∈ { , } , Y j k isindependent from Y j k and Y s,k = ( Y k , s ( Y k − E (cid:2) Y k (cid:3) ) + (cid:112) − s ( Y k − E (cid:2) Y k (cid:3) + E (cid:2) Y k (cid:3) )) . Then, by Lemma A.3, for each s ∈ [0 , t ],DC( Y s,k ) = DC( Y k , sY k + (cid:112) − s Y k ) ≥ DC( Y k , sY k ) + DC( Y k )DC( (cid:112) − s Y k ) ≥ (1 − s ) dim( F )+1 DC( Y k )DC( Y k )= (1 − t ) n +1 DC( f k ( x ) , d x f k | T x F )DC( f k ( x ) , d x f k | T x F ) . Since ( f k ) k ∈ N converges in law to f , and since ( f ( x ) , d x f ) is non-degenerate for each x ∈ M (and since B is compact), there exist k ∈ N and a constant c > t ∈ [0 , s ∈ [0 , t ], x ∈ F and x ∈ F , as long as k ≥ k ,DC( Y s,k ) ≥ (1 − t ) n +1 c. In particular | Λ k ( s ; x , x ) γ s,k ; x ,x (0) | ≤ Cc − / (1 − t ) − ( n +1) / . Hence the integrand in theright hand side of (5.2) is uniformly integrable, so the dominated convergence theorem applies.All in all, letting k → ∞ in both sides of (5.2) yields P (cid:2) f t ∈ ˆ A × ˆ A ] − P (cid:2) f ∈ ˆ A × ˆ A ](5.6) = (cid:88) F ∈F , F ∈F (cid:90) t (cid:90) F × F K ( x , x ) × Λ( s ; x , x ) γ s ; x ,x (0)dv F ( x ) dv F ( x )d s, where Λ is defined in (5.5).Let us now finish off the proof. Fix F ∈ F and F ∈ F . By the second point of Lemma 5.1,for each i ∈ { , } , each s ∈ [0 , t ] and each x i ∈ F i , a.s. (under the conditioning present in Λ)det( H F i x i f is ) (cid:103) Piv xi ( ˆ A i ) ( f is ) = det( H F i x i f is ) Piv xi ( ˆ A i ) ( f is )since the only place at which the two pivotal events do not coincide is where the determinant ofthe Hessian vanishes. Moreover, Piv x i ( ˆ A i ) splits as the disjoint union of Piv + x i ( ˆ A i ) and Piv − x i ( ˆ A i ).All in all, a.s. under the conditioning used in Λ, σ ( f s , f s ) Piv x ( ˆ A ) × Piv x ( ˆ A ) ( f s , f s )= (cid:16) Piv + x ( ˆ A ) × Piv + x ( ˆ A ) ( f s , f s ) + Piv − x ( ˆ A ) × Piv − x ( ˆ A ) ( f s , f s ) (cid:17) − (cid:16) Piv + x ( ˆ A ) × Piv − x ( ˆ A ) ( f s , f s ) + Piv − x ( ˆ A ) × Piv + x ( ˆ A ) ( f s , f s ) (cid:17) . In particular, Λ( s ; x , x ) γ s ; x ,x (0) = I + s ( x , x ) − I − s ( x , x ) , where I ± s are the signed pivotal intensity functions from Definition 2.11. Thus, (5.6) yields P [ f ∈ ˆ A × ˆ A ] − P [ f ∈ ˆ A × ˆ A ]= (cid:88) F ∈F , F ∈F (cid:90) F × F K ( x , x ) × (cid:90) t (cid:0) I + s ( x , x ) − I − s ( x , x ) (cid:1) d s dv F ( x )dv F ( x ) . By the definition of ˆ A and ˆ A as well as ( f t ) t ∈ [0 , , P [ f ∈ ˆ A × ˆ A ] − P [ f ∈ ˆ A × ˆ A ] = P [ A ∩ A ] − P [ A ] P [ A ] , so letting t → P [ A ∩ A ] − P [ A ] P [ A ] = (cid:90) B × B K ( x , x ) (cid:0) d π + ( x , x ) − d π − ( x , x ) (cid:1) , where dπ σ ( x , x ), for σ ∈ { + , −} , are the signed pivotal measures from Definition 2.11. (cid:3) To complete the section, we verify that Corollary 1.1 is indeed a special case of Theorem 2.14:
Proof of Corollary 1.1.
Recall that B and B are closed boxes, F and F are their interiors,and F ij , for j ∈ { , , , } and i ∈ { , } , are their four sides. Together with the corners of theboxes, which we denote F ij for j ∈ { , , , } and i ∈ { , } , the set of F ij form a tame (andaffine) stratification of B . Moreover, the events A and A are indeed topological events sincestratified isotopies preserve crossings, and so Theorem 2.14 applies to these events, yielding anexact formula for Cov[ A , A ].Let us next analyse the terms in this formula. By Remark 2.12, the corners do not contributeto the sum over strata. Further, by Remark 2.13, since A i are both increasing events, the setsPiv − x i ( A i ) are empty and so the pivotal measure only contains positively pivotal events. Finallynotice that, for each i = 1 , t ∈ (0 ,
1) and ( x , x ) ∈ ( B , B ), under the conditioning that x and x are stratified critical points at level 0 of f t and f t respectively, the fields f t and f t COVARIANCE FORMULA FOR TOPOLOGICAL EVENTS OF SMOOTH GAUSSIAN FIELDS 37 have a.s. no other critical points at level 0 (by the non-degeneracy assumption). Moreover, ifthe Hessians of f it at x i do not degenerate, the x i are non-degenerate stratified critical pointsof f it . The pivotal event for A i is then equivalent to the existence of a path in { f it ≥ } joining‘left’ to ‘right’ and a path in { f it ≤ } joining ‘top’ to ‘bottom’, both passing through x i . (cid:3) Proofs of the applications
In this section we give proofs for the applications that are discussed in Sections 1 and 2, inparticular Theorem 2.15 and Corollaries 1.2, 1.6, 2.17, and 2.19.6.1.
Strong mixing for topological events.
Proof of Theorem 2.15.
Let c denote a constant, that can change line-to-line, that depends onlyon d . By Theorem 2.14 and the definition of α top , after replacing K with its absolute value,and dropping the condition that f t and f t lie in the pivotal sets, it suffices to show that, for all t ∈ [0 , γ t ( x , x ) E t ; x ,x (cid:2) | det( H F x f t ) det( H F x f t ) | (cid:3) is bounded above by the maximum, over i, j, k ∈ { , } , of c E (cid:2) (cid:107) H F i x i f (cid:107) | d x i f | F i = 0 (cid:3) d i (cid:112) det(∆( x , x )) max (cid:110) , (cid:16) K ( x j , x j ) det( d x k ⊗ d x k K | F i × F i ) (cid:112) det(∆( x , x )) (cid:17) d i (cid:111) . Let ∆ t ( x , x ) denote the covariance matrix of (2.2) and let Ω t ( x , x ) denote the covariancematrix of ( d x f t | F , d x f t | F ). Applying Lemma A.4 to the matrices X = H F x f t and X = H F x f t and the vectors Y = ( f t ( x ) , f t ( x )) and Z = ( d x f t | F , d x f t | F ), (6.1) is boundedabove by the maximum, over i, j, k ∈ { , } , of c E (cid:2) (cid:107) H F i x i f it (cid:107) | d x i f it | F i = 0 (cid:3) d i (cid:112) det(∆ t ( x , x )) max (cid:110) , (cid:16) Var[ f jt ( x j )] (cid:112) det(Ω t ( x , x )) (cid:112) det(∆ t ( x , x )) (cid:17) d i (cid:111) . Recall that f it is equal in law to f , and that, by Lemma A.5,det(∆ t ( x , x )) ≥ det(∆ ( x , x )) = det(∆( x , x ))and det(Ω t ( x , x )) ≤ det(Ω ( x , x )) = Π k =1 , DC[ d x k f | F k ] ≤ max k =1 , DC[ d x k f | F k ] , where DC( X ) is the determinant of the covariance of X in orthonormal coordinates. SinceVar[ f ( x j )] = K ( x j , x j ) and DC[ d x k f | F k ] = det( d x k ⊗ d x k K | F i × F i ) , we have the desired result. (cid:3) Proof of Corollary 1.2.
The non-degeneracy condition in the statement of Corollary 1.2 is equiv-alent to Condition 2.7, and so we are in the setting of Theorem 2.15. First we argue that thereexists constants c , c >
0, depending only on d , κ (0), and the Hessian of κ at 0, such that, ifmax α ∈ N d : | α |≤ sup x ∈ B ,x ∈ B | ∂ α κ ( x − x ) | < c , then for any affine sets F and F the covariance matrix of the Gaussian vector(6.2) ( f ( x ) , ∇ f | F ( x ) , f ( x ) , ∇ f | F ( x ))has a determinant bounded below by c . Let Σ ( x , x ) denote this matrix, and observe thatΣ ( x , x ) = (cid:20) M M M T M (cid:21) where, by stationarity, M ii = (cid:20) κ (0) 00 H F i κ (cid:21) , and M depends only on the value and second derivatives of κ at x and x (here H F denotesthe Hessian at the point x in an orthonormal basis of the linear span of F ). The result thenfollows by the continuity, on the set of strictly positive-definite matrices, of the determinantwith respect to the entry-wise sup-norm.Combined with the stationarity of f , under the assumption thatmax α ∈ N d : | α |≤ sup x ∈ B ,x ∈ B | ∂ α κ ( x − x ) | < c , the quantity c F ,F ( x , x ) in Theorem 2.15 can be bound above by c max i (cid:26) E (cid:2) (cid:107) H F i f (cid:107) | ∇ f | F i (0) = 0 (cid:3) dim( F i ) (cid:27) , for some c >
0. Since this is a finite quantity, we have proved the result.To verify the observation in Remark 1.3 note that we have already established that c dependsonly on d , κ (0), and the Hessian of κ at 0. Next, since all norms on R d are equivalent, E (cid:2) (cid:107) H F i f (cid:107) | ∇ f | F i (0) = 0 (cid:3) ≤ c d max j ,j E (cid:2) ( H F ,F f ) j ,j | ∇ f | F i (0) = 0 (cid:3) , where c d is a constant depending only on the dimension d . By stationarity, and since conditioningon part of a Gaussian vector reduces the variance of all coordinates, this is at most c d max j ,j E (cid:2) ( H F ,F f ) j ,j (cid:3) ≤ c d max j ,j ∂ κ (0) ∂x j ∂x j . Finally, applying the Cauchy–Schwartz inequality in Fourier space,max j ,j ∂ κ (0) ∂x j ∂x j ≤ max j ∂ κ (0) ∂x j and we have the result. (cid:3) Proof of Corollary 2.17.
Observe that each of f n | B ∪ B satisfies Condition 2.7, since B ∪ B does not include antipodal points. Note also that a condition analogous to (1.4) holds; moreprecisely, as n → ∞ ,sup x ∈ B ,x ∈ B (cid:107) ( K n ( x , x ) , d x K n ( x , x ) , d x K n ( x , x ) , d x ⊗ d x K n ( x , x )) (cid:107) ∞ → , which, as in the proof of Theorem 2.15, implies that, as n → ∞ ,sup x ∈ F ,x ∈ F (cid:12)(cid:12)(cid:12) det(∆ ( x , x ))det(∆ ( x , x )) − (cid:12)(cid:12)(cid:12) → , where ∆ t ( x , x ) is the covariance matrix, for the field f n , that is defined in the proof of Theo-rem 2.15 (note that we have omitted the dependence on n in the notation). Observe also thatthe scale s n = 1 / √ n at which the Kostlan ensemble converges to a local limit is a polynomial,and so all derivatives of K n on the diagonal ( x, x ) grow at most polynomially, uniformly over x .Along with the discussion in Remark 2.16, we deduce thatsup F ∈F ,F ∈F sup x ∈ F ,x ∈ F c F ,F ( x , x )grows at most polynomially as a function of n , where c F ,F ( x , x ) is the constant appearing inTheorem 2.15 applied to f n (again we omit the dependence on n in the notation). Since on theother hand sup x ∈ B ,x ∈ B | K n ( x , x ) | COVARIANCE FORMULA FOR TOPOLOGICAL EVENTS OF SMOOTH GAUSSIAN FIELDS 39 decays exponentially in n (recall that B and B are contained within an open hemisphere), wededuce the result from Theorem 2.15. (cid:3) Lower concentration for topological counts.
Our proof of the lower concentrationresults in Corollaries 1.6 and 2.19 essentially follows the proof of [44, Theorem 1.4]. We makeuse of the following simple lemma:
Lemma 6.1.
Let B i ⊂ M be a sequence of disjoint stratified sets, and let A i ∈ σ top ( B i ) betopological events such that sup i P [ A i ] < h < . Then | P [ ∩ i A i ] − Π i P [ A i ] | ≤ − h sup n ∈ N α top (cid:0) B n , ∪ j>n B j (cid:1) . Proof.
By the definition of α top | P ( A n ∩ ( ∩ j>n A j )) − P ( A n ) P ( ∩ j>n A j ) | ≤ α top (cid:0) B n , ∪ j>n B j (cid:1) . Iterating this inequality for n = 1 , , . . . and using that P ( A n ) < h we get the upper bound(1 + h + h + . . . ) × sup n ∈ N α top (cid:0) B n , ∪ j>n B j (cid:1) , which is equal to the desired upped bound. (cid:3) Proof of Corollary 1.6.
First observe that we may assume the infimum in (1.8) is eventuallyattained in the set r ∈ [ g s , s/g s ] for some function g s → ∞ as s → ∞ , since otherwise theright-hand side of (1.8) is bounded from below, and we may then choose c > ε >
0, a function g s ∈ (0 , √ s ) such that g s → ∞ , and a mesoscopic parameter g s 2. If this holds, then we can also find at least ν ( s/r ) d mesoscopic cubes M i with this property that are separated by a distance r , where ν = c d × ε Vol( B )2 c N − ε , and c d > h ∈ (0 , P [ N ( M i ) /r d ≤ c N − ε/ < h eventually as s → ∞ . Hence, by Lemma 6.1, for every choice of ν ( s/r ) d mesoscopic cubes M i which are r -separated, the probability that N ( M i ) /r d ≤ c N − ε/ h ν ( s/r ) d + 11 − h α r,s , where α r,s = max n α top ( M n , ∪ i>n M i ). There are at most 2 (Vol( B )+ o (1))( s/r ) d ways to choose cubes M i , hence by the union bound P (cid:2) N ( sB ) /s d ≤ ( c N − ε )Vol( B ) (cid:3) ≤ (Vol( B )+ o (1))( s/r ) d (cid:16) h ν ( s/r ) d + 11 − h α r,s (cid:17) . Let F and F be the standard stratifications of M n and ∪ i>n M i . Clearly, there is a constant c > |F | ≤ c and |F | ≤ cν ( s/r ) d . In bothstratifications the strata with the largest volume are interiors of mesoscopic cubes that havevolume r d . By Corollary 1.2 this implies that there is a constant c > α r,s ≤ c r d s d ¯ κ ( r ) . Combining these estimates, taking c B = 2 log 2Vol( B ) and choosing h small enough, we see thatfor every C > P (cid:2) N ( sB ) /s d ≤ ( c N − ε )Vol( B ) (cid:3) ≤ c (cid:0) e − C ( s/r ) d + e c B ( s/r ) d r d s d ¯ κ ( r ) (cid:1) , provided s is large enough. This proves the result for r ∈ [ g s , s/g s ]. As mentioned in thevery beginning of the proof, by choosing sufficiently large c we can extend the estimate to all r ∈ [0 , s ]. (cid:3) Proof of Corollary 2.19. This follows closely the proof of Corollary 1.6. We first treat the casethat B is contained in an open hemisphere. Equip S d with a marked pole x , and for r ∈ (0 , 1] let rB denote the symmetric spherical cap centred at x with volume r d , considered as a stratifiedset via the stratification F B = { int( B ) , ∂B } . Fix ε > g n such that g n → ∞ as n → ∞ . Define a mesoscopic scale g n / √ n < r < /g n and consider placing (Vol( B ) + o (1)) /r d disjoint copies of rB inside B . Following exactly the proof of Corollary 1.6, we deduce theexistence of c , c > P [ N ( B ) /n d/ ≤ ( c N − ε )Vol( B )] ≤ c ( e − c r − d + α r,n ) , where α r,n denotes the supremum of α -mixing coefficients α top ( B , B ) among all pairs of disjointstratified sets B and B contained in B and separated by a distance at least r . By Theorem 2.15(see also the proof of Corollary 2.17), there exist k, c , c > α r,n ≤ c n k e − c r n . Setting r = n − / (2+ d ) yields the desired bound.In the general case, we simply choose a finite number of disjoint stratified sets B i that are eachcontained within an open hemisphere. Since by super-additivity N ( B ) /n d/ ≤ c N Vol( B ) − ε implies that N ( B i ) /n d/ ≤ c N Vol( B i ) − ε/k for some B i , the argument goes through in this caseas well. (cid:3) Decorrelation for topological counts. Corollary 2.20 is a direct consequence of thefollowing general result, applied to the random variables X = N ( B ) and Y = N ( B ): Proposition 6.2 (See [25, Theorem 17.2.2]) . Let X and Y be random variables and define the α -mixing coefficient associated to their σ -algebras α ( X, Y ) = sup A ∈ σ ( X ) ,B ∈ σ ( Y ) | P [ A ∩ B ] − P [ A ] P [ B ] | . Suppose further that E [ X δ ] < c and E [ Y δ ] < c for positive constants δ, c > . Then | Cov( X, Y ) | ≤ c / (2+ δ ) α ( X, Y ) / (2+ δ ) . Appendix A. Gaussian computations In this section we gather results about finite-dimensional Gaussian vectors. If X is a Gaussianvector in finite-dimensional vector space equipped with a scalar product, let DC( X ) be thedeterminant of its covariance. Lemma A.1. Let X, Y be jointly Gaussian vectors such that Y is non-degenerate. Then DC( X | Y ) does not depend on Y , and DC( X, Y ) = DC( X | Y )DC( Y ) . Proof. This is an easy consequence of the Gaussian regression formula ([3, Proposition 1.2]). (cid:3) COVARIANCE FORMULA FOR TOPOLOGICAL EVENTS OF SMOOTH GAUSSIAN FIELDS 41 Lemma A.2. Given two independent Gaussian vectors X and Y of the same dimension, DC( X + Y ) ≥ DC( X ) + DC( Y ) . Proof. In terms of covariance matrices, this amounts to saying that given A, B two symmetricnon-negative matrices of the same size,det( A + B ) ≥ det( A ) + det( B ) , which follows from the Minkowski inequality. (cid:3) Lemma A.3. Let X, Y, Z be jointly Gaussian vectors such that Y and Z have the same dimen-sion and Z is independent of ( X, Y ) . Then DC( X, Y + Z ) ≥ DC( X, Y ) + DC( X )DC( Z ) . Proof. Let us assume that X is non-degenerate; the general case follows by continuity. ByLemma A.1 DC( X, Y + Z ) = DC( X )DC( Y + Z | X ) . Applying Lemma A.2,DC( Y + Z | X ) ≥ DC( Y | X ) + DC( Z | X ) = DC( Y | X ) + DC( Z ) , with the final equality since Z is independent of X . HenceDC( X, Y + Z ) ≥ DC( X )DC( Y | X ) + DC( X )DC( Z ) = DC( X, Y ) + DC( X )DC( Z ) , where the equality holds by Lemma A.1. (cid:3) Lemma A.4. Let X and X be respectively d × d and d × d random matrices, and let Y = ( Y , Y ) ∈ R and Z ∈ R d + d be random vectors. Suppose that ( Y, Z ) is a non-degenerateGaussian vector and, conditionally on Z = 0 , X and X have entries that are jointly Gaussianwith Y . Let ϕ Y,Z denote the density of ( Y, Z ) . Then there exists a constant c > , dependingonly on d and d , such that (A.1) ϕ Y,Z (0) E [ | det( X )det( X ) | | Y = 0 , Z = 0] is bounded above by the maximum, over i ∈ { , } , of c E (cid:2) (cid:107) X i (cid:107) op | Z = 0 (cid:3) d i (cid:112) DC( Y, Z ) max (cid:110) , (cid:16) max k Var (cid:2) Y k (cid:3) (cid:112) DC( Z ) (cid:112) DC( Y, Z ) (cid:17) d i (cid:111) , where (cid:107) · (cid:107) op denotes the ( L -)operator norm.Proof. Let c denote a positive constant, depending only on d and d , that may change fromline to line. In the proof we use repeated the fact that conditioning on part of a Gaussian vectorreduces the variance of all coordinates. By the Cauchy-Schwarz inequality and an elementarybound on the determinant, (A.1) is bounded above by c ϕ Y,Z (0) max i,j ,j E (cid:2) ( X i ) d i j ,j | Y = 0 , Z = 0 (cid:3) . Since a normally distributed random variable Z ∼ N ( µ, σ ) satisfies E [ Z d i ] ≤ c max { µ d i , σ d i } , and since the variance of a random variable is less than its second moment, E [( X i ) d i j ,j | Y = 0 , Z = 0] ≤ c max (cid:110) E (cid:2) ( X i ) j ,j | Z = 0 (cid:3) d i , E (cid:2) ( X i ) j ,j | Y = 0 , Z = 0 (cid:3) d i (cid:111) . Let Σ Y | Z and Σ Z denote the covariance matrices of Y | Z and Z respectively. By conditioningon Z = 0 and applying Lemma A.1, we have that ϕ Y,Z (0) = c e − E [ Y | Z =0] T Σ − Y | Z E [ Y | Z =0] (cid:112) DC( Y | Z ) e − E [ Z ] T Σ − Z E [ Z ] (cid:112) DC( Z ) ≤ c e − E [ Y | Z =0] T Σ − Y | Z E [ Y | Z =0] (cid:112) DC( Y, Z ) ≤ c (cid:112) DC( Y, Z ) . Since, moreover, max j ,j E (cid:2) ( X i ) j ,j | Z = 0 (cid:3) ≤ E [ (cid:107) X i (cid:107) | Z = 0] , it suffices to show thatsup (cid:96) ∈ R (cid:110) E (cid:2) ( X i ) j ,j | Y − E [ Y | Z = 0] = (cid:96), Z = 0 (cid:3) d i e − (cid:96) T Σ − Y | Z (cid:96) (cid:111) is bounded above by c E (cid:2) ( X i ) j ,j | Z = 0 (cid:3) d i max (cid:110) , (cid:16) max k Var (cid:2) Y k (cid:3)(cid:112) DC( Z ) (cid:112) DC( Y, Z ) (cid:17) d i (cid:111) . To show this, decompose Σ − Y | Z = U T Λ − U , where U = ( u k ,k ) is a 2 × λ k ) is the 2 × Y | Z . Abbreviating W = ( w k ) := U Cov[( X i ) j ,j Y k | Z = 0] and replacing (cid:96) by U (cid:96) , by the Gaussian regression formula([3, Proposition 1.2]) we have thatsup (cid:96) (cid:110) E (cid:2) ( X i ) j ,j | Y − E [ Y | Z = 0] = (cid:96), Z = 0 (cid:3) d i e − (cid:96) T Σ − Y | Z (cid:96) (cid:111) = sup (cid:96) (cid:110)(cid:0) E [( X i ) j ,j | Z = 0] + W T Λ − (cid:96) (cid:1) d i e − (cid:96) T Λ − (cid:96) (cid:111) ≤ c max (cid:40) sup (cid:96) (cid:110) E [( X i ) j ,j | Z = 0] d i e − (cid:96) T Λ − (cid:96) (cid:111) , sup (cid:96) (cid:110)(cid:0) W T Λ − (cid:96) (cid:1) d i e − (cid:96) T Λ − (cid:96) (cid:111)(cid:41) ≤ c max (cid:40) E [( X i ) j ,j | Z = 0] d i , sup (cid:96) (cid:110)(cid:0) W T Λ − (cid:96) (cid:1) d i e − (cid:96) T Λ − (cid:96) (cid:111)(cid:41) . Differentiating in (cid:96) , the maxima of the expression on the right is attained at (cid:96) = ( (cid:96) , (cid:96) ) = ±√ d i √ w λ − + w λ − (cid:0) w , w (cid:1) , ( w , w ) (cid:54) = (0 , , (0 , , ( w , w ) = (0 , , and yields a maximum value of(2 d i /e ) d i ( w λ − + w λ − ) d i ≤ c (cid:16) max k w k max k λ − k (cid:17) d i . Since the eigenvalues of a positive-definite real symmetric matrix are bounded by a constanttimes the maximum diagonal entry,max k λ − k = max k λ k det(Λ) ≤ c max k Var (cid:2) Y k | Z = 0 (cid:3) det(Σ Y | Z ) ≤ c max k Var (cid:2) Y k (cid:3) det(Σ Y | Z ) = c max k Var (cid:2) Y k (cid:3) DC( Z )DC( Y, Z ) , where in the last step we used Lemma A.1. Moreover, since U has entries bounded above inabsolute value by one (being orthogonal), and by the Cauchy-Schwarz inequality,max k | w k | ≤ c max k | Cov[( X i ) j ,j Y k | Z = 0] | ≤ c E [( X i ) j ,j | Z = 0] / max k Var (cid:2) Y k (cid:3) / . Combining we have the result. (cid:3) COVARIANCE FORMULA FOR TOPOLOGICAL EVENTS OF SMOOTH GAUSSIAN FIELDS 43 Lemma A.5. Let ( Y , Y ) denote a ( d + d ) -dimensional non-degenerate Gaussian vector. Foreach t ∈ [0 , , define Y t = ( Y , tY + √ − t ˜ Y ) where ˜ Y is a copy of Y independent of ( Y , Y ) .Then DC ( Y ) ≤ DC ( Y t ) ≤ DC ( Y ) . Proof. Observe that DC ( Y t ) has the block form (cid:20) A tBtB T C (cid:21) , where A and C are (strictly) positive-definite. Since A is positive-definite and BC − B T issymmetric and positive-definite, there exists a P such that A = P T P and BC − B T = P T DP, where D = Diag(( d i ) i ) is a positive diagonal matrix. Hence DC ( Y t ) = det( C ) det( A − t BC − B T ) = det( C ) det( P ) Π i (1 − t d i )which, since d i > 0, is decreasing in t ∈ [0 , (cid:3) Appendix B. Proof of Piterbarg’s formula In the proof of Piterbarg’s formula, we will use the classical fact that the density function ϕ ( x ; Σ) of a (possibly non-centred) Gaussian vector with covariance Σ satisfies(B.1) 12 ∂ ∂x i ϕ ( x ; Σ) = ∂∂ Σ ii ϕ ( x ; Σ) and ∂ ∂x i ∂x j ϕ ( x ; Σ) = ∂∂ Σ ij ϕ ( x ; Σ) , i (cid:54) = j. Proof of Lemma 2.22. Let ( f i ) i ≥ and ( g i ) i ≥ be sequences of smooth compactly supported func-tions on R m that converge to A and B in the sense of tempered distributions. Following theproof of [40, Theorem 1.4], by writing the derivative with respect to t in terms of derivativeswith respect to the elements of the covariant matrix, and then by using the identity (B.1) andintegrating by parts, we obtain ddt E [ f i ( X t ) g i ( Y t )] = m (cid:88) k =1 (cid:90) R m ∂ x k f i ( x ) ∂ y k g i ( y ) γ t ( x, y )d x d y = m (cid:88) k =1 (cid:90) R m f i ( x ) g i ( y ) ∂ x k ∂ y k γ t ( x, y )d x d y ;(all the other terms disappear since the only covariances that depend on t are Cov( X t,k , Y t,k ) = t ).Passing to the limit as i → ∞ gives that ddt P [ Z t ∈ A × B ] = ddt E [ A ( X t ) B ( Y t )] = m (cid:88) k =1 (cid:90) R m A ( x ) B ( y ) ∂ x k ∂ y k γ t ( x, y )d x d y. By Gauss’s theorem, applied both in the x k and in the y k variables we have (cid:90) R m A ( x ) B ( y ) ∂ x k ∂ y k γ t ( x, y )d x d y = (cid:90) ∂A × ∂B ν A ( x ) k ν B ( y ) k γ t ( x, y ) d x d y where ν A ( x ) k is the k -th component of ν A ( x ), and d x on the right-hand side of the equationis the volume element on ∂ A (and similarly for B and y ). Since (cid:80) mk =1 ( ν A ( x )) k ( ν B ( y )) k = (cid:104) ν A ( x ) , ν B ( y ) (cid:105) , ddt P [ Z t ∈ A × B ] = (cid:90) ∂A × ∂B (cid:104) ν A ( x ) , ν B ( y ) (cid:105) γ t ( x, y )d x d y, which proves the first part of the statement. To prove the last part of the lemma let us consider X to be a translation of the standardGaussian vector by µ . Let Y be an independent copy of X . We can define X t = X and Y t = t ( X − µ ) + √ − t ( Y − µ ) + µ . It is easy to see that these vectors satisfy the assumptionsin the first part of the lemma. Note that in this case Z = ( X, Y ) and Z = ( X, X ). Integratingwith respect to t from 0 to 1 we have (cid:90) (cid:90) ∂A × ∂B (cid:104) ν A ( x ) , ν B ( y ) (cid:105) γ t ( x, y )d x d y d t = P [( X, X ) ∈ A × B ] − P [( X, Y ) ∈ A × B ] . Since P [( X, X ) ∈ A × B ] = P [ X ∈ A ∩ B ] and P [( X, Y ) ∈ A × B ] = P [ X ∈ A ] P [ X ∈ B ], thisproves the second part of the statement. (cid:3) Appendix C. On stratified critical points Here we prove two elementary lemmas about stratified critical points. Recall that M is asmooth manifold and ( B, F ) is a stratified set of M . Lemma C.1. Let ( u k , x k ) k ∈ N be a sequence in C ( M ) × B converging to a limit ( u, x ) ∈ C ( M ) × B . Assume that, for each k ∈ N , x k is a stratified critical point of u k . Then x is a stratifiedcritical point of u . Lemma C.1 implies that the discriminant D B is C -closed. Moreover, taking u k = u for all k ,it implies that the set of stratified critical points of u in B is compact. Proof of Lemma C.1. Without loss of generality, we may assume that there exist F, F (cid:48) ∈ F suchthat x k ∈ F for each k ∈ N and x ∈ F (cid:48) . If F (cid:48) = F then the sequence ( u k | F ) k converges to u | F in C so d x u | F = 0 and x is a stratified critical point of u . Otherwise, F (cid:48) < F and d x u vanisheson T x F | F (cid:48) which contains T x F (cid:48) , so x is a critical point of u | F (cid:48) . (cid:3) Lemma C.2. Let u ∈ C ( M ) and let x ∈ B be a non-degenerate stratified critical point of u .Then x is isolated in the set of stratified critical points in B . Lemma C.2 shows that Definition 3.1 is the natural definition of non-degenerate critical pointsin the setting of stratified sets. Proof of Lemma C.2. Let x be a stratified critical point of u belonging to F (cid:48) ∈ F . Assume thatthere exists a sequence ( x k ) k ∈ N of stratified critical points of u distinct from x converging to x ;let us show that x is degenerate. 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