Perturbation of the expected Minkowski functional for weakly non-Gaussian isotropic fields on a bounded domain
PPERTURBATION OF THE EXPECTED MINKOWSKIFUNCTIONAL FOR WEAKLY NON-GAUSSIANISOTROPIC FIELDS ON A BOUNDED DOMAIN
SATOSHI KURIKI AND TAKAHIKO MATSUBARA
Abstract.
The Minkowski functionals (MF) including the Euler char-acteristic (EC) statistics are standard tools for morphological analysisin cosmology. Motivated by cosmic research, we consider the Minkowskifunctional of the excursion set for a weakly non-Gaussian isotropic smoothrandom field on an arbitrary dimensional compact domain. The weaknon-Gaussianity is represented by the N -point correlation function ofthe order O ( ν N − ), where ν (cid:28) O ( ν ) includ-ing the skewness and kurtosis. The resulting formula reveals the localpower property of the Minkowski functional as a statistic for testingGaussianity. Moreover, up to an arbitrary order in ν , the perturbationformula for the expected Minkowski functional is shown to be a linearcombination of the Euler characteristic density function multiplied bythe Lipschitz–Killing curvature of the index set, which has the sameform as the Gaussian kinematic formula (GKF). The application of theobtained perturbation formula in cosmic research is discussed. Introduction
The Minkowski functional (MF) is one of the fundamental concepts inintegral and stochastic geometry. This is a series of geometric quantitiesdefined for a bounded set in Euclidean space. In the 2-dimensional case, theMinkowski functional of the set M is a triplet consisting of the area Vol ( M ),the length of the boundary Vol ( ∂M ), and the Euler characteristic χ ( M ).The Minkowski functional measures morphological future of M in a differentway from conventional moment-type statistics, and has been used in variousscientific fields.In cosmology, the Minkowski functional was introduced around in the1990’s, and was first used to analyze the large-scale structure of the uni-verse, and then the cosmic microwave background (CMB). In particular, theMinkowski functional for the excursion set of the smoothed CMB map wasfirstly analyzed by [SB97] (cf. [SG98]). The CMB radiation provides rich in-formation on the very early stages of the universe. Its signal is recognized as Key words and phrases.
Euler characteristic; Gaussian kinematic formula; testingGaussianity. a r X i v : . [ m a t h . S T ] N ov Figure 1.
A random field (left), excursion set (middle), andtriangularization (right).an isotropic and nearly Gaussian random field. However, hundreds of infla-tionary models are available to deduce the various types of non-Gaussianity.The Minkowski functional is used for the selection of such candidate models.More precisely, let X ( t ), t ∈ E ⊂ R n , be such a random field. Its sup-levelset with a threshold v , E v = { t ∈ E | X ( t ) ≥ v } = X − ([ v, ∞ )) , is referred to as the excursion set (Figure 1). Then, we can calculate theMinkowski functional curves {M j ( E v ) } j =0 ,...,n as a function of v . When X ( t ) is Gaussian, the expected Minkowski functional density (i.e., the MFper unit volume) has been well known to be described in the Hermite polyno-mial (e.g., [Tom86]). Based on this fact, the departure between the sampleMinkowski functional and the expected Minkowski functional is used as ameasure of non-Gaussianity.On the other hand, the expected Minkowski functional for a non-Gaussianrandom field is unknown (except for the Gaussian related fields). The non-Gaussianity of a random field is measured by the cumulant generating func-tionallog E (cid:104) exp (cid:16)(cid:90) E θ ( t ) X ( t )d t (cid:17)(cid:105) = (cid:88) N N ! (cid:90) E N θ ( t ) · · · θ ( t N )cum( X ( t ) , . . . , X ( t N ))d t . . . d t N . To express weakly non-Gaussian random fields, we assume a weak non-Gaussianity through the N -point correlation function (the N -th cumulant)cum( X ( t ) , . . . , X ( t N )) = O (cid:0) ν N − (cid:1) , N ≥ , or equivalent conditions on higher-order spectra, where ν (cid:28) ν is the variationcoefficient of the temperature measurement. Such a weak non-Gaussianrandom field is realized by the weak non-linear filtering (e.g., the Wiener-chaos or Volterra series) of a Gaussian random field (see Section 3.5). Under this assumption, [Mat03, HKM06] provided the perturbation formula forthe expected Minkowski functional up to O ( ν ) including the skewness whenthe dimension is n ≤
3, while [Mat10] provided the formula up to O ( ν )including the skewness and kurtosis when n = 2. Then, they compared thetheoretical Minkowski functionals under several candidate inflation modelswith observed and simulated CMB data to evaluate the goodness-of-fit.The primary motivation of the paper is to complete the perturbationformulas when the dimension n is arbitrary. This not only completes theexisting research, but also reveals the local power of the Minkowski func-tional for testing Gaussianity against contiguous alternatives. Moreover, weshall characterize undetectable non-Gaussianities by the Minkowski func-tional statistic. As previously stated, although the Minkowski functional ispopular in morphological research, its performance as a test statistic has notyet to be studied.In reality, the index set (i.e., survey area) E is a bounded domain, andthe boundary corrections should always be incorporated. The Gaussiankinematic formula (GKF) established by [TA09, AT07, AT11] provides theframework of the expected Minkowski functional on a bounded domain. TheGKF states that when the random field X ( t ) is a Gaussian random field,or a Gaussian related field, the expected Minkowski functional is written asa linear combination of the Lipschitz–Killing curvatures L j ( E ) of E multi-plied by the Euler characteristic density Ξ j ( v ) depending on the marginaldistribution of X ( t ). Because of its simplicity, the GKF has been appliedto cosmology recently ([FMHM15], [PvV + +
20] and references therein for recent developments.In this paper, we will not pursue in this direction, but cite some papers fromthese areas for geometric preliminary.The contributions of the paper can be summarized as follows: (i) Theexpected Euler characteristic density for an isotropic field is provided as anasymptotic expansion up to O ( ν ) (Theorem 3.1). (ii) The undetectable non-Gaussianity by the Minkowski functional is characterized (Theorem 3.2).(iii) The perturbation formula for the expected Minkowski functional ona bounded domain is proved to have the same form as the GKF up toan arbitrary order (Theorem 3.3). (iv) As a crucial tool in perturbation analysis, we pose identities on the Hermite polynomial (Lemmas A.1, A.3,and Corollary A.1).The authors are preparing another cosmic paper [MK20], where the per-turbation formula for the Euler characteristic density up to O ( ν ) is derivedby an approach different from the one used in this paper. The final for-mulas are presented in terms of the higher-order spectra for the purpose ofconvenience in cosmic research.This paper is organized as follows. Section 2 is the preliminary part. Theexpected Euler characteristic formula via Morse’s theorem, and its simplifiedversion under the isotropic assumption are briefly summarized (Sections 2.1and 2.2). Geometric notions and tools including the Minkowski functionalsare introduced (Section 2.3). Section 3 covers the main results. We firstintroduce the isotropic N -point correlation functions to represent the weaknon-Gaussianity, and then discuss their properties (Sections 3.1, 3.2). Theperturbation formula for the Euler characteristic density up to O ( ν ) isderived (Section 3.3). We also identify the undetectable derivatives of the N -point correlation functions for arbitrary N . The expected Minkowskifunctional formula is proved to have the same form as the GKF (Section3.4). We consider a weak non-Gaussian random field and determine theperformance of the perturbation formula by a simulation study (Section 3.5).In Section 4, we discuss how the perturbation formula is used in the studyof cosmology. In the Appendix, the identities of the Hermite polynomial areproved (Section A.1), and the regularity conditions are summarized (SectionA.2). 2. Preliminaries from integral/stochastic geometry
Throughout the paper, we assume that X ( t ) is a random field on E ⊂ R n with mean 0, variance 1, and smooth sample path t (cid:55)→ X ( t ) in the followingsense: X ( t ), ∇ X ( t ) = ( X i ( t )) ≤ i ≤ n , and ∇ X ( t ) = ( X ij ( t )) ≤ i,j ≤ n exist andare continuous with respect to t = ( t , . . . , t n ) a.s., where X i ( t ) = ∂∂t i X ( t ) , X ij ( t ) = ∂ ∂t i ∂t j X ( t ) . In addition, we assume that X ( · ) is isotropic. That is, the arbitrary finitemarginal distribution { X ( t ) } t ∈ E (cid:48) , where E (cid:48) ⊂ R n is a finite set, is invariantunder the group of rigid motion of t .2.1. Expected Euler characteristic via Morse’s theory.
In this sub-section, we summarize the expected Euler characteristic formula withoutassuming that X ( · ) is isotropic. Major materials were taken from [AT07],[AT11], [TK02], and [SW08].For the index set E , we make two assumptions: Assumption 2.1. (i) E is a C piecewise smooth manifold [TK02] , orequivalently, a C stratified manifold [AT07] . Specifically, E is disjointly divided as E = n (cid:71) d =0 ∂ d E, where ∂ d E is a finite union of d -dimensional open C submanifold. ∂ n E =int E is the interior of E , and ∂ n − E (cid:116) · · · (cid:116) ∂ E = ∂E is the boundary of E .(ii) E has a positive reach (positive critical radius). The definition of the reach (critical radius) is provided in Section 2.3.The formula for the reach can be found in [Fed59, Theorem 4.18]. See alsoSection 2.3 of [KT09] for its derivation. Roughly speaking, positive-reachmeans the local convexity.For each point t ∈ ∂ d E , E has a tangent cone C t = C t ( E ) ⊂ R n . Fromthe assumption of the positive-reach, C t ( E ) is always convex. (The converseis not necessarily true.) The dual of C t ( E ) in R n , N t = N t ( E ) = (cid:8) x ∈ R n | (cid:104) x, y (cid:105) R n ≤ , ∀ y ∈ C t ( E ) (cid:9) , is referred to as the normal cone of E at t .Let f be a real-valued C function on R n . The gradient and Hesse matrixof f are denoted by ∇ f and ∇ f , respectively. The restriction of f on ∂ d E is denoted by f | ∂ d E . The gradient and Hesse matrix of f | ∂ d E with respectto a standard coordinate s = ( s , . . . , s d ) of ∂ d E at t are denoted by ∇ f | ∂ d E and ∇ f | ∂ d E , respectively.The point t = t ∗ ∈ ∂ d E such that ∇ f | ∂ d E ( t ∗ ) = 0 is called the criticalpoint of f | ∂ d E . Definition 2.1. f is called Morse on E if, at any critical point t ∗ ∈ ∂ d E of f | ∂ d E ,(i) ∇ f | ∂ d E ( t ∗ ) is of full rank, and(ii) ∇ f ( t ∗ ) (cid:54) = 0 if d < n . The following proposition is a version of Morse’s theorem for a piecewisesmooth manifold of positive-reach. sgn( · ) is the sign function, and {·} isthe indicator function. Proposition 2.1 ([AT07, Corollary 9.3.3]) . Suppose that f is Morse on E .Then, (1) χ ( f − (( −∞ , v ])) = n (cid:88) d =0 (cid:88) t ∗ sgn det (cid:0) ∇ f | ∂ d E ( t ∗ ) (cid:1) { f ( t ∗ ) ≤ v } {−∇ f ( t ∗ ) ∈ N t ( E ) } , where the summation (cid:80) t ∗ runs over the set of critical points of f | ∂ d E . We will use − X as a Morse function under regularity conditions. Usingthe Dirac measure δ ( · ) on ∂ d E with respect to the volume element d s = (cid:81) di =1 d s i = dvol d ( t ) of ∂ d E , (1) with f := − X is formally rewritten as χ ( E v ) = n (cid:88) d =0 (cid:90) ∂ d E sgn det( −∇ X | ∂ d E ) { X ( t ) ≥ v } {∇ X ( t ) ∈ N t } δ ( −∇ X | ∂ d E ) | J t | d s = n (cid:88) d =0 (cid:90) ∂ d E det( −∇ X | ∂ d E ) { X ( t ) ≥ v } {∇ X ( t ) ∈ N t } δ ( ∇ X | ∂ d E )dvol d ( t ) , where J t = det( ∂ ( −∇ X | ∂ d E ) /∂s ) = det( −∇ X | ∂ d E ) is the Jacobian.Let Q t be an n × ( n − d ) matrix such that Q t Q (cid:62) t is the orthogonal projectiononto T t ( ∂ d E ) ⊥ , the orthogonal complement of the tangent space of ∂ d E .Then, N t ⊂ T ⊥ t ( ∂ d E ), and hence ∇ f ∈ N t ⇔ Q (cid:62) t ∇ f ∈ Q (cid:62) t N t . By takingthe expectation and changing the expectation and integral, we have E [ χ ( E v )] = n (cid:88) d =0 (cid:90) ∂ d E E [ − det( ∇ X | ∂ d E ) { Q t ∇ X ( t ) ∈ Q t N t } δ ( ∇ X | ∂ d E )]dvol d ( t )(2) = n (cid:88) d =0 (cid:90) ∂ d E (cid:20)(cid:90) R n + 12 d ( d +1) det( − H ) { x ≥ v } { V ∈ Q (cid:62) t N t } × p t ( x, , V , H )d x d V d H (cid:21) dvol d ( t ) , where p t ( x, , V , H ) is the probability density of ( X ( t ) , ∇ X | ∂ d E ( t ) , Q (cid:62) t ∇ X ( t ) , ∇ X | ∂ d E ( t )) ∈ R d +( n − d )+ d ( d +1) for a fixed t .The regularity conditions that assure − X is a Morse function, and vali-date the integration including the Dirac delta function are given in [AT07,Theorem 12.1.1], and are summarized in the Appendix.2.2. Expected Euler characteristic for isotropic random fields.
Thusfar, we have not assumed that X ( t ) is isotropic. Hereafter, we assume that X ( t ) is isotropic, and simplify formula (2) under the assumption. We some-times drop the argument t , because the marginal distribution is irrelevantto t .The isotropic property implies that the marginal moment is irrelevant to t . Without loss of generality, we let(3) E [ X ( t )] = 0 , E [ X ( t ) ] = 1 . The covariance function is a function of the distance between the two points:(4) E ( X ( t ) X ( t )) = cum( X ( t ) , X ( t )) = ρ (cid:0) (cid:107) t − t (cid:107) (cid:1) , ρ (0) = 1 . To describe the smoothness of the covariance function, we let it be a functionof the squared distance. Under the regularity condition for exchanging thederivative ∂/∂t i and the expectation E [ · ], we have from (3) that E [ X i ] = 0, E [ X ij ] = 0, E [ X i X ] = 0, and E [ X i X j ] + E [ XX ij ] = 0 (see Section A.2). Therefore, ∇ X ( t ) is uncorrelated with X ( t ) for each fixed t . Moreover,from (4), we have E [ X i X j ] = ∂ ∂t i ∂t j ρ (cid:0) (cid:107) t − t (cid:107) (cid:1)(cid:12)(cid:12) t = t = − ρ (cid:48) (0) δ ij . We change the variable from ∇ X to R = ( R ij ) ≤ i,j ≤ n = ∇ X + γXI n , γ = − ρ (cid:48) (0) , so that R is uncorrelated to X and ∇ X , which simplifies the whole manip-ulation.For a fixed t ∈ ∂ d E , let P t be an n × d matrix such that P t P (cid:62) t is theorthogonal projection matrix onto T t ( ∂ d E ). Let Q t be an n × ( n − d ) matrixdefined before. Then, P t P (cid:62) t + Q t Q (cid:62) t = I n . We write the tangent cone andthe normal cone as C t ( E ) = R d ⊕ P t C t, , N t ( E ) = Q t N t, , respectively, where C t, and N t, are proper convex cones in R n − d .Around t ∈ ∂ d E , we consider a geodesic coordinate system t i = φ i ( s ), s = ( s a ) ≤ a ≤ d s.t. (cid:104) ∂φ/∂s a , ∂φ/∂s b (cid:105)| t = δ ab , and (cid:104) ∂φ/∂s a , ∂ φ/∂s b ∂s c (cid:105)| t =0. Let P t = ( ∂φ i /∂s a ) ≤ i ≤ n, ≤ a ≤ d . Lemma 2.1.
Suppose that t ∈ ∂ d E is a critical point of X ( t ) restricted to ∂ d E . That is, V = P (cid:62) t ∇ X ( t ) = 0 . Then, ∇ X | ∂ d E ( t ) = P (cid:62) t ∇ X ( t ) P t − S t ( V ) , V = Q (cid:62) t ∇ X ( t ) , and S t ( V ) is the second fundamental form of ∂ d E at t with respect to thedirection V .Proof. When t is a critical point, ∂X ( φ ( s )) ∂s a = n (cid:88) k =1 X k ∂φ k ( s ) ∂s a = ( P t ∇ X ( t )) a = 0for 1 ≤ a ≤ d , and( ∇| ∂ d E X ( t )) ab = ∂ X ( φ ( s )) ∂s a ∂s b = n (cid:88) k,l =1 X kl ( t ) ∂φ k ( s ) ∂s a ∂φ l ( s ) ∂s b + n (cid:88) k =1 X k ( t ) ∂ φ k ( s ) ∂s a ∂s b =( P (cid:62) t ∇ X ( t ) P t ) ab + (cid:68) Q t V , ∂ φ ( s ) ∂s a ∂s b (cid:69) . (cid:3) Lemma 2.2.
Let V and V be defined in Lemma 2.1. V = V / (cid:107) V (cid:107) isdistributed uniformly on the unit sphere of Q (cid:62) t T t ( ∂E d ) ⊥ , independently from ( X, V , (cid:107) V (cid:107) , R ) , where R = P (cid:62) t RP t . Proof.
The assumption of isotropic property implies the invariance of distri-bution under (
X, V , V , R ) (cid:55)→ ( X, V , P V , R ), where P ∈ O ( n − d ).The conditional distribution of V when ( X, V , R ) is given is orthogo-nally invariant, and the conditional distribution of V is a uniform dis-tribution on the unit sphere of Q (cid:62) t T t ( ∂E d ) ⊥ independently from ( X, V , R )and (cid:107) V (cid:107) . (cid:3) By choosing an appropriate orthonormal coordinate again, we can assumethat S t ( V ) = S t (cid:0) V (cid:1) (cid:107) V (cid:107) = diag (cid:0) σ t, (cid:0) V (cid:1) , . . . , σ t,k ( V ) (cid:1) (cid:107) V (cid:107) , where σ ti ( V ), i = 1 , . . . , k , are principal curvatures of ∂ d E at t with respectto the normal direction Q t V ∈ N t ( E ). For a subset I = { i , . . . , i k } ⊂{ , . . . , d } ( i < · · · < i k ) such that | I | = k , let I = { , . . . , d } \ I , and R I = (( R ) ij ) i,j ∈ I , S t,I (cid:0) V (cid:1) = diag (cid:0) σ t,i (cid:0) V (cid:1) , . . . , σ t,i k ( V ) (cid:1) . The determinant in (2) is expanded asdet (cid:0) − R + γXI d + S t ( V ) (cid:1) = d (cid:88) e =0 (cid:88) I : | I | = e det( − R I + γXI d − e ) det S t,I (cid:0) V (cid:1) (cid:107) V (cid:107) e . We substitute this into (2), and find that V is uniformly distributed on theunit sphere S ( R n − d ) of R n − d independently from X, R I , V and (cid:107) V (cid:107) . Let R = (( R ) ij ) ≤ i,j ≤ d − e . Then, E [ χ ( E v )] = n (cid:88) d =0 d (cid:88) e =0 L n − d,e Ξ d,e ( v ) , (5)where Ξ d,e ( v ) = (cid:90) ∞ x = v (cid:20) (cid:90) Sym( d − e ) det (cid:0) − R + γxI d − e (cid:1) (6) × (cid:90) R n − d (cid:107) V (cid:107) e p ( x, (0 , V ) , R )d V d R (cid:21) d x, and L n − d,e = (cid:90) ∂E d E (cid:2) ( ξ ∈ N t, )tr e S t ( ξ ) (cid:3) dvol d ( t ) , ξ ∼ Unif (cid:0) S ( R n − d ) (cid:1) (7)with tr e S t ( ξ ) = (cid:88) I : I ⊂{ ,...,d } , | I | = e det S t,I ( ξ ) . When the leading term from int( E ) is dominant and the contribution ofthe boundaries ∂ d E ( d < n ) is negligible, (5) is approximated as E [ χ ( E v )] ≈ L n, Ξ n ( v ) , where(8) Ξ n ( v ) = (cid:90) ∞ x = v (cid:20)(cid:90) Sym( n ) det (cid:0) − R + γxI n (cid:1) p ( x, , R )d R (cid:21) d x,p ( x, , R ) is the probability density of ( X, V, R ), and L n, = (cid:90) E d t = Vol n ( E ) . We call Ξ n ( v ) in (8) as the Euler characteristic density. In Section 3, weobtain Ξ n = Ξ n, in (8), Ξ d,e in (6), and E [ χ ( E v )] in (5), in turn. Becauseof the assumption of isotropicity, we can let P (cid:62) t = ( I d , Q (cid:62) t = (0 , I n − d )included in the integrals (6) and (8).2.3. Tube, critical radius, and Minkowski functional.
The criticalradius is a geometric quantity for a subset M in Euclidean space R n . Thisis defined via the tube about M below.Tube about M with radius ρ is defined by a set of points whose distancefrom M is less than or equal to M :(9)Tube( M, ρ ) = (cid:8) x ∈ R n | dist( x, M ) ≤ ρ (cid:9) , dist( x, M ) = min y ∈ M (cid:107) y − x (cid:107) . Suppose that the radius ρ increased from 0 to infinity. Unless M is convex,there exists a finite value of ρ = ρ c ( M ) such that the tube Tube( M, ρ ) has aself-overlap. This threshold is referred to as the critical radius or the reach of M : ρ c ( M ) = inf (cid:8) ρ ≥ | Tube(
M, ρ ) has a self-overlap (cid:9) .M is said to be of positive-reach if ρ c ( M ) is strictly positive. The classicalSteiner formula states that, when M is a C piecewise smooth manifold ofpositive-reach, for all ρ ∈ [0 , ρ c ( M )], the volume of the tube (9) is expressedas a polynomial in ρ :(10) Vol n (Tube( M, ρ )) = n (cid:88) j =0 ω n − j ρ n − j L j ( M ) = n (cid:88) j =0 ρ j (cid:18) nj (cid:19) M j ( M ) , where ω d = Vol d ( B d ) = π d/ Γ( d/ R d . The Minkowski functional M j ( M )of M , and the Lipschitz–Killing curvature L j ( M ) of M are defined as thecoefficients of the polynomial. L j ( M ) is defined independently from thedimension n of the ambient space. There are variations of the definitions ofthe Minkowski functional. The definition in (10) is by [SW08, Section 14.2].Owing to the Gauss-Bonnet theorem, the Minkowski functional of thelargest degree is proportional to the Euler characteristic (EC) of M : χ ( M ) = L ( M ) = M n ( M ) /ω n . The next two propositions are used in Section 3.4. LetΩ d = Vol d − ( S d − ) = 2 π d/ Γ( d/ R d . Note that ω d = Ω d /d . Proposition 2.2 (Crofton’s intersection formula, [SW08, Theorem 5.1.1]) . Let A ( n, k ) be the set of k -dimensional affine subspaces in R n . Let µ n,k be theinvariant measure on A ( n, k ) with respect to the isotropic group normalizedby µ n,k ( { L ∈ A ( n, k ) | L ∩ B n (cid:54) = ∅} ) = ω n − k . Then, (cid:90) A ( n,n − k ) L j ( M ∩ L )d µ n,n − k ( L ) = c n,j,k L n + j − k ( M ) , where c n,j,k = Γ (cid:0) k +12 (cid:1) Γ (cid:0) k + j +12 (cid:1) Γ (cid:0) j +12 (cid:1) Γ (cid:0) n +12 (cid:1) . Proposition 2.3.
Let E = (cid:70) nd =0 ∂ d E ⊂ R n be the C piecewise smoothmanifold of positive-reach. Then, the Lipschitz–Killing curvature of E is L d ( E ) = n (cid:88) e = d L n − e,e − d Ω n − e Ω n − d , where L n − e,e − d is defined in (7).Proof. According to the volume formula of tube,Vol n (Tube( E, ρ )) = n (cid:88) d =0 (cid:90) ∂E d (cid:90) r<ρ (cid:90) v ∈ S ( N t, ) det (cid:0) I d + rS t ( v ) (cid:1) r n − d − d v d r d t, where d v is the volume element of S ( N t, ) = { v/ (cid:107) v (cid:107) | v ∈ N t, } . Note thatd v/ Ω n − d is the uniform distribution on S ( N t, ), (cid:82) v ∈ S ( N t, ) tr e ( S t ( v ))d v = L n − d,e Ω n − d , andVol n (Tube( E, ρ )) = n (cid:88) d =0 d (cid:88) e =0 L n − d,e Ω n − d (cid:90) r<ρ r n − d + e − d r = n (cid:88) d =0 d (cid:88) e =0 L n − d,e Ω n − d Ω n − d + e × ω n − d + e ρ n − d + e = n (cid:88) d =0 (cid:32) n (cid:88) e = d L n − e,e − d Ω n − e Ω n − d (cid:33) ω n − d ρ n − d = n (cid:88) d =0 L d ( E ) ω n − d ρ n − d . (cid:3) Perturbation of the Minkowski functional
Throughout the paper, we assume that X ( t ) ∈ R , t ∈ E ⊂ R n , is a zero-mean, unit-variance random field with continuous derivatives ∇ X ( t ) and ∇ X ( t ) with probability one. We also assume that X ( t ) is isotropic (exceptfor Section 2.1).3.1. Weakly non-Gaussian random fields.
To describe the weak non-Gaussianity, we introduce the N -point correlation functions defined as ajoint cumulant of the N variable ( X ( t ) , . . . , X ( t N )), where t i ’s are arbitrarypoints. In cosmology, the weakly nonlinear evolution of cosmic fields ismodeled such that the N -point correlation is of the order O ( ν N − ), where ν (cid:28) /ν is the sample size. Therefore, wewill discuss the perturbation theory in the framework of the CLT ([Pet75,BR10]).Because of the isotropic property, we assume the N -point correlation func-tion: cum( X ( t ) , . . . , X ( t N )) = ν N − κ ( N ) (cid:0) ( (cid:107) t a − t b (cid:107) ) ≤ a
4) are notsymmetric.To evaluate the Euler characteristic densities Ξ n in (8) or Ξ d,e in (6), weneed to identify the joint distribution of ( X, ∇ X, ∇ X ) and then ( X, V, R )with V = ∇ X , and R = ∇ X + γXI n .Using the parameters T = ( τ i ) ∈ R n and Θ = ( θ ij ) ∈ Sym( n ) such that(11) θ ij = 1 + δ ij τ ij ( i ≤ j ) , the characteristic function of 1+ n + n ( n +1) / (cid:0) n +22 (cid:1) dimensional randomvariables ( X, ∇ X, ∇ X ) is ψ ( is, iT, i Θ), where ψ ( s, T, Θ) = E (cid:104) e sX + (cid:80) i τ i X i + (cid:80) i ≤ j τ ij X ij (cid:105) = E (cid:104) e sX + (cid:104) T, ∇ X (cid:105) +tr(Θ ∇ X ) (cid:105) (12)is the moment generating function (mgf). Then, the mgf of ( X, ∇ X, R ) is E (cid:104) e sX + (cid:104) T, ∇ X (cid:105) +tr(Θ ∇ R ) (cid:105) = ψ ( (cid:101) s, T, Θ) , (cid:101) s = s + γ tr(Θ) . Under smoothness conditions, the cumulant generating function log ψ ( s, T, Θ)has a Taylor serieslog ψ ( s, T, Θ) = (cid:88) u,v,w ≥ ν u + v + w − u ! v ! w ! K u,v,w ( s, T, Θ) , where, for N = u + v + w , ν N − K u,v,w ( s, T, Θ)(13)= cum (cid:0) sX, . . . , sX (cid:124) (cid:123)(cid:122) (cid:125) u , (cid:104) T, ∇ X (cid:105) , . . . , (cid:104) T, ∇ X (cid:105) (cid:124) (cid:123)(cid:122) (cid:125) v , tr(Θ ∇ X ) , . . . , tr(Θ ∇ X ) (cid:124) (cid:123)(cid:122) (cid:125) w (cid:1) = ν N − s u (cid:18) u + v (cid:89) b = u +1 (cid:104) T, ∇ t b (cid:105) N (cid:89) c = u + v +1 tr (cid:0) Θ ∇ t c (cid:1)(cid:19) κ ( N ) (cid:0) ( (cid:107) t a − t b (cid:107) ) a
Lemma 3.1. K u, v,w ( s, T, Θ) is a linear combination of (14) s u (cid:89) j ≥ ( T (cid:62) Θ j T ) v j (cid:89) k ≥ tr(Θ k ) w k , where v j and w k are non-negative integers satisfying (cid:88) j ≥ v j = v, (cid:88) j ≥ jv j + (cid:88) k ≥ kw k = w. Proof.
By considering the orthogonal transformation t (cid:55)→ P t , P ∈ O ( n ), wehave the invariance in distribution ( X, ∇ X, ∇ X ) = d ( X, P ∇ X, P ∇ XP (cid:62) ).Because of the multilinearity of the cumulant, K u,v,w ( s, T, Θ) = s u K u,v,w (1 , T, Θ)and K u,v,w (1 , P (cid:62) T, P (cid:62) Θ P ) = K u,v,w (1 , T, Θ). By letting P = − I n , we seethat K u,v,w (1 , T, Θ) = 0 when b is odd. When b is even, K u,v,w (1 , T, Θ) is aquadratic function of T , and can be regarded as a function of ( t i t j ) ≤ i,j ≤ n = T T (cid:62) ∈ Sym( n ).As an extension of the zonal polynomial, [Dav80] introduced an invariantpolynomial of two symmetric matrices A, B ∈ Sym( n ) which is invariantunder the transformation ( A, B ) (cid:55)→ ( P (cid:62) AP, P (cid:62) BP ). s u K u,v,w (1 , T, Θ) is an invariant polynomial in ( T T (cid:62) , Θ), which is a linear combination of (14)as shown in (4.8) of [Dav80]. (cid:3)
In the limit ν ↓
0, uncorrelated random variables X , ∇ X , and ∇ X + γI n = R are independently distributed as the Gaussian distributions X ∼ N (0 , ∇ X ∼ N n (0 , γI n ), and R ∈ Sym( n ) is a zero-mean Gaussian ran-dom matrix with the covariance structurecum( R ij , R kl ) = α
12 ( δ ik δ jl + δ il δ jk ) + βδ ij δ kl , where α = 2 ρ (cid:48)(cid:48) (0) , β = ρ (cid:48)(cid:48) (0) − ρ (cid:48) (0) . The probability density function (pdf) of the limiting R with respect tod R = (cid:81) i ≤ j d R ij is proportional toexp (cid:110) − α tr (cid:0) ( R − n tr( R ) I n ) (cid:1) − n ( α + nβ ) tr( R ) (cid:111) . This is a proper density if(15) α > , α + nβ > , or equivalently , ρ (cid:48)(cid:48) (0) > nn + 2 ρ (cid:48) (0) > . This random matrix R is refereed to as the Gaussian orthogonal invariantmatrix ([CS18]).3.2. Derivatives of point correlation functions.
Here, we will evaluate K u,v,w ( s, T, Θ) in detail, specifically up to N = u + v + w ≤
4. Lemma3.1 states these are linear combinations of (14) with coefficients the partialderivatives of the point correlation function evaluated at the origin(16) κ ( N )( i ,...,i N − ,N ) (0 , . . . , , where κ ( N )( i ,...,i N − ,N ) ( x , . . . , x N − ,N ) = (cid:18) (cid:89) ≤ a
4, as well as their abbreviated notations. Because κ (3) ( x , x , x ) issymmetric in its arguments, there are three 1st derivatives with the samevalue κ (3)(1 , , (0 , ,
0) = κ (3)(0 , , (0 , ,
0) = κ (3)(0 , , (0 , , multiplicity .For the derivative (16), the diagram is defined as an undirected graphwith vertex set { , . . . , N } and edge set { (1 , , . . . , (1 , (cid:124) (cid:123)(cid:122) (cid:125) i times , (1 , , . . . , (1 , (cid:124) (cid:123)(cid:122) (cid:125) i times , . . . . . . } .
563 412 563 412
Figure 2.
Diagrams without loop (left) and with a loop (right)This edge set is a multiset, which means that multiple edges connecting apair of two vertices are allowed. The vertices of degree 0 (vertices withoutedges) can be omitted from the diagram.The diagram is characterized as follows.
Lemma 3.2.
The diagram of (16) is an undirected graph with a maximumof N vertices. Each connected component is a tree, or graph containing oneloop with a length greater than or equal to .Proof. Let x ab = (cid:107) t a − t b (cid:107) / (cid:80) ni =1 ( t ia − t ib ) /
2. Since ∂/∂t ia = (cid:80) a (cid:48) (cid:54) = a ( t ia − t ia (cid:48) ) ∂/∂x aa (cid:48) , we have ∂/∂t ia | t = ··· = t N = 0, and ∂ / ( ∂t ia ∂t jb ) | t = ··· = t N = δ ij × (cid:40)(cid:80) a (cid:48) (cid:54) = a ∂/∂x aa (cid:48) ( a = b ) , − ∂/∂x ab ( a (cid:54) = b ) . From this observation, we find that the derivative (cid:104) T, ∇ t b (cid:105)(cid:104) T, ∇ t b (cid:105) yieldsthe edge ( b , b ), while the derivative tr(Θ ∇ t c ) yields the edge ( c, c (cid:48) ), where c (cid:48) is any vertex other than c . The diagram is constructed by using theseedges as the building blocks. We see that a loop-free diagram consists of one( b , b )-type edges and ( c, c (cid:48) )-type edges, and a diagram containing a loopconsists only of ( c, c (cid:48) )-type edges. (For example, (cid:104) T, ∇ t (cid:105)(cid:104) T, ∇ t (cid:105) tr(Θ ∇ t )tr(Θ ∇ t )tr(Θ ∇ t )tr(Θ ∇ t ) produces Figure 2 (left), and tr(Θ ∇ t )tr(Θ ∇ t )tr(Θ ∇ t )tr(Θ ∇ t )tr(Θ ∇ t )tr(Θ ∇ t ) produces Figure 2 (right).) Hence, theresulting diagram is a union of connected graphs, and each connected graphhas at most one loop. (cid:3) The last column in Table 1 shows the number of loops in the diagram.Using the abbreviated notations listed in Table 1, the cumulants K u,v,w ( s, T, Θ)in (13) are evaluated. As an example, we consider νK , , ( s, T, Θ) =cum( (cid:104) T, ∇ X (cid:105) , (cid:104) T, ∇ X (cid:105) , tr(Θ ∇ X ))= n (cid:88) i,j,k,l =1 τ i τ j θ ij cum( X i , X j , X kl ) , Table 1.
Derivatives of correlation functions κ ( N ) , N = 2 , , ρ ρ (cid:48) (0) 1 0 ρ ρ (cid:48)(cid:48) (0) 1 1 κ κ (3) (0 , ,
0) 1 0 κ κ (3)(1 , , (0 , ,
0) 3 0 κ κ (3)(2 , , (0 , ,
0) 3 1 κ κ (3)(1 , , (0 , ,
0) 3 0 κ κ (3)(2 , , (0 , ,
0) 6 1 κ κ (3)(1 , , (0 , ,
0) 1 1 (cid:101) κ κ (4)(0 , , , , , (0 , , , , ,
0) 1 0 (cid:101) κ κ (4)(1 , , , , , (0 , , , , ,
0) 6 0 (cid:101) κ κ (4)(2 , , , , , (0 , , , , ,
0) 6 1 (cid:101) κ a κ (4)(1 , , , , , (0 , , , , ,
0) 12 0 (cid:101) κ aa κ (4)(1 , , , , , (0 , , , , ,
0) 3 0 (cid:101) κ c κ (4)(1 , , , , , (0 , , , , ,
0) 4 1 (cid:101) κ d κ (4)(1 , , , , , (0 , , , , ,
0) 4 0 (cid:101) κ a κ (4)(1 , , , , , (0 , , , , ,
0) 12 0 (cid:101) κ b κ (4)(2 , , , , , (0 , , , , ,
0) 24 1 (cid:101) κ ab κ (4)(2 , , , , , (0 , , , , ,
0) 6 1 (cid:101) κ bb κ (4)(2 , , , , , (0 , , , , ,
0) 3 2 (cid:101) κ f κ (4)(2 , , , , , (0 , , , , ,
0) 12 1 (cid:101) κ b κ (4)(2 , , , , , (0 , , , , ,
0) 24 1 (cid:101) κ d κ (4)(2 , , , , , (0 , , , , ,
0) 12 1 (cid:101) κ c κ (4)(1 , , , , , (0 , , , , ,
0) 3 1 (cid:101) κ q κ (4)(1 , , , , , (0 , , , , ,
0) 12 1 and cum( X i , X j , X kl )= ∂ ∂t i ∂t j ∂t k ∂t l νκ (3) (cid:0) (cid:107) t − t (cid:107) , (cid:107) t − t (cid:107) , (cid:107) t − t (cid:107) (cid:1)(cid:12)(cid:12) t = t = t = ν (cid:0) − κ δ ij δ kl + κ ( δ ik δ jl + δ il δ jk ) (cid:1) , hence we obtain the result K , , ( s, T, Θ) = − κ (cid:107) T (cid:107) tr(Θ) + 2 κ T (cid:62) Θ T .This manipulation can be conducted by the symbolic computation. We listthe cumulants K u,v,w ( s, T, Θ) up to the 4-th order below.
The 2nd cumulants. K , , = s , K , , = γ (cid:107) T (cid:107) , K , , = 0 , cum(tr(Θ R ) , tr(Θ R )) = α tr(Θ ) + β tr(Θ) , cum( sX, tr(Θ R )) = cum( (cid:104) T, ∇ X (cid:105) , tr(Θ R )) = 0 , where α = 2 ρ , and β = ρ − ρ . The 3rd cumulants: K , , = C s , K , , = C s (cid:107) T (cid:107) , K , , = s F tr(Θ) ,K , , = F s tr(Θ) + F s tr(Θ ) , K , , = J (cid:107) T (cid:107) tr(Θ) + J T (cid:62) Θ T,K , , = K tr(Θ) + K tr(Θ )tr(Θ) + K tr(Θ ) , where C = κ , C = − κ , F = 2 κ , F = 3 κ + κ , F = 2 κ ,J = − κ , J = 2 κ , K = 2( κ + 3 κ ) , K = 12 κ , K = − κ . The 4th cumulants. K , , = (cid:101) C s , K , , = (cid:101) C s (cid:107) T (cid:107) , K , , = (cid:101) C (cid:107) T (cid:107) , K , , = (cid:101) F s tr(Θ) ,K , , = (cid:101) F s tr(Θ) + (cid:101) F s tr(Θ ) , K , , = (cid:101) J s (cid:107) T (cid:107) tr(Θ) + (cid:101) J sT (cid:62) Θ T,K , , = (cid:101) K s tr(Θ) + (cid:101) K s tr(Θ )tr(Θ) + (cid:101) K s tr(Θ ) ,K , , = (cid:101) L (cid:107) T (cid:107) tr(Θ) + (cid:101) L T (cid:62) Θ T tr(Θ) + (cid:101) L (cid:107) T (cid:107) tr(Θ ) + (cid:101) L T (cid:62) Θ T,K , , = (cid:102) M tr(Θ) + (cid:102) M tr(Θ )tr(Θ) + (cid:102) M tr(Θ ) + (cid:102) M tr(Θ )tr(Θ) + (cid:102) M tr(Θ ) , where (cid:101) C = (cid:101) κ , (cid:101) C = − (cid:101) κ , (cid:101) C = 3 (cid:101) κ aa , (cid:101) F = 3 (cid:101) κ , (cid:101) F = 6 (cid:101) κ a + 2 (cid:101) κ aa + (cid:101) κ , (cid:101) F = 2 (cid:101) κ , (cid:101) J = − (2 (cid:101) κ a + (cid:101) κ aa ) , (cid:101) J = 2 (cid:101) κ a , (cid:101) K = 12 (cid:101) κ a + 2 (cid:101) κ c + 4 (cid:101) κ d + 3 (cid:101) κ ab + 6 (cid:101) κ b , (cid:101) K = 6( (cid:101) κ ab + 2 (cid:101) κ b ) , (cid:101) K = − (cid:101) κ c , (cid:101) L = − (6 (cid:101) κ a + 2 (cid:101) κ d + (cid:101) κ ab ) , (cid:101) L = 4(2 (cid:101) κ a + (cid:101) κ d ) , (cid:101) L = − (cid:101) κ ab , (cid:101) L = − (cid:101) κ a , (cid:102) M = 3(2 (cid:101) κ c + 8 (cid:101) κ q + 8 (cid:101) κ b + 4 (cid:101) κ d + 4 (cid:101) κ f + (cid:101) κ bb ) (cid:102) M = 12(4 (cid:101) κ b + 2 (cid:101) κ d + 2 (cid:101) κ f + (cid:101) κ bb ) , (cid:102) M = 12 (cid:101) κ bb , (cid:102) M = − (cid:101) κ q , (cid:102) M = 48 (cid:101) κ c . Perturbation of the Euler characteristic density.
This sectionattempts to obtain the perturbation formula for the Euler characteristicdensity in (8), which is expressed by ( X, ∇ X, R ). In the previous section, weobtained the Taylor series of the cumulant generating function log ψ ( s, T, Θ)in (12) up to O ( ν ). The moment generating function of ( X, ∇ X, R ) was ψ ( s + γ tr(Θ) , T, Θ). We will conduct the formal calculation, however, itsvalidity condition is provided in Section A.2.The cumulant generating function islog ψ ( (cid:101) s, T, Θ) = log ψ X ( s ) + log ψ V ( T ) + log ψ R (Θ)+ νQ ,n ( (cid:101) s, T, Θ) + ν Q ,n ( (cid:101) s, T, Θ) + o ( ν ) , where (cid:101) s = (cid:101) s ( s, Θ) = s + γ tr(Θ), ψ X ( s ) = e s , ψ V ( T ) = e γ (cid:107) T (cid:107) , ψ R (Θ) = e α tr(Θ )+ β tr(Θ) , and Q k,n ( (cid:101) s, T, Θ) = (cid:88) u + v + w = k u ! v ! w ! K u,v,w ( (cid:101) s, T, Θ) , k = 3 , . Therefore, ψ ( (cid:101) s, T, Θ) = ψ X ( s ) ψ V ( T ) ψ R (Θ) (cid:16) νQ ,n ( (cid:101) s, T, Θ) + ν Q ,n ( (cid:101) s, T, Θ)(17) + 12 ν Q ,n ( (cid:101) s, T, Θ) + o ( ν ) (cid:17) . In the following calculation, we will ignore the term o ( ν ). This formalmanipulation is validated in Section A.2.Define ψ V =0 ( (cid:101) s, Θ) = (cid:90) R (cid:90) Sym( n ) e sx +tr(Θ R ) p ( x, , R )d R d x (18) = 1(2 π ) n (cid:90) R n ψ ( (cid:101) s, iT, Θ)d T. Recall that the general term of the expansion in (17) is (14). The inte-gration with respect to d T is conducted as an expectation with respect tothe Gaussian random variable T ∼ N n (0 , γ − ), and multiplying it by thenormalizing factor(19) 1(2 π ) n (cid:90) R n e − γ (cid:107) T (cid:107) d T = (2 πγ ) − n/ . Let ( n ) m = n ( n − · · · ( n − m + 1) = Γ( n + 1)Γ( n − m + 1)be the falling factorial. Since (cid:107) T (cid:107) is independent of (cid:81) j ≥ ( T (cid:62) Θ j T / (cid:107) T (cid:107) ) v j ,the expectation of the factor of (14) including T becomes E T (cid:20)(cid:89) j ≥ ( T (cid:62) Θ j T ) v j (cid:21) = E T (cid:2) (cid:107) T (cid:107) v (cid:3) E T (cid:20) (cid:81) j ≥ ( T (cid:62) Θ j T ) v j (cid:107) T (cid:107) v − v ) (cid:21) (20) = E T (cid:2) (cid:107) T (cid:107) v (cid:3) E T (cid:2)(cid:81) j ≥ ( T (cid:62) Θ j T ) v j (cid:3) E T (cid:2) (cid:107) T (cid:107) v − v ) (cid:3) =( − /γ ) v ( − ( n/ v − v )) v × ζ v ,v ,... (Θ) , where v = (cid:80) j ≥ v j , and ζ v ,v ,... (Θ) = E (cid:20)(cid:89) j ≥ ( ξ (cid:62) Θ j ξ ) v j (cid:21) , ξ ∼ N n (0 , I n )is a polynomial in tr(Θ j ), j ≥
1. For example, ζ (Θ) = tr(Θ), ζ (Θ) =tr(Θ ), ζ , (Θ) = tr(Θ) + 2tr(Θ ). Note that ζ v ,v ,... (Θ) does not includethe dimension n explicitly.For the terms in Q , Q and Q in (17) including T , we can rewrite theterms according to the rules below: (cid:107) T (cid:107) (cid:55)→ γ − n, (cid:107) T (cid:107) (cid:55)→ γ − n ( n + 2) ,T (cid:62) Θ k T (cid:55)→ γ − tr(Θ k ) , k = 1 , , (cid:107) T (cid:107) · T (cid:62) Θ T (cid:55)→ γ − ( n + 2)tr(Θ) , ( T (cid:62) Θ T ) (cid:55)→ γ − [tr(Θ) + 2tr(Θ )] , (21)and multiply the normalizing factor (19). Then, we have ψ V =0 ( (cid:101) s, Θ) = ψ X ( s ) ψ R (Θ)(2 πγ ) − n/ (cid:16) ν (cid:101) Q ,n ( (cid:101) s, Θ) + ν (cid:101) Q ,n ( (cid:101) s, Θ)(22) + 12 ν (cid:101) Q (2)3 ,n ( (cid:101) s, Θ) + o ( ν ) (cid:17) , where (cid:101) Q ,n ( (cid:101) s, Θ), (cid:101) Q ,n ( (cid:101) s, Θ), and (cid:101) Q (2)3 ,n ( (cid:101) s, Θ) are Q ,n ( (cid:101) s, T, Θ), Q ,n ( (cid:101) s, T, Θ), Q ,n ( (cid:101) s, T, Θ) with terms including T replaced according to the rules (21). Recall that the integral we are in the process of obtaining is Ξ n ( v ) in (8).From the definition of ψ V =0 ( i (cid:101) s, Θ) in (18), we see that (8) is evaluated as(23) Ξ n ( x ) = (cid:90) ∞ x (cid:20) det( − D Θ + γxI n ) (cid:12)(cid:12)(cid:12) Θ=0 π (cid:90) R e − isx ψ V =0 ( i (cid:101) s, Θ)d s (cid:21) d x, where D Θ is an n × n symmetric matrix differential operator defined by(24) ( D Θ ) ij = 1 + δ ij ∂∂ (Θ) ij = ∂∂τ ij ( i ≤ j ) . The integral in (23) with respect to d s is easily evaluated by(25) 12 π (cid:90) R e − isx ψ X ( s )( is ) k d s = H k ( x ) φ ( x ) , where φ ( x ) = 1 √ π e − x / is the probability density of the standard Gaussian distribution N (0 , H k ( x ) = φ ( x ) − ( − d / d x ) k φ ( x )is the Hermite polynomial of degree k . For the derivatives with respect toΘ, we use the lemma below. The main part of the proof is given in SectionA.1. Proposition 3.1. det( − D Θ + γxI n ) (cid:0) ψ R (Θ)tr(Θ c ) · · · tr(Θ c k ) (cid:1)(cid:12)(cid:12)(cid:12) Θ=0 (26) = ( − m γ n − m ( − / m − k ( n ) m H n − m ( x ) , where m = (cid:80) ki =1 c i .Proof. Recall that γ = (cid:112) α/ − β . (49) in Lemma A.2 with β := β/γ ,Θ := γ Θ, and x := − x yields (26). (cid:3) Now, we are ready to complete the whole calculations.Step 1. Expand the inside of the parenthesis in (22). The resulting mgf willbe ψ X ( s ) ψ R (Θ) times polynomial in s and tr(Θ k ), k ≤ φ ( x )multiplied by a polynomial in x k , and H k ( x ).Step 3. By using the three-term relation xH k +1 ( x ) = H k +1 ( x ) + kH k − ( x ) , we arrange the result of Step 2 to be φ ( x ) multiplied by a linearcombination of H k ( x ).Step 4. Taking the integration (cid:82) ∞ x d x by using (cid:90) ∞ x H k ( x ) φ ( x )d x = H k − ( x ) φ ( x ) , we complete the integral (23). Let H − ( x ) = φ ( x ) − (cid:82) ∞ x φ ( x )d x . These processes are simple rewritings of terms that can be conducted bysymbolic calculation. The results are summarized in the theorem below.
Theorem 3.1.
Under Assumptions A.1 and A.3, it holds that γ − n/ Ξ n ( x ) =(2 π ) − n/ φ ( x ) × (cid:16) H n − ( x ) + ν ∆ ,n ( x ) + ν ∆ ,n ( x ) (cid:17) + o ( ν ) , (27) where ∆ ,n ( x ) = γ − κ n ( n − H n − ( x ) − γ − κ nH n ( x ) + κ H n +2 ( x ) , (28)∆ ,n ( x ) = (cid:16) − γ − (3 (cid:101) κ a + (cid:101) κ d ) + γ − κ ( n − (cid:17) n ( n − n − H n − ( x )+ (cid:16) γ − (cid:0)(cid:101) κ aa ( n −
2) + 4 (cid:101) κ a ( n − (cid:1) − γ − κ κ ( n − n − (cid:17) nH n − ( x )+ (cid:16) − γ − (cid:101) κ + γ − (cid:0) κ ( n −
2) + 2 κ κ ( n − (cid:1)(cid:17) nH n +1 ( x )+ (cid:16) (cid:101) κ − γ − κ κ n (cid:17) H n +3 ( x ) + κ H n +5 ( x ) . Proof.
For the regularity conditions, see Section A.2. (cid:3)
This theorem shows the kind of non-Gaussianity captured by the Eulercharacteristic/Minkowski functional approach. Formula (27) includes ρ = 1, ρ = − γ , κ , κ , κ , (cid:101) κ , (cid:101) κ , (cid:101) κ a , (cid:101) κ aa , (cid:101) κ a , (cid:101) κ d ,and excludes ρ , κ , κ , κ , (cid:101) κ , (cid:101) κ c , (cid:101) κ b , (cid:101) κ ab , (cid:101) κ bb , (cid:101) κ f , (cid:101) κ b , (cid:101) κ d , (cid:101) κ c , (cid:101) κ q .According to Table 1, the diagrams of the excluded parameters containloops, while those of the included parameters are loop-free. Theorem 3.2claims that this observation holds for any cumulant κ ( N ) in the perturbationof an arbitrary order. LetΠ K (Θ) = tr(Θ) K − ( − K − tr(Θ K ) ( K ≥ . Theorem 3.2.
Suppose that the diagram of the derivative of the N -pointfunction κ ( N ) , (29) (cid:18) (cid:89) ( a,b ) ∂∂x ab (cid:19) κ ( N ) (0 , . . . , , has a loop of length K ≥ , where (cid:81) ( a,b ) runs over all edges of the diagram.Then, (29) does not appear in the asymptotic expansion of arbitrary order.Proof. We first assume that the diagram is connected. Let (cid:101) K ( ≤ N ) be thenumber of vertices of the diagram. When there exists a loop in the diagram, as seen in Lemma 3.2, the edge set of the diagram is disjointly divided as L (cid:116) T , where L = { (1 , , (2 , , . . . , ( K − , K ) , ( K, } is a loop of length K , and T is a union of the tree edges. For example,in Figure 2 (right), K = 3, (cid:101) K = 6, L = { (1 , , (2 , , (1 , } , and T = { (1 , , (4 , , (4 , } . When a loop is included, as in the proof of Lemma3.2, the derivative (29) appears only from the operation(30) (cid:101) K (cid:89) a =1 tr(Θ ∇ t a ) κ ( N ) (cid:0) ( (cid:107) t a − t b (cid:107) ) a
0) and tr(Θ k ) ( k ≥ T ∼ N n (0 , γ − I n ), they willbe a polynomial in tr(Θ k ) ( k ≥ K , after taking the expectation with respect to T , the coefficient of(29) will be a polynomial in tr(Θ k ) ( k ≥
1) with a factor Π K (Θ). It willvanish at the final stage because of Proposition 3.1,det( − D Θ + γxI n ) (cid:0) ψ R (Θ)tr(Θ c ) · · · tr(Θ c k )Π K (Θ) (cid:1)(cid:12)(cid:12)(cid:12) Θ=0 =( − m + K γ n − ( m + K ) ( − / ( m + K ) − ( k + K ) ( n ) m + K H n − ( m + K ) ( x ) − ( − K − ( − m + K γ n − ( m + K ) ( − / ( m + K ) − ( k +1) ( n ) m + K H n − ( m + K ) ( x )=0 , where m = (cid:80) ki =1 c i .Finally we prove (32). First, since T is a union of trees,(33) (cid:18) (cid:101) K (cid:89) a =1 ∂ ∂t i a a ∂t j a a (cid:19) ∆ L ∆ T = (cid:18) K (cid:89) a =1 ∂ ∂t i a a ∂t j a a (cid:19) ∆ L (cid:101) K (cid:89) a = K +1 δ i a j a , where ∆ L = (cid:89) ( a,b ) ∈ L (cid:0) (cid:107) t a − t b (cid:107) (cid:1) , ∆ T = (cid:89) ( a,b ) ∈ T (cid:0) (cid:107) t a − t b (cid:107) (cid:1) . Let u ia = t ia − t ia − ( t i = t iK ). Since ∂/∂t ia = ∂/∂u ia − ∂/∂u ia − , K (cid:89) a =1 ∂ ∂t i a a ∂t j a a ∆ L = K (cid:89) a =1 (cid:18) ∂ ∂u ia ∂u ja + ∂ ∂u ia − ∂u ja − − ∂ ∂u ia ∂u ja − − ∂ ∂u ia − ∂u ja (cid:19) ∆ L (34) = 2 (cid:18) K (cid:89) a =1 ∂ ∂u i a a ∂u j a a (cid:19) ∆ L − (cid:18) (cid:88) chain K (cid:89) a =1 ( − ∂ ∂u i a a ∂u j a a (cid:19) ∆ L , where (cid:80) chain runs over the set such that { ( i , j ) , . . . , ( i K , j K ) } forms anundirected chain. Hence, we have (cid:18) (cid:101) K (cid:89) a =1 tr(Θ ∇ t a ) (cid:19) ∆ L ∆ T = n (cid:88) i ,...,i (cid:101) K ,j ,...,j (cid:101) K =1 θ i j · · · θ i K (cid:48) j K (cid:48) (cid:18) (cid:101) K (cid:89) a =1 ∂ ∂t i a a ∂t j a a (cid:19) ∆ L ∆ T =2tr(Θ) (cid:101) K − K (cid:0) tr(Θ) K − ( − K tr(Θ K ) (cid:1) because of (33) and (34). (cid:3) Expected Minkowski functional on a bounded domain.
In theprevious subsection, we have obtained the perturbation formula for the Eulercharacteristic density Ξ n ( v ) in (8). Here, we evaluate Ξ d,e ( v ) in (6), and E [ χ ( E v )] in (5), in turn. They are shown to be expressed by the functionΞ n ( v ). The argument in this subsection is valid for an arbitrary order inwhich the asymptotic expansion is validated.The joint mgf of ( X, V , V , R ) expressing Ξ d,e ( v ) is E (cid:20) exp (cid:16) sX + ( T (cid:62) , T (cid:62) ) (cid:18) V V (cid:19) + tr(Θ R ) (cid:17)(cid:21) = ψ (cid:18)(cid:101) s, (cid:18) T T (cid:19) , (cid:18) Θ
00 0 (cid:19)(cid:19) , where ψ is defined by (12) and (cid:101) s = s + γ tr(Θ ).Define ψ V =0 ( s, T , Θ ) = (cid:90) R (cid:90) Sym( d ) (cid:90) R n − d e sx + (cid:104) T ,V (cid:105) +tr(Θ R ) × p ( x, (0 , V ) , R )d V d R d x. Then,(35) ψ V =0 ( (cid:101) s, T , Θ ) = 1(2 π ) d (cid:90) R d ψ (cid:18)(cid:101) s, (cid:18) iT T (cid:19) , (cid:18) Θ
00 0 (cid:19)(cid:19) d T . Using this, Ξ d,e ( v ) in (6) is expressed as(36)Ξ d,e ( v ) = (cid:90) ∞ x = v (cid:20) det (cid:0) − D Θ + γxI d − e (cid:1)(cid:12)(cid:12)(cid:12) Θ =0 π (cid:90) R e − isx (cid:101) ψ e ( i (cid:101) s, Θ )d s (cid:21) d x, where (cid:101) ψ e ( (cid:101) s, Θ ) = (cid:90) R n − d (cid:107) V (cid:107) e π ) n − d (cid:90) R n − d e − i (cid:104) T ,V (cid:105) ψ V =0 ( (cid:101) s, iT , Θ )d T d V . (37)In the following, we will establish the statement below. Lemma 3.3.
Under Assumption A.2, (38) (cid:101) ψ e ( (cid:101) s, Θ ) = Ω n − d Ω n − d + e × ( ψ V =0 ( (cid:101) s, Θ) with n := d − e and Θ := Θ )+ o ( ν s − ) , where ψ V =0 ( (cid:101) s, Θ) is defined in (18).Proof. Under this assumption, the formal asymptotic expansion is validatedup to o ( ν s − ). We evaluate (35) and (37). There are three integrals withrespect to d T , d T , and d V . After expanding the o (1) terms in the exponen-tial function, the integral (35) with respect to d T becomes the expectationwith respect to T ∼ N d (0 , γ − I d ), and multiplying it by the factor1(2 π ) d (cid:90) R d e − γ (cid:107) T (cid:107) d T = (2 πγ ) − d/ . For the integrals (37) with respect to d T and d V , we need to evaluate J e,k ( γ ) = (cid:90) R n − d (cid:107) V (cid:107) e π ) n − d (cid:90) R n − d e − i (cid:104) T ,V (cid:105) e − γ (cid:107) T (cid:107) (cid:107) T (cid:107) k d T d V . Noting that1(2 π ) n − d (cid:90) R n − d e − i (cid:104) T ,V (cid:105) e − γ (cid:107) T (cid:107) d T = 1(2 πγ ) ( n − d ) / e − γ (cid:107) V (cid:107) , and J e, ( γ ) = (cid:90) R n − d (cid:107) V (cid:107) e πγ ) ( n − d ) / e − γ (cid:107) V (cid:107) d V = γ e/ e/ Γ( n − d + e )Γ( n − d ) = (2 πγ ) e/ Ω n − d Ω n − d + e , we have J e,k ( γ ) /J e, ( γ ) = (cid:16) − ∂∂γ (cid:17) k J e, ( γ ) /J e, ( γ )= (cid:16) − ∂∂γ (cid:17) k γ e/ /γ e/ = ( − /γ ) k ( e/ k . Therefore, rewriting the terms(39) (cid:107) T (cid:107) k (cid:55)→ ( − /γ ) k ( e/ k , and multiplying them by the factor J e, ( γ ) yields the integral J e,k ( γ ). When we consider the asymptotic expansion of an arbitrary order, thegeneral term has of the form of (14), where T = (cid:18) T T (cid:19) , and Θ = (cid:18) Θ
00 0 (cid:19) .The factor including T and T is (cid:89) j ≥ ( T (cid:62) Θ j T ) v j = ( (cid:107) T (cid:107) + (cid:107) T (cid:107) ) v (cid:89) j ≥ ( T (cid:62) Θ j T ) v j . We take the expectation under T ∼ N d (0 , γ − I d ). As in the derivation of(20), the expectation of the right-hand side is v (cid:88) i =0 (cid:18) v i (cid:19) E T (cid:20) (cid:107) T (cid:107) v − i ) (cid:89) j ≥ ( T (cid:62) Θ j T ) v j (cid:21) (cid:107) T (cid:107) i = v (cid:88) i =0 (cid:18) v i (cid:19) ( − /γ ) v − i ( − ( d/ v − v )) v − i × ζ v ,v ,... (Θ ) × (cid:107) T (cid:107) i , where v = (cid:80) j ≥ v j . Applying the term-rewriting rule (39) for (cid:107) T (cid:107) i , thisbecomes v (cid:88) i =0 (cid:18) v i (cid:19) ( − /γ ) v − i × ( − ( d/ v − v )) v − i ( − /γ ) i ( e/ i × ζ v ,v ,... (Θ )(40) = ( − /γ ) v ( − (( d − e ) / v − v )) v × ζ v ,v ,... (Θ ) . Here we used the binomial formula for the falling factorial( x + y ) m = m (cid:88) i =0 (cid:18) mi (cid:19) ( x ) m − i ( y ) i . (40) is equal to (20) with n and Θ replaced by d − e and Θ , respectively.Therefore, we have (cid:101) ψ e ( (cid:101) s, Θ ) = ψ X ( s ) ψ R (Θ )(2 πγ ) − d/ J e, ( γ ) × (cid:16) ν (cid:101) Q ,d − e ( (cid:101) s, Θ ) + ν (cid:101) Q ,d − e ( (cid:101) s, Θ ) + 12 ν (cid:101) Q (2)3 ,d − e ( (cid:101) s, Θ ) + · · · (cid:17) , where (cid:101) Q , (cid:101) Q , and (cid:101) Q (2)3 are defined in (22). Except for the factor (2 πγ ) − e/ J e, ( γ )= Ω n − d / Ω n − d + e , (cid:101) ψ e ( (cid:101) s, Θ ) is exactly the same as the ψ V =0 ( (cid:101) s, Θ) in (22)with n := d − e and Θ := Θ . (cid:3) Therefore, from (38) combined with (23) and (36), we haveΞ d,e ( v ) = Ω n − d Ω n − d + e Ξ d − e ( v ) + o ( ν s − ) . Substituting this into (5), E [ χ ( E v )] = n (cid:88) d =0 d (cid:88) e =0 L n − d,e Ω n − d Ω n − d + e Ξ d − e ( v ) + o ( ν s − )= n (cid:88) d =0 (cid:32) n (cid:88) e = d L n − e,e − d Ω n − e Ω n − d (cid:33) Ξ d ( v ) + o ( ν s − )= n (cid:88) d =0 L d Ξ d ( v ) + o ( ν s − ) . Theorem 3.3.
Assume Assumptions A.2 and A.3. Then, (41) E [ χ ( E v )] = n (cid:88) d =0 L d ( E )Ξ d ( v ) + o ( ν s − ) . L d is the Lipschitz–Killing curvature of E defined in (10). Note that if p ( x, V, R ) has the moments of arbitrary order, then the re-mainder term of (41) is o ( ν s ), where s is an arbitrary integer.The Minkowski functional for the excursion set M k ( E v ) can be evaluatedvia the Crofton formula (Proposition 2.2). Let L ∈ A ( n, n − k ), and let X | L be the restriction of X on L , i.e., a random field on E ∩ L . Since X isisotropic, when L is given, we have E [ χ (( E ∩ L ) v )] = n − k (cid:88) d =0 L d ( E ∩ L ) Ξ d ( v ) + o ( ν s − ) ( E ∩ L (cid:54) = ∅ ) . We perform an integration with respect to (cid:82) A ( n,n − k ) dµ n,n − k ( L ). Noting that( E ∩ L ) v = E v ∩ L , we have c n, ,k E [ L k ( E v )] = n − k (cid:88) d =0 c n,d,k L k + d ( E ) Ξ d ( v ) + o ( ν s − ) ω k . Let (cid:20) k + dk (cid:21) = c n,d,k c n, ,k = Γ( k + d +12 )Γ( )Γ( k +12 )Γ( d +12 ) . Now, we have the formula for the expected Lipschitz–Killing curvature ofthe excursion set, which is proportional to the Minkowski functional.
Theorem 3.4.
Under Assumptions A.2 and A.3, for k = 0 , , . . . , n , E [ L k ( E v )] = ω − n − k (cid:18) nk (cid:19) E [ M n − k ( E v )](42) = n − k (cid:88) d =0 (cid:20) k + dk (cid:21) L k + d ( E ) Ξ d ( v ) + o ( ν s − ) . Remark 3.1.
The Gaussian Kinematic formula (GKF) established by [TA09,AT07, AT11] states that the expected Euler characteristic for the Gaussianrelated field is written as a linear combination of the Lipschitz–Killing curva-ture L d ( E ) multiplied by the Euler characteristic density ρ d ( v ) defined by themarginal distribution of the field. Here, the Gaussian related field extendedby [PTV19] is a random field defined by X ( t ) = F ( Z (1) ( t ) , . . . , Z ( p ) ( t )) , t ∈ E, where Z (1) , . . . , Z ( p ) are independent smooth random fields on E with mar-ginal mean 0 and variance 1. The expression (42) is the same as the GKFformula, because γ n/ L d ( E ) is the Lipschitz–Killing curvature defined by themetric induced by the covariance Cov( ∇ X ( t )) = γI n . This similarity sug-gests the possibility of the extension of the GKF formula. Remark 3.2.
The perturbation term ∆ ,n ( x ) in (28) of Theorem 3.1 deter-mines the local power of the Minkowski functional for testing Gaussianity.Suppose a setting that we have N i.i.d. random fields X (1) ( · ) , . . . , X ( N ) ( · ) ,and let E ( i ) v be the excursion set of X ( i ) ( · ) . Define the test statistic T N = 1 √ N N (cid:88) i =1 (cid:0) L k ( E ( i ) v ) − µ k,v (cid:1) /σ k,v , where µ k,v = E [ L k ( E v )] and σ k,v = Var ( L k ( E v )) are mean and varianceunder the Gaussian hypothesis ν = 0 , respectively. Let δ k,v = n − k (cid:88) d =0 (cid:20) k + dk (cid:21) L k + d ( E ) γ − d/ (2 π ) − d/ φ ( v )∆ ,d ( v ) . Then, under the contiguous alternative ν = c/ √ N , because Var( L k ( E v )) =Var ( L k ( E v ))(1 + o (1)) , the limiting distribution of T N when N → ∞ is N (cid:0) cδ k,v /σ k,v , . A numerical example.
As an example of a weakly non-Gaussianrandom field, we consider a Gaussian random field by adding a quadraticform using an isotropic kernel. By choosing a Gaussian kernel, the N -pointcorrelation functions can be expressed explicitly. We set the index set E tobe a square E = [0 , e ] × · · · × [0 , e n ] with the Lipschitz–Killing curvature L d = (cid:81) i < ···
Euler characteristic curves for a Gaussian ran-dom field (left) and a non-Gaussian random field (right).(solid: simulation, dotted: no correction, dot-dashed: skew-ness, dashed: skewness+kurtosis)where DC = { ( e , e , e ) : directed chain connecting 4 vertices 1 , , , } ( | DC | = 24), and L = { ( e , e , e , e ) : undirected looppassing through 4 vertices 1 , , , } ( | L | = 3). For exam-ple, ((13) , (23) , (24)) ∈ DC and ((13) , (23) , (24) , (14)) ∈ L . From theseformulas, the derivatives of the N -point correlation functions at the originare evaluated explicitly.Figure 3 depicts the simulated Euler characteristic curves (solid line) andthe expected Euler characteristic curves without correction (dotted line),with corrections using the 3-point correlation κ (3) (dot-dashed line), andwith corrections using 3- and 4-point correlations κ (3) , κ (4) (dashed line).In this study, we set the 2-dimensional index set as E = [0 , , and theparameter in (43) g = 50. For the non-Gaussian case, we set as δ = 0 . τ = 0 . E = [0 , is divided into N × N parts( N = 200), and the data is generated at the lattice points { ( i /N, i /N ) | i , i ∈ Z } by the R package RandomFields . For the convolution, the Rpackage imagine was used.We approximate the excursion set by a triangle complex as in Figure1 (right) and compute the approximate Euler characteristic. The Eulercharacteristic for the threshold v is χ ( v ) = v ) − v ) + v ) . To calculate the Euler characteristic curve, we start form the initial value χ ( v ) = 0 when v = ∞ , and then change the value of v from ∞ to −∞ . χ ( v )changes at a critical threshold v = v ∗ satisfying X ( t ) = v ∗ at a vertex t . Then, we calculate the difference χ ( v ∗ + 0) − χ ( v ∗ ) by counting the numbersof the appearing/disappearing edges and triangles, all of which are connectedto the newly appearing vertex at v ∗ . They are easy to identify.This procedure is based on the same principle as the standard proof ofMorse’s theory [Mil63]. The extension to 3 or higher dimensional case isstraightforward.4. Cosmic data analysis: Future prospects
Finally, we discuss briefly how the obtained formula will be utilized in thecontext of cosmic research.In cosmology, the research is often based on data by simulator. In orderto check the validity of a cosmic model, the Minkowski functional curvesare calculated based on simulation data generated by the assumed cosmicmodel, and are compared with the observed Minkowski functional curvesto see the goodness-of-fit. If the goodness-of-fit is not good, we concludethat the assumed model is not true. Because the computational cost ofthe simulator is very high, the perturbation formulas will be very helpful inresearch if it provides the same information as simulators.The Euler characteristic density Ξ n ( v ) in (8) when the dimension is n = 2was derived and examined by [Mat10]. To check the accuracy of the formulaΞ ( v ), he considered a simple model for the temperature map of the CMBradiation: The weak non-Gaussian temperature fluctuations in the direction θ (spherical coordinates), denoted by X ( θ ) = ( T ( θ ) − E [ T ( θ )]) / E [ T ( θ )], aregenerated by X ( θ ) = X G ( θ ) − f NL X G ( θ ) + 9 g NL X G ( θ ) − (monopole + dipole) , where X G ( θ ) = ( T G ( θ ) − E [ T G ( θ )]) / E [ T G ( θ )] is the auxiliary temperaturefluctuation of a Gaussian random field with a Saches–Wolfe power spec-trum. The local relation between X G and X is a simple assumption ofthe toy model just for the numerical demonstration, and is not necessarilyaccurate in reality. This toy model is derived from the following assump-tions: the primordial non-Gaussianity is given by a local-type ansatz withparameters f NL and g NL , and the CMB temperature fluctuations are givenby an asymptotic formula of the Sachs–Wolfe effects on large-angular scalesbeyond the horizon at recombination epoch. Then, a radial Gaussian filteris applied to X ( θ ).In this setting, 100,000 realizations of the non-Gaussian CMB tempera-ture map are generated with random seeds for the Gaussian random field X G ( θ ), and Minkowski functionals for filtered X ( θ ) are calculated in eachrealizations, and finally they are averaged over. The numerical results arecompared with the theoretical curves as functions of the threshold. It isshown that the formula for Minkowski functionals M , M and M areextremely accurate within small sample variances among the 100,000 real-izations (Figure 1 of [Mat10]). In general, the CMB temperature fluctuations with a primordial non-Gaussianity is not given by the simply local relationof the toy model discussed above. They can have much complicated higher-order cumulants. Our formula is applicable to these general cases of weaklynon-Gaussian fields with arbitrary forms of cumulants.In this paper, we have obtained the Minkowski functionals in arbitrarydimensions n , and on a bounded domain. In cosmology, the cases n = 1 , , Appendix
A.A.1.
Identities on the Hermite polynomial.
We pose the identities forthe Hermite polynomial, which plays a crucial role in the derivation of theperturbation.For an n × n symmetric matrix A , the principle minor matrix correspond-ing to the indices K ⊂ { , . . . , n } is denoted by A [ K ] = ( a ij ) i,j ∈ K . Note that A = A [ { , . . . , n } ].We will first prove the lemma below first. Recall that Θ = ( θ ij ) and D Θ = ( d ij ) are defined in (11) and (24), respectively. Lemma A.1. (45)det( xI + D Θ ) (cid:0) e tr(Θ ) tr(Θ c ) · · · tr(Θ c (cid:96) ) (cid:1)(cid:12)(cid:12)(cid:12) Θ=0 = ( − / m − (cid:96) ( n ) m H n − m ( x ) , where m = (cid:80) (cid:96)i =1 c i .Proof. Because of the expansion formuladet( xI n + D Θ ) = n (cid:88) k =0 x n − k (cid:88) K : K ⊂{ ,...,n } , | K | = k det( D Θ[ K ] ) , the left-hand side of (45) is a polynomial in x with coefficients in the form(46) det( D Θ[ K ] ) (cid:0) e tr(Θ ) tr(Θ c ) · · · tr(Θ c (cid:96) ) (cid:1)(cid:12)(cid:12)(cid:12) Θ=0 . By the symmetry, it suffices to consider the case K = { , . . . , k } . Let Θ k =Θ[ { , . . . , k } ].By the definition of the determinant,det( D Θ k ) = (cid:88) σ ∈ S k sgn( σ ) d σ (1) · · · d nσ ( k ) , where S k is the permutation group on { , . . . , k } .tr(Θ c ) is a linear combination of the terms of the form θ j j θ j j · · · θ j c − j c θ j c j .The form(47) d i σ ( i ) · · · d i e σ ( i e ) (cid:0) θ j j θ j j · · · θ j c − j c θ j c j (cid:1)(cid:12)(cid:12)(cid:12) Θ=0 (1 ≤ i < · · · < i e ≤ k )does not vanish iff e = c , the map (cid:18) i · · · i c σ ( i ) · · · σ ( i c ) (cid:19) forms a cycle of length c , and( j , . . . , j c ) = ( i h , σ ( i h ) , σ ( i h ) . . . , σ c − ( i h )) or ( i h , σ − ( i h ) , . . . , σ − ( c − ( i h ))for some h = 1 , . . . , c (i.e., there are 2 c ways). The value of (47) is (1 / c ifit does not vanish.The form(48) d i σ ( i ) · · · d i e σ ( i e ) e tr(Θ ) (cid:12)(cid:12)(cid:12) Θ=0 (1 ≤ i < · · · < i e ≤ k )does not vanish iff e is even, and the map (cid:18) i · · · i e σ ( i ) · · · σ ( i e ) (cid:19) is a productof e/ σ ) d σ (1) · · · d kσ ( k ) (cid:0) e tr(Θ ) tr(Θ c ) · · · tr(Θ c (cid:96) ) (cid:1)(cid:12)(cid:12)(cid:12) Θ=0 does not vanish iff σ (in the cycle product form) is factorized as (cid:96) cycles oflength c i , i = 1 , . . . , (cid:96) , and e/ e = k − (cid:80) (cid:96)i =1 c i = k − m even. Noting that the number of cycles made from distinct c atoms is ( c − σ is (cid:18) k − c c (cid:19) ( c − × (cid:18) k − c − c c (cid:19) ( c − × · · ·× (cid:18) k − c − · · · − c (cid:96) − c (cid:96) (cid:19) ( c (cid:96) − × ( k − m )!!= ( k ) m (cid:81) c i × ( k − m )!2 ( k − m ) ( k − m )! = k !( (cid:81) c i )2 ( k − m ) ( k − m )! . The sign of σ issgn( σ ) = (cid:96) (cid:89) i =1 ( − c i − × ( − ( k − m ) / = ( − m − (cid:96) +( k − m ) / . Therefore,(46) = k !( (cid:81) c i )2 ( k − m ) ( k − m )! × (cid:96) (cid:89) i =1 (cid:0) c i × (1 / c i (cid:1) × k !2 m − (cid:96) +( k − m ) / ( k − m )! . Now we show that the left-hand side of (45) is n (cid:88) k = m, k − m :even x n − k (cid:18) nk (cid:19) k !2 m − (cid:96) +( k − m ) / ( k − m )! ( − m − (cid:96) +( k − m ) / = ( − / m − (cid:96) [ n − m ] (cid:88) k (cid:48) =0 x n − m − k (cid:48) n !2 k (cid:48) ( n − m − k (cid:48) )! k (cid:48) ! ( − k (cid:48) (cid:16) k (cid:48) = k − m (cid:17) = ( − / m − (cid:96) ( n ) m H n − m ( x ) . (cid:3) Lemma A.2.
For any β , det( xI + D Θ ) (cid:0) e (1+ β )tr(Θ )+ β tr(Θ) tr(Θ c ) · · · tr(Θ c (cid:96) ) (cid:1)(cid:12)(cid:12)(cid:12) Θ=0 (49) = ( − / m − (cid:96) ( n ) m H n − m ( x ) , where m = (cid:80) (cid:96)i =1 c i .Proof. Using the expansion about β = 0, e β (tr(Θ )+ tr(Θ) ) = (cid:88) k ≥ β k k ! (cid:88) ≤ h ≤ k (cid:18) kh (cid:19) tr(Θ ) k − h tr(Θ) h (1 / h , the left-hand side of (49) becomes a series in β . The coefficient of β k /k ! is (cid:88) ≤ h ≤ k (cid:18) kh (cid:19) ( − / ( m +2 k ) − ( (cid:96) + k + h ) ( n ) m +2 k H n − ( m +2 k ) ( x )(1 / h = 0except for the case k = 0. (cid:3) Let B ∈ Sym( n ) be an n × n symmetric random matrix whose momentgenerating function is(50) E (cid:2) e tr(Θ B ) (cid:3) = e (1+ β )tr(Θ )+ β tr(Θ) , β > − n + 2 . The range of β is given in (15). When β ≥
0, such B is constructed as(51) B = (cid:112) β ) A + (cid:112) βξI n , where A = ( a ij ) ∈ Sym( n ) is a Gaussian orthogonal ensemble (GOE) ran-dom matrix such that a ii ∼ N (0 , , a ij = a ji ∼ N (0 , / , independently , and denoted by A ∼ GOE( n ), and ξ ∼ N (0 ,
1) independently from A .Lemma A.3 is rewritten in terms of the expectation with respect to B . Lemma A.3.
Let B = ( b ij ) ∈ Sym( n ) be defined in (51). Then, E (cid:2) tr( D c B ) · · · tr( D c (cid:96) B ) det( xI + B ) (cid:3) = ( − / m − (cid:96) ( n ) m H n − m ( x ) , where m = (cid:80) (cid:96)i =1 c i , and D B is an n × n matrix differential operator whose ( i, j ) -th element is ( D B ) ij = 1 + δ ij ∂∂b ij ( i ≤ j ) . Proof.
Recall that ψ B (Θ) = E (cid:2) e tr(Θ B ) (cid:3) is given in (50). By multiplyingtr(Θ c ) · · · tr(Θ c (cid:96) ), and noting that tr( D cB ) e tr(Θ B ) = tr(Θ c ) e tr(Θ B ) , we have E (cid:2) tr( D c B ) · · · tr( D c (cid:96) B ) e tr(Θ B ) (cid:3) = ψ B (Θ)tr(Θ c ) · · · tr(Θ c (cid:96) ) . Then, we apply the derivative operator det( xI n + D Θ ) at Θ = 0. Sincedet( xI n + D Θ ) e tr(Θ B ) (cid:12)(cid:12)(cid:12) Θ=0 = det( xI n + B ) , we have E (cid:2) tr( D c B ) · · · tr( D c (cid:96) B ) det( xI n + B ) (cid:3) = det( xI n + D Θ ) (cid:0) ψ B (Θ)tr(Θ c ) · · · tr(Θ c (cid:96) ) (cid:1)(cid:12)(cid:12)(cid:12) Θ=0 . (cid:3) The Hermite polynomial is characterized as follow.
Corollary A.1.
Let A ∼ √ n ) . Let Θ and D Θ be defined in (11)and (24), respectively. Then, H n ( x ) = E [det( xI n + A )] = det( xI n + D Θ ) e tr(Θ ) (cid:12)(cid:12)(cid:12) Θ=0 . A.2.
Regularity conditions.
A.2.1.
Exchangeability of E [ · ] and ∂/∂t i . The covariance function is denotedby R ( s, t ) = E [ X ( s ) X ( t )]. If “ ∂ ∂s i ∂t i R ( s, t ) exists and is continuous at s = t ”,then the l.i.m. derivative X ∗ i ( t ) of X ( t ) exists, and holds that(52) E [ X ∗ i ( s ) X ∗ j ( t )] = ∂ ∂s i ∂t i R ( s, t )(e.g., [Adl81, Theorem 2.2.2 (p.27)]). This holds for the subsequence thatconverges to the a.s. limit. Therefore, under the assumption that the a.s.derivatives X i ( t ) exist, (52) is equal to E [ X i ( s ) X j ( t )]. When R ( s, t ) = ρ ( (cid:107) s − t (cid:107) ), the condition is restated as “ ρ (cid:48) ( x ) exists and is continuousat x = 0”.For our purpose, we need to apply the assumptions to κ ( N ) so that higher-order moments for higher-order derivatives exist. However, these require-ments are included in the conditions for the asymptotic expansion discussedbelow.A.2.2. Conditions for asymptotic expansion.
We begin by summarizing theasymptotic expansion for the moment and the probability density functionin the i.i.d. setting by [Pet75] (univariate case) and [BR10] (multivariatecase).Let q ν ( x ), x ∈ R k , be a probability density with the characteristic function ψ ν ( it ) = ψ ( it ) /ν . Let φ k ( x ; Σ) be the probability density of the Gaussiandistribution N k (0 , Σ), Σ >
0. Assume that the s -th moment exists under q . Then,log ψ ν ( t ) = ν − log ψ (1) ( νt ) = 12 t (cid:62) Σ t + s (cid:88) j =3 j ! ν j − (cid:88) i : | i | = j c i t i + ν − r s ( νt ) , where t = ( t , . . . , t k ), and i = ( i , . . . , i k ) is a multi-index such that t i =( t ) i · · · ( t k ) i k , and r s ( t ) is a function such that r s ( t ) = o ( | t | s − ). Let (cid:98) ψ ( s ) ν ( t ) = e t (cid:62) Σ t (cid:18) s (cid:88) j =3 ν j − F j − ( t ) (cid:19) , where the inside of the parenthesis is the expansion ofexp (cid:18) s (cid:88) j =3 j ! ν s − (cid:88) i : | i | = j c i t i (cid:19) about ν = 0 up to the order of ν s − . Here F j − ( t ) is a polynomial in t ofdegree 3( j − q ( s ) ν ( x ) = φ k ( x ; Σ) (cid:18) s (cid:88) j =3 ν j − G j − ( x ) (cid:19) , x = ( x , . . . , x k ) , exists and is denoted by (cid:98) ψ ( s ) ν ( t ). Proposition A.1 ([BR10, Theorem 19.1 (p.189), Theorem 19.2 (p.192)]) . For a sufficiently small ν , assume that the probability density q ν ( x ) existsand is bounded. Assume that E [ (cid:107) X (cid:107) s ] < ∞ . Then, sup x ∈ R k (1 + (cid:107) x (cid:107) s ) (cid:12)(cid:12) q ν ( x ) − (cid:98) q ( s ) ν ( x ) (cid:12)(cid:12) = o ( ν s − ) ( ν → . Write x = ( x , x ), x ∈ R k , x ∈ R k ( k + k = k ). Let f ( x ) be | f ( x ) | ≤ C (1 + (cid:107) x (cid:107) s ). x is assumed to be fixed (a constant vector). Corollary A.2.
For s ≥ s + k + 1 and for s ≤ s , (cid:90) R k f ( x ) q ν ( x , x )d x = (cid:90) R k f ( x ) (cid:98) q ( s ) ν ( x , x )d x + o ( ν s − )+ O ( ν s − ) { s q ( s ) ν ( x , x ) − (cid:98) q ( s ) ν ( x , x ) is φ k (( x , x ); Σ) × (a polynomial in ( x , x )) , the integral exists, and the coefficients of the polynomials are multiples of ν s +1 − , . . . , ν s − (if s < s ), or 0 (if s = s ). Hence, the second term is O ( ν s − ) when s < s . (cid:3) To apply Corollary A.2 to Theorems 3.1 and 3.3, let x = ( X, V , R ), x = V , and f ( x ) = { X ≥ x } det( − R + γXI ) (cid:107) V (cid:107) e . Then, k = 1 + d +( d − e )( d − e +1) / k = n − d , and s = d − e + e = d . Note that 1+ s + k =1+ d +1+ d +( d − e )( d − e +1) / ≤ n +1+ n + n ( n +1) / n +1)( n +4) /
2. InTheorem 3.1, we choose s = 4. The choice s = ( n + 1)( n + 4) / n + 1)( n + 4) / ≥ s when n ≥
1. In Theorem 3.3, we can choose s = s ≥ ( n + 1)( n + 4) / Assumption A.1.
The joint density of ( X (0) , ∇ X (0) , R (0)) , where R (0) = ∇ X (0) − γX (0) I n , denoted by p ( x, V, R ) , exists and is bounded. It has the ( n + 1)( n + 4) / -th moments. Assumption A.2.
In addition to Assumption A.1, p ( x, V, R ) has the s -thmoments where s ≥ ( n + 1)( n + 4) / . A.2.3.
Conditions for the Euler characteristic formula.
The regularity con-ditions are summarized in the general (non-isotropic) setting in [AT07, The-orem 12.1.1 (p.302)] or [AT11, Theorem 4.1.1 (p.61)]. For an open set B ⊂ E ⊂ R n , the modulus of continuity is defined as ω ( η ) = max i,j (cid:18) sup s,t ∈ B, (cid:107) s − t (cid:107)≤ η | X ( s ) − X ( t ) | , sup s,t ∈ B, (cid:107) s − t (cid:107)≤ η | X i ( s ) − X i ( t ) | , sup s,t ∈ B, (cid:107) s − t (cid:107)≤ η | X ij ( s ) − X ij ( t ) | (cid:19) . Under the assumption of the existence of the bounded probability density p ( x, V, R ), which will be needed in the asymptotic expansion, the sufficientconditions are summarized as follows. Assumption A.3. (i) X ( · ) , ∇ X ( · ) , and ∇ X ( · ) are a.s. continuous withfinite variances.(ii) There exists a probability density p ( x, y, z ) of ( X (0) , ∇ X (0) , ∇ X (0)) =( x, y, z ) . p ( x, y, z ) is bounded and continuous ∀ x, z and in the neighborhoodof y = 0 .(iii) The joint density of ∇ X | ∂ d E (0) = V = y with ( X, V , R ) =( x, y , z ) given exists, and is uniformly bounded.(iv) The marginal density of Y evaluated at Y = y = 0 , p ( y ) | y =0 , ispositive.(v) E (cid:2) |∇ X (0) | n (cid:3) < ∞ .(vi) Modulus of continuity: For any ε > , P ( ω ( η ) > ε ) = o ( η n ) as η ↓ . Acknowledgments.
The authors are grateful to Shiro Ikeda and TsutomuT. Takeuchi, the organizers of a workshop on Minkowski functionals atIPMU (June 20th, 2017), which initiated this project. They also appreciateSoham Sarkar for his help in developing the prototype R code for calculatingthe Euler characteristic, and Nakahiro Yoshida for his valuable comments.
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