Convergence of Combinatorial Gravity
aa r X i v : . [ g r- q c ] F e b Convergence of Combinatorial Gravity
Christy Kelly ∗ , Carlo Trugenberger , Fabio Biancalana School of Engineering and Physical Sciences, Heriot-Watt University, Edinburgh, UK. SwissScientific, Geneva, Switzerland.February 2021
Abstract
We present a new regularisation of Euclidean Einstein gravity in terms of (sequences of) graphs. In par-ticular, we define a discrete Einstein-Hilbert action that converges to its manifold counterpart on sufficientlydense random geometric graphs (more generally on any sequence of graphs that converges to the manifoldin the sense of Gromov-Hausdorff). Our construction relies crucially on the Ollivier curvature of optimaltransport theory. Our methods also allow us to define an analogous discrete action for Klein-Gordon fields.These results may be taken as the basis for a combinatorial approach to quantum gravity where we seek togenerate graphs that approximate manifolds as metric-measure structures.
While Gauss discovered intrinsic geometry with his theorema egregium in 1827, and Riemann intuited mani-folds in the 1850s, it was not until the 1930s that a more or less modern mathematical definition of a differentiablemanifold was made [33, 69, 82]; also c.f. [89, 91] for important contributions to this line of development and,e.g. [15, 75] for more recent reviews. The intervening period saw the development of general relativity byEinstein (and others) [30] which was a fantastic corroboration of Gauss’ original intuition—exemplified by hisfamous measurement of the large Brocken-Hohehagen-Inselberg triangle—that the geometry of space was amatter of empirical determination. Indeed the general programme of relativity theory surely represents one ofthe highpoints of the intersection between physics and geometry in the modern period.Recently there has been active mathematical research in an area which might be loosely—and somewhatparadoxically—called discrete differential geometry examining discrete analogues of smooth notions, drivenby applications in computer science—especially computer graphics and mesh processing [18, 24, 25], but alsomore loosely as a natural extension of methods of discrete topology (discrete Morse theory and combinatorialalgebraic topology [53, 73]) to e.g. topological data analysis [22]—network geometry [17, 19, 59] and quan-tum gravity. Indeed, as a quite general principle, it is desirable to find coarse formalisms for gravity sincequantum fluctuations of spacetime are expected to ruin smooth structure. In approaches where spacetime isfundamentally discrete, this coarseness obviously must be promoted to full discreteness [65]; even without funda-mental (physical) discreteness, however, it may nonetheless be desirable to formulate a discrete—and not simplycoarse—approximation of gravity as a nonperturbative regularisation of the smooth theory in the context of theasymptotic safety scenario [29]. Indeed, in a gravitational context, the use of discrete methods dates back to atleast Regge’s classic paper [68] where he introduced the eponymous calculus that allowed for the calculation ofcurvature in terms of deficit angles on manifold triangulations. Regge’s somewhat heuristic account has beenrigorously confirmed by Cheeger, M¨uller and Schrader in their demonstration of the convergence of curvature onsuitable triangulations in a manifold [23]. This has led to the development of simplicial formalisms for quantumgravity [39], the foremost of which is the dynamical triangulations approach [7, 11, 55].The Euclidean dynamical triangulations approach[7]—often in the form of a matrix model [32]—was assidu-ously pursued in the 1980s and 1990s in two-dimensions, after it became clear that the scaling limit of this modelwas quantum Liouville theory [26, 27, 52]; following an observation of Polyakov, this made it simultaneosuly aregularisation of 2D-gravity coupled to conformal matter and of noncritical string theory [66]. Taking a gravi-tational interpretation, it was realised in the process of this work that the emergent structure of 2 D -quantumEuclidean spacetime was that of a Brownian sphere , a topological 2-sphere with Hausdorff dimension 4 andspectral dimension 2 [5, 8]. These findings have recently been placed on a firm mathematical footing: thereis a large body of literature in this direction, but most relevant results are referred to in [58] which focuseson the key proof of the equivalence between quantum Liouville gravity and the Brownian sphere; for rigorousspectral dimension results see [38, 54]. In other dimensions, however, the Euclidean dynamical triangulations Historical honesty demands that we note that this was probably never intended as a test of spacetime geometry, the Euclideanstructure of which, it seems, Gauss never seriously doubted. He of course recognised the logical possibility of non-Euclideangeometries and remarked that the measurement could be regarded as a corroboration of the Euclidean nature of space, but it seemsthe measurement was required for more mundane reasons relating to the geodetic survey of Hanover Gauss had been commissionedto carry out. See [20] for more details. branched polymerphase of topological dimension 1, Hausdorff dimension 2 and spectral dimension 4 / causal dynamical triangulations [11, 55] programme appears to do much better in this regard. In thisapproach one typically fixes not only the topology of spacetime but also a preferred foliation. These modelstypically have a rich phase structure and have much improved behaviour with regards to the structure of thescaling limit [4, 9, 10, 12, 28].An alternative approach to quantum gravity that also makes much of the causal structure of spacetime is causal set theory ; see [79] for a recent review. The basic insight is that geometric structures on spacetime maybe encoded in causal relations and related topologies, an insight that in modern form dates back to at leastZeeman in the 1960s, who showed that the Lorentz group (augmented by dilatations) was the group of causalautomorphisms on Minkowski space and found a suitable topology to encode this information [93, 94]. A decadelater, these results were vastly generalised by Hawking, King and McCarthy [41] and Malament [57] who showedthat causal structure on a Lorentzian manifold encodes both the differential and conformal structure. This workmotivates the causal set Hauptvermutung which purports that any Lorentzian manifold should be characterisedby an essentially unique causal set, i.e. a locally finite poset describing the causal structure of spacetime.A significant development in this regard is the demonstration that the
Benincasa-Dowker action —essentiallydefined by counting short causal chains in a causal set—converges to the Einstein-Hilbert action on causal setsgenerated by Poisson processes in Lorentzian manifolds (‘sprinklings’) [14].Returning to a Euclidean setting, a major recent development in Riemannian geometry using ideas fromoptimal transport theory [83, 84] has been a synthetic characterisation of Ricci curvature using the metric-measure structure of the manifold [56, 59, 61, 62, 63, 76, 77, 78, 84]: a Riemannian manifold M is a metricspace when equipped with the geodesic distance ρ ; it is made into a metric-measure space when equipped witha random walk, i.e. a probability measure at each point. One natural choice of random walk is the uniformrandom walk at scale δ , i.e. for each point p ∈ M we have the measure µ δp ( E ) = vol( B M δ ( p ) ∩ E )vol( B M δ ( p )) , (1)where vol is the natural volume measure on M and B M δ ( p ) is the open ball of radius δ centred at p . Thisinsight has allowed for the definition of coarse notions of curvature valid in generic metric-measure spaces. Onenotion due to Sturm [76] and independently Lott and Villani [56] is related to the so-called L -transport cost ;this is perhaps the canonical mathematical example of a synthetic curvature, but is ill-equipped for use indiscrete spaces. On the other hand an alternative notion due to Ollivier [62, 63, 64] is well-defined for discretemetric-measure spaces and indeed has widely gained traction for a variety of applications in the network theorycommunity: c.f. e.g. [31, 45, 60, 67, 71, 72, 74, 85, 86, 87, 88, 90]. The Ollivier curvature in a (completeseparable) metric space ( X, ρ X ) with random walk { µ x } x ∈ X is defined as κ X ( x, y ) = 1 − W ( µ x , µ y ) ρ X ( x, y ) , (2)where W is the L -transport cost or Wasserstein distance , a metric on the space of Borel probability measureson X . This definition captures the intuition that in a positively curved space, the average distance betweentwo nearby balls will be closer than their centres. This entire line of development is a natural extension of theprogramme of metric geometry [21, 36], a coarse generalisation of many ideas in Riemannian geometry usingthe fact that many results of the smooth theory rely only on its structure as a metric space or length space.The precise relation between the Ollivier and Ricci curvatures is as follows: let M be a Riemannian manifoldof dimension D , and let p and q ∈ M be points of M such that ρ M ( p, q ) = ℓ ; for ℓ sufficiently small, there is aunique unit-speed geodesic between p and q which has tangent vector V at p . Then: κ δ M ( p, q ) = δ Ric p ( V, V )2( D + 2) + O ( δ ( δ + ℓ )) (3)where Ric p is the Ricci tensor at p , and κ δ M is the Ollivier curvature of M ; the superscript δ indicates that eachpoint p ∈ M is equipped with the uniform measure µ δp (equation 1).In the context of Euclidean gravity the potential ramifications are clear: the Ollivier curvature may be usedto specify a new regularisation of Euclidean Einstein gravity in terms of discrete structures such as graphs,regarded as discrete metric spaces; we have begun to pursue this line of thought in [47, 48, 81], calling theassociated Ollivier curvature based model combinatorial quantum gravity due to a formal analogy with Einstein2ravity. Moreover combinatorial quantum gravity appears to have a continuous phase transition between a phaseof random graphs and a phase evincing emergent geometric structure. Gorsky and Valba [35] have recentlyexamined an approximation to combinatorial quantum gravity—studied briefly in [47]—and showed that theresulting model displays quite a different phase structure; indeed in this approximate model there is a first-ordertransition between a random graph phase and a phase characterised by weakly interacting hypercubes. Otherthan combinatorial quantum gravity, Loll and Klitgaard have recently advocated the use of a quantum Riccicurvature , inspired by the Ollivier curvature, as a quasilocal observable in nonperturbative quantum gravity[49, 50, 51]; Gorard has also advocated the use of the Ollivier curvature in the context of Wolfram’s recent workon graphs [34, 92].There are some important caveats, however, to the simple application of Ollivier curvature as a direct graph-theoretic regularisation of the Ricci curvature: the metric-measure structure of a graph is certainly not equal tothe metric-measure structure of a Riemannian manifold in general, and as such the Ollivier curvature evaluatedin a graph will typically be quite different from the Ollivier curvature evaluated in a manifold. The upshot, ofcourse, is that combinatorial quantum gravity as defined in [47, 48, 81] does not converge to classical gravityin general; note that we believe these models may nonetheless have something relevant to say about Euclideanquantum gravity because we obtain useful results on the Ricci flat sector, where the action of combinatorialquantum gravity does agree with the classical action. Since the Einstein-Hilbert action is defined as the integralover spacetime of the scalar curvature R , it is clear that even if the difference between graph and manifoldmetric-measure structures can be overcome, specifying a convergent model of combinatorial gravity will involveboth approximating spacetime integrals (via graph vertex sums) and finding some way of carrying out a traceat each point—since R = tr(Ric).An important development in this direction was the recent demonstration that the Ollivier curvature ofa random geometric graph in a Riemannian manifold converges to the Ollivier curvature of the underlyingmanifold [42, 43]. The essential idea is that for a sufficiently dense sampling of a Riemannian manifold M ,a random geometric graph G will approximate M as a metric space. Furthermore it can be shown that theuniform measures: m δu ( E ) = | B Gδ ( u ) ∩ E || B Gδ ( u ) | , (4) u ∈ G , approximate the uniform measure µ δu as defined in 1, where this means that the Wasserstein distance W M ( µ δu , m δu ) is small. In this way the associated Ollivier curvature κ δG ( u, v ) should converge to the manifoldOllivier curvature κ δ M . Note that we are identifying the graph G ∼ = ι ( G ) ⊆ M where ι : G ֒ → M is the naturalimbedding of G into M .While this is essentially sufficient for a kinematic characterisation of a Riemannian manifold in terms ofa graph, the key step showing that the Wasserstein distance W M ( µ δu , m δu ) is small relies heavily on the factthat random geometric graphs are obtained via Poisson processes in Riemannian manifolds, since it involvesan extension of certain probabilistic grid matching results due to Talagrand and others; see [80] for a review ofthese results. In a dynamical model of random graphs with no underlying manifold—or at least the underlyingmanifold indeterminate—the convergence result for random geometric graphs thus does not suffice. Moreover,the construction of van der Hoorn et al. picks out somewhat more than necessary an initial point and tangentvector, and from a gravitational perspective it is desirable to have a more background independent proof.The crucial notion we need to extend the convergence result above is that of the Gromov-Hausdorff distance [21, 36]. This is a metric ρ GH on the space of isometry classes of compact metric and so defines a notion ofconvergence between compact metric spaces. In particular, any length space—metric space where the distancebetween two points is the shortest length of an admissible curve between those points—is obtained as theGromov-Hausdorff limit of a graph. For our purposes it is sufficient to treat the Gromov-Hausdorff distance be-tween two compact metric spaces between two compact metric spaces in terms of near isometries . In particular,let ( X, ρ X ) be a metric space; for any A ⊆ X the ε -thickening of A is defined as the set A ε = [ a ∈ A B Xε ( a ) . (5)Then a set A ⊆ X is said to be an ε -net in A iff X ⊆ A ε . Now let ( X, ρ X ) and ( Y, ρ Y ) be metric spaces; wedefine the distortion of a map f : X → Y as the quantitydis( f ) = sup ( x ,x ) ∈ X × X | ρ Y ( f ( x ) , f ( x )) − ρ X ( x , x ) | . (6)An ε -isometry between X and Y is a mapping f : X → Y such that dis( f ) < ε and f ( X ) is a ε -net in Y .The connection between near isometries and the Gromov-Hausdorff distance is that there is a O ( ε )-isometrybetween two compact metric spaces ( X, ρ X ) and ( Y, ρ Y ) iff the Gromov-Hausdorff distance between X and Y is O ( ε ). 3he Gromov-Hausdorff metric gives a canonical sense in which we may interpret the ‘proximity’ of com-pact metric spaces, and we say that a graph approximates a (compact) manifold M as a metric space if theGromov-Hausdorff distance between them is small. A priori it is not clear that Gromov-Hausdorff proximityimplies anything more than metric closeness. Indeed, Ollivier used a notion of measured
Gromov-Hausdorffconvergence in which the Ollivier curvature respected limits; the key point to note was that the stability of theOllivier curvature under the limit required both metric and measure theoretic proximity, while the latter is notguaranteed for general Gromov-Hausdorff convergence. From this perspective, the point-wise convergence resultof van der Hoorn et al. amounts to a demonstration of measured Gromov-Hausdorff convergence on appropriatesequences of random geometric graphs. This involves showing, firstly, that the random geometric graphs doindeed converge to the underlying manifold in the sense of Gromov-Hausdorff and secondly showing that thediscrete and continuous measures are sufficiently similar to ensure measured Gromov-Hausdorff convergence;specifically, if ℓ is the distance between the points u and v and δ is the radius of the balls defining the measures m δu etc. then sufficiently similar reduces to the requirement that: W M ( ι ∗ m δu , µ δι ( u ) ) ≪ δ ℓ (7)and similarly for v . In particular, the semidiscrete discrepancy needs to be smaller than third-order in smallterms δ and ℓ . Note that for random geometric graphs it is possible to choose δ = ℓ .In a more general dynamical context we can imagine a sequence of graphs G N → M in the sense of Gromov-Hausdorff; for N sufficiently large, the natural imbedding ι may be replaced by an arbitrary ε ( N )-isometry ι ( N ) and G N . Strengthening Gromov-Hausdorff convergence to measured Gromov-Hausdorff convergence onceagain involves, mutatis mutandis , the demonstration of 7. However, in this context there is no guarantee thatthe grid-matching results of Talagrand and others maintain and other methods must be used to show that thesemidiscrete discrepancy is small. It turns out that this remains possible if we use Euclidean semidiscrete optimaltransport results [40] and pushforward these Euclidean results to the manifold setting using the exponentialmap; we describe precisely how this works in more detail in appendix A. It appears, however, that a price mustbe paid for working in this more general context viz. we seem to need δ ≪ ℓ .We now seek to extend this result from point-wise convergence of the Ollivier curvature to the convergenceof a discrete Einstein-Hilbert action. It will be sufficient to prove two lemmas: Lemma 1.
Let M be a Riemannian D -manifold and let A ⊆ M be a ε -net in M . For any smooth function f : M → R , let g : A → R be a function such that | f ( a ) − g ( a ) | = O ( σ ) . (8) Then for ε = N − /D sufficiently small we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z M d vol M ( x ) f ( x ) − N X a ∈ A ω D g ( a ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = O (cid:16) max (cid:16) σ, N − D (cid:17)(cid:17) (9) where ω D is the volume of the unit D -ball in R D . The proof of this lemma is essentially a direct application of the definition of integration on manifolds alongwith the Taylor theorem; see, e.g. [46]. In particular, noting that the set { B M ε ( a ) } a ∈ A is an open cover of M ,we may write Z M d vol M ( x ) f ( x ) = X a ∈ A Z B M ε ( a ) d vol M ( x ) ρ a ◦ f ( a ) (10)where { ρ a } a ∈ A is a partition of unity such that ρ a ( a ) = 1 for all a ∈ A . Taylor-expanding the integrand andnoting that vol M ( B ε ( a )) = ε D ω D (1 + O ( ε )) (11)then essentially proves the desired result. This lemma allows us to approximate integrals over a manifold by asum over an ε -net.The second fundamental lemma involves taking the trace; this is required since the integrand of the Einstein-Hilbert action is the scalar curvature R = tr(Ric), while the Ollivier curvature allows us to approximate Ric.Specifically we have Lemma 2.
Let M be a Riemannian manifold, let A ⊆ M be an ε -net in M . For some a ∈ A , let T : T a M × T a M → R be a bilinear form on the tangent space T a M and let κ be a partial functionon M × M such that for some sufficiently small ℓ, r > with r ≪ ℓ we have | T ( V, V ) − κ ( a, b ) | = O ( σ ) (12)4 ll b ∈ S M ℓ,r ( a ) ∩ A where S M ℓ,r ( a ) := B M ℓ + r ( a ) /B M ℓ − r ( a ) and where V is the normalised velocity of the uniquegeodesic connecting a and b . Let n := | S M ℓ,r ( a ) ∩ A | and define d : S D − ⊆ R D → R as the scale-factor of thespherical volume element: d ( ϕ , ..., ϕ D − ) = sin D − ( ϕ ) sin D − ( ϕ ) · · · sin( ϕ D − ) . (13) Then if we define: tr a M ( T ) := 2 π D − ω D n X b ∈ S M ℓ,r ( a ) ∩ A d (cid:18) exp − a ( b ) || exp − a ( b ) || (cid:19) κ ( a, b ) , (14) we have | tr ( T ) − tr a M ( T ) | = O (cid:16) max (cid:16) σ, n − D − (cid:17)(cid:17) (15) as long as ε ≪ ℓn − D − − ℓ . (16)We present a full proof of this lemma in appendix B. The idea is that we may approximate the trace bysumming over a uniform grid of points in the sphere S D − ⊆ R D ; we then show that we may transport thepoints in the pushforward of this grid of points under the exponential map to the set A ∩ S M ℓ,r ( a )) for any a ∈ A for ℓ > Theorem 3.
Let G be a graph with N vertices, let M be a Riemannian D -manifold and let ι : G → M be a ε ( N ) -isometry. Let ε ( N ) = N − D δ ( N ) := N − a ℓ ( N ) := N − b r ( N ) := N − c , (17) and let S Gℓ ( N ) ,r ( N ) ( u ) := B Gℓ ( N )+ r ( N ) ( u ) /B Gℓ ( N ) − r ( N ) ( u ) for all u ∈ G ; also let us define: A f dis ( G ; δ, ℓ ) = ω D D X u ∈ G f π D − ( D + 2) ω D δ ( N ) | S Gℓ ( N ) ,r ( N ) ( u ) | X v ∈ S Gℓ ( N ) ,r ( N ) ( u ) κ δG ( u, v ) , (18) where ω D is the volume of the D -dimensional Euclidean unit ball. For N sufficiently large, any choice of numbers a, b > such that aD + bD < < aD D + ( D − b < b < a, c < D (19) ensures that (cid:12)(cid:12)(cid:12) A f dis ( G ; δ, ℓ ) − A f ( M )) (cid:12)(cid:12)(cid:12) = O (cid:16) max (cid:16) N − a (3+2 b ) , N − − ( c +( D − b ) D − (cid:17)(cid:17) (20) holds and is small for sufficiently large N , where A f ( M ) = Z M d D xf ( R ) . (21)We prove this statement in appendix C below. As an immediate corollary we thus have: Corollary 4.
For any space of (weighted) graphs Ω , there is a discrete f ( R ) -gravity action A f dis : Ω → R givenby equation 18 such that A f dis ( ω N ) → A f ( M ) for any sequence of graphs { ω N } N ∈ N + ⊆ Ω such that | ω N | = N and ω N → M sufficiently rapidly in the sense of Gromov-Hausdorff. There are four intrinsic scales in this problem: the Gromov-Hausdorff error ε , the radius δ of the balls definingthe the uniform measures in G and M , the distance ℓ between points on which we evaluate the curvature andthe characteristic thickness of the shell r in which we evaluate the trace; note that it is possible to identify δ and r , though there is perhaps no particular reason to do so. The three distinct scales ε , δ and ℓ are requiredto guarantee convergence at the level of Ricci curvature, while the fourth scale r is needed to obtain the scalar curvature, i.e. to guarantee the convergence of the trace as per lemma 2. The precise dependence of ε on thenumber of points in the graph N is fixed by lemma 1. The basic idea of the main theorem 3 is then that if5e have a sequence of graphs G N → M in the sense of Gromov-Hausdorff sufficiently quickly, i.e. such that ρ GH ( G N , M ) < ε ( N ) = N − /D for all sufficiently large N , then we may pick scales δ ( N ) and ℓ ( N ) such that A f dis ( G ) → A f ( M ).Now, recall that the Euclidean Klein-Gordon Lagrangian density is L KG ( ϕ, d ϕ ) = 12 tr(d ϕ ⊗ d ϕ ) + 12 mϕ . (22)If we have points a, b ∈ M such that ρ M ( a, b ) < δ for δ sufficiently small, then we have a unit tangent vector U at a for the unique geodesic connecting a and b ; by definition this satisfies: (cid:12)(cid:12)(cid:12)(cid:12) d ϕ ( U ) − ϕ ( b ) − ϕ ( a ) ρ M ( a, b ) (cid:12)(cid:12)(cid:12)(cid:12) = O ( δ ) . (23)The following then follows immediately from an application of lemmas 1 and 2: Proposition 5.
Let ω n → M in the sense of Gromov-Hausdorff as above and let f n : ω n → R be a sequence offunctions that converges pointwise to a smooth function ϕ . If ϕ is a Klein-Gordon field, i.e. if it extremises theKlein-Gordon action A KG , then there is a discrete Klein-Gordon action A DKG such that A DKG ( f n ) → A KG ( ϕ ) . We have thus shown that we may approximate gravitational and free scalar dynamics with graphs that areproximal as metric spaces to Riemannian manifolds. What then of the criterion of Gromov-Hausdorff proximity?In a recent paper we have defined a statistical model of random 3-regular graphs using the Ollivier curvaturein which we observe the generation of the smooth 1-dimensional compact geometry without boundary viz. thecircle, as the Gromov-Hausdorff limit of classical (action minimising) configurations of the model. In this sensewe have shown that it is possible to (spontaneously) obtain random graphs that are Gromov-Hausdorff proximalto continuum geometries, in related models of random graphs using the Ollivier curvature. Note that while themodel in [48] does not converge in general, it agrees with our model on the Ricci flat sector; this is preciselythe sector where we obtain our positive results in [48] related to the generation of random flat geometries andin relation to the existence of a second-order phase transition. Together with the numerical analysis of [47, 81],this amounts to evidence for the claim that nonconvergent combinatorial quantum gravity defines a UV-fixedpoint in line with the asymptotic safety scenario. It remains to be seen whether this apparent UV-fixed pointpersists in more general convergent models.As mentioned previously, Gorsky and Valba [35] have recently studied an approximation of the nonconvergentmodel of combinatorial quantum gravity introduced in [47, 81] and have found quite different results for thephase structure and nature of the observed phase transition. Moreover, Gorsky and Valba claim that the modelstudied in [35] is the same as the model we consider in [47, 48]. It is worth stressing that this apparent conflictis in fact the result of a misunderstanding: Gorsky and Valba study a model based on the action: A MF ( G ) = 8 k (cid:18) k ( k − N − (cid:3) ( G ) (cid:19) (24)where G is a k -regular bipartite graph (i.e. each vertex of G has k neighbours and G has no odd cycles) with N vertices satisfying a constraint we have dubbed the hard core or independent short cycle constraint; (cid:3) ( G )denotes the number of squares in G . In fact we also briefly examine this model in section 3.3.1 of [47], and toa certain extent in [81] (though we shall have more to say about the role of this model in [81] shortly). Ourintention in [47, 48, 81] has always been to study a model A ( G ) = − X u ∈ G X v ∼ u κ G ( u, v ) (25)where u ∼ v denotes that v neighbours u and κ G is the Ollivier curvature of the graph G , but for contingentreasons we have only made this action the starting point of our paper [48]. To be sure in [47] we ultimatelydiffer from 25 by a factor of the degree k , which we take to be a fixed parameter, and as such the differenceis essentially negligible. In [81] the aim is to derive the action 25 from the mean field action 24, though this‘derivation’ follows from an error that we detail below. In particular, we know that in bipartite k -regular graphssatisfying the independent short cycle condition we have κ G ( u, v ) = − k [( k − − (cid:3) uv ] + (26)for any edge uv ∈ E ( G ), where (cid:3) uv denotes the number of squares supported on the edge uv and [ α ] + :=max( α,
0) for any α ∈ R . Note the subscript + in the expression; this somewhat innocuous symbol will havesurprisingly significant ramifications for the behaviour of our model. It is required because it expresses the6onexistence of certain transport plans at sufficiently high clustering. It turns out that by taking into accountthe expression 26, the action 25 can be expressed: A ( G ) = A MF ( G ) + X uv ∈ P ( G ) ( (cid:3) uv − ( k − P ( G ) = { uv ∈ E ( G ) : (cid:3) uv > ( k − } . (28)That is to say the model studied by Gorsky and Valba comes from combinatorial quantum gravity under theapproximation that P ( G ) = ∅ . In [47], we referred to this as a mean field approximation because the term A MF depends only on global quantities. This mean field approximation is also equivalent to ignoring the subscript+ in the expression 26 (compare with equation 2.6 of [35]).The basic idea is that A MF promotes the formation of squares; at fixed N there is nothing to opposethis tendency in A MF and in this sense it is not surprising that there is a first-order transition to a phasecharacterised by separated (if we allow configurations to be disconnected) or weakly interacting (if we demandthat configurations remain connected) hypercubes, since the latter maximise the number of squares per edge atgiven degree. Indeed in [47] we observed separated hypercubes, which we somewhat inappropriately referred toas baby universes : compare figure 5 of [47] with figure 4 of [35]. On the other hand, the term defined as a sumover P ( G ) acts in opposition to A MF , once the degree of clustering reaches a certain point. It is this whichensures that the for k = 2 D , configurations of in the limit β → ∞ are approximately locally isomorphic to Z D and is also probably responsible for ensuring that the phase transition becomes second-order.The responsibility for this misunderstanding surely lies with us. In [81] it was—mistakenly—believed thatthe hard core condition was sufficient to ensure that the subscript + in equation 26 could be neglected, i.e.that the maximum number of squares on an edge in a k -regular graphs was ( k −
2) and that the mean fieldapproximation was generally valid; that this is not true is easily seen by examples such as the tesseract, a4-regular graph with 3 > k − β → ∞ phase has distracted fromthe fact that the action studied in [47] does not correspond to the action 25. The situation in [81] is even moreunfortunate; the difference between the mean field and exact action only becomes relevant for configurationsthat are highly clustered on average. However, in [81] simulations began with toroidal graphs at β = ∞ andinvestigated the behaviour of the model as β was decreased ; as β is decreased we also expect clustering todecrease and we expect both actions 25 and 24 to behave similarly. It is for this reason that both the weaklyinteracting hypercubes and the hysteresis effects observed by Gorsky and Valba were not observed in [81]. A The Semidiscrete Discrepancy
In this section we show that it is possible to choose scales ε , δ and ℓ such that the inequality 7 is satisfied; weassume that the reader is familiar with the basic requisite definitions of optimal transport theory: c.f. e.g. [83,84]. Let M be a Riemannian manifold, A ⊆ M a ε -net in M and fix a point a ∈ A ; we assume that ε ≪ δ ℓ .The required inequality 7 follows as long as W M ( m δa , µ δa ) ≪ δ ℓ. (29)The discrete measure m δa is concentrated on the set A a ( δ ) := A ∩ B M δ ( a ), while the support of µ δa is the closureof B M δ ( a ).To verify this inequality we begin by considering the seemingly unrelated problem of calculating W R D ( m Q , λ δD ) λ δD = λ D / (2 δ ) D is the uniform measure on ( − δ, δ ) D — λ D is the standard Lebesgue measure on R D —and m Q isthe empirical measure defined by a set of points Q ⊆ ( − δ, δ ) D . Known Euclidean semidiscrete optimal transportresults tell us that if we choose Q to be a uniform grid in R D , there is a (deterministic) optimal transport plan ξ from m Q to λ δD given by sending the entire mass in each Voronoi cell V q about each point q ∈ Q to q [40].Each Voronoi cell has volume O ( | Q | − ) while the mass is moved a maximum distance O ( R ) , R := | Q | − /D , sowe have W R D ( m Q , λ δD ) = T ρ R D ( ξ ) ≤ X q ∈ Q O ( | Q | − ) · O ( R ) = O ( R ) , (30)where we assume that R ≪ δ ℓ . Now let Q ( δ ) := Q ∩ B R D δ (0) and let V ( δ ) := S q ∈ Q ( δ ) V q . Since λ D (( V ( δ ) /B R D δ (0)) ∪ ( B R D δ (0) /V ( δ ))) λ D ( B R D δ (0)) ≤ λ D ( B R D δ + O ( R ) (0)) − λ D ( B R D δ −O ( R ) (0)) λ D ( B R D δ (0)) = O ( R ) , (31)7he optimal transport plan ξ induces a transport plan ξ δ from m Q ( δ ) to µ δ where m Q ( δ ) is the empirical measureon the set Q ( δ ) and µ δ is the uniform measure on B R D δ (0), up to a negligible error of order O ( R ) arising fromboundary defects. The transport plan ξ δ again has cost O ( R ) so we again have W R D ( m Q ( δ ) , µ δ ) = O ( R ) . (32)To make these Euclidean results relevant to the manifold context we need to pushforward with the expo-nential map. In particular,(exp a ) ∗ m Q ( δ ) = m exp a ( Q ( δ )) µ δa = (exp a ) ∗ µ δ (33)where m exp a ( Q ( δ )) is the empirical measure in M defined by the set exp a ( Q ( δ )). It turns out that for anyEuclidean transport plan ξ ∈ Π( m Q ( δ ) , µ δ ), the pushforward(exp a ) ∗ ξ := (exp a × exp ) ∗ ξ (34)is a transport plan from m exp a ( Q ( δ )) to µ δa . Moreover, recalling that in a manifold with bounded sectionalcurvature (i.e. in any compact manifold) we have the expansion ρ M (exp a ( x ) , exp a ( y )) = ρ R D ( x, y ) (cid:16) O (cid:16) { | x | , | y | } (cid:17)(cid:17) (35)where { | x | , | y | } denotes terms that are second-order in products of | x | and | y | , we note that W M ( m exp a ( Q ( δ )) , µ δa ) = inf ζ ∈ Π( m exp a ( Q ( δ )) ,µ δa ) Z B M δ ( a ) × B M δ ( a ) d ζ ( x, y ) ρ M ( x, y ) ≤ inf ξ ∈ Π( m Q ( δ ) ,µ δ ) Z M×M d(exp a ) ∗ ξ ( x, y ) ρ M ( x, y )= inf ξ ∈ Π( m δ ,µ δ ) Z B R Dδ (0) × B R Dδ (0) d ξ ( x, y ) ρ R D ( x, y ) (cid:16) O (cid:16) { | x | , | y | } (cid:17)(cid:17) ≤ W R D ( m Q ( δ ) , µ δ ) (cid:0) O ( δ ) (cid:1) = O ( R ) . (36)Since A is a ε -net in M , then up to negligible adjustments due to boundary defects, we may assume thateach element of exp a ( Q ( δ )) lies within a distance ε of some element of A a ( δ ). At the same time, the expansion35 ensures that the distortion of the exponential map on a geodesic ball is controlled to third order by the radiusof the ball: dis(exp a | B R D δ (0)) = O ( δ ) . (37)Thus if δ + ε ≪ R, (38)each element of exp a ( Q ( δ )) lies within a distance ε of a unique element of A a ( δ ) because the distance betweentwo points of exp a ( Q ( δ )) is then of order O (2( R − δ )) ≫ ε . By choosing Q such that n := | Q ( δ ) | = | A a ( δ ) | , wethen have a bijective correspondence between the elements of exp a ( Q ( δ )) and A a ( δ ). This defines a deterministictransport plan from m δa to m exp a ( Q ( δ )) with cost O ( ε ). Hence, using the subadditivity of the metric, we have W M ( m δa , µ δa ) ≤ W M ( m δa , m exp a ( Q ( δ )) ) , + W M ( m exp a ( Q ( δ )) , µ δa ) = O (max( ε, R )) . (39)Both R and ε ≪ δ ℓ by assumption so the inequality 29—and hence the inequality 7—is satisfied.It remains to show that it is possible to choose Q such that R ≪ δ ℓ , δ + ε ≪ R and n = | Q ( δ ) | = | A a ( δ ) | .Clearly, up to a positive number less than 1 we have, | Q ( δ ) | = ω D | Q | where ω D is the volume of the unitEuclidean D -ball, so we choose | Q | = n/ω D . Then n − /D = O ( R ). Since—up to negligible boundary defects—the set S b ∈ A a ( δ ) B ε ( b ) covers B M δ ( a ), n satisfies δ D = O ( nε D ) , (40)i.e. n − /D = O ( ε/δ ); thus the constraints R ≪ δ ℓ and δ + ε ≪ R are satisfied if ε ≪ δ ℓ and δ + εδ ≪ ε .Since δ ≪ εδ ≪ ε and the latter constraint reduces to δ ≪ ε , i.e. the inequality 7 is satisfied as long as δ ≪ ε ≪ δ ℓ (41)are both satisfied; together these imply δ ≪ ℓ . These constraints are clearly satisfied given the specification 17if the scale constraints 19 are satisfied. 8 Approximating the Trace
In this section we prove lemma 2. There are essentially two steps: first we show that we may approximate thetrace by summing over a uniform grid U of points in the Euclidean unit ( D − S D − ; this is a discreteanalogue of the fact that tr( T ) = 1 ω D Z S D − d vol S D − ( u ) T ( u, u ) (42)for any bilinear form T : R D × R D → R . We then push this result forwards with the exponential map to obtaina result valid in manifolds. B.1 Traces via Sums in Euclidean Spheres
Consider the following set-up: let T : R D × R D → R be a bilinear form, for any sufficiently large positive integer n , let U ⊆ S D − be a set with 2 m D − -points where m := ⌊ ( n/ / ( D − ⌋ , and let κ : U → R be a mapping suchthat | T ( u, u ) − κ ( u ) | = O ( σ ) . (43)For any function d : S D − → [ − ,
1] let us definetr d κ ( T ) = 2 π D − ω D n X u ∈ U d ( u ) κ ( u ) . (44)We wish to show that for appropriate choices of U and d we have (cid:12)(cid:12) tr d U ( T ) − tr( T ) (cid:12)(cid:12) = O (cid:16) max (cid:16) σ, n − D − (cid:17)(cid:17) (45)Let us begin by definingtr d U ( T ) = π D − ω D m D − X u ∈ U d ( u ) κ ( u ) Tr d U ( T ) = π D − ω D m D − X u ∈ U d ( u ) T ( u, u ) . (46)Then, by subadditivity we have: (cid:12)(cid:12) tr d U ( T ) − tr( T ) (cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) tr U ( T ) − tr d U ( T ) + tr d U ( T ) − Tr d U ( T ) + Tr d U ( T ) − tr( T ) (cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12) tr U ( T ) − tr d U ( T ) (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) tr d U ( T ) − Tr d U ( T ) (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) Tr d U ( T ) − tr( T ) (cid:12)(cid:12)(cid:12) . (47)But noting that 2 m D − ≤ n ≤ m + 1) D − = 2 m D − + O ( m D − ) , (48)we see that (cid:12)(cid:12)(cid:12) tr d U ( T ) − tr d U ( T ) (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) π D − ω D n X u ∈ U d ( u ) κ ( u ) − π D − ω D m D − X u ∈ U d ( u ) κ ( u ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = tr d U ( T ) (cid:12)(cid:12)(cid:12) n m D − − (cid:12)(cid:12)(cid:12) = O ( m − ) . (49)Similarly, by subadditivity (cid:12)(cid:12)(cid:12) tr U ( T ) − tr d U ( T ) (cid:12)(cid:12)(cid:12) = π D − ω D m D − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X u ∈ U d ( u )( κ ( u ) − T ( u, u )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ π D − ω D m D − X u ∈ U | d ( u ) | · | κ ( u ) − T ( u, u ) | = O ( σ ) . (50)9hus (cid:12)(cid:12) tr d U ( T ) − tr( T ) (cid:12)(cid:12) ≤ O (max( σ, n − / ( D − )) + (cid:12)(cid:12)(cid:12) Tr d U ( T ) − tr( T ) (cid:12)(cid:12)(cid:12) (51)We thus turn to the calculation of (cid:12)(cid:12)(cid:12) Tr d U ( T ) − tr( T ) (cid:12)(cid:12)(cid:12) ; this requires a specific choice of U and d . We begin byspecifying U : let µ = ( µ , ..., µ D − ) be a multi-index with µ , ..., µ D − ∈ { , ..., m } and µ D − ∈ { , ..., m } .We shall let U = { u µ } , where in spherical coordinates we have: u µ = (cid:16) µ πm , µ πm , · · · , µ D − πm , µ D − π m (cid:17) . (52)Note that we have exactly one element of U for each multi-index µ while there are clearly 2 m D − differentmulti-indices µ so | U | = 2 m D − as required. If { e k } ≤ k ≤ D is the standard basis for R D , the assignment 52 for u µ is equivalent to the specification u µ = D X k =1 u kµ e k (53a) u kµ = sin (cid:16) µ πm (cid:17) sin (cid:16) µ πm (cid:17) · · · sin (cid:16) µ k − πm (cid:17) cos (cid:16) µ k πm (cid:17) (53b) u Dµ = sin (cid:16) µ πm (cid:17) sin (cid:16) µ πm (cid:17) · · · sin (cid:16) µ D − πm (cid:17) sin (cid:16) µ D − πm (cid:17) (53c)where k ∈ { , ..., D − } . This implies thatTr d U ( T ) = π D − ω D m D − X µ d ( u µ ) T ( u µ , u µ )= π D − ω D m D − X µ D X k =1 D X ℓ =1 u kµ u ℓµ T ( e k , e ℓ )= π D − ω D m D − X k = ℓ T ( e k , e ℓ ) X µ d ( u µ ) u kµ u ℓµ + π D − ω D m D − D X k =1 T ( e k , e k ) X µ d ( u µ )( u kµ ) (54)Thus the problem reduces down to solving: σ ( k, ℓ ) = π D − m D − X µ d ( u µ ) u kµ u ℓµ σ ( k ) := σ ( k, k ) . (55)We now specify d ; in particular we take it to be scale factor of the spherical volume element: d ( u µ ) = sin D − (cid:16) µ πm (cid:17) sin D − (cid:16) µ πm (cid:17) · · · sin (cid:16) µ D − πm (cid:17) . (56)With this definition, we find that σ ( k, ℓ ) factorises into a products of terms depending only on µ k ; in particular,let us expand the multi-index µ = ( µ , ..., µ D − ). Then since u kµ = D − Y ℓ =1 u kµ ( µ ℓ ) (57)where u kµ ( µ ℓ ) is given u kµ ( µ ℓ ) = sin (cid:0) µ ℓ πm (cid:1) , ℓ < k cos (cid:0) µ ℓ πm (cid:1) , ℓ = k < D , ℓ > k , (58)and noting that the sum over multi-indices can be expressed X µ = m X µ =1 · · · m X µ D − =1 2 m X µ D − =1 , (59)we can use distributivity of addition to write σ ( k, ℓ ) = D − Y j =1 σ jk,ℓ σ jk,ℓ = πm m X µ j =1 sin D − − j (cid:16) µ j πm (cid:17) u kµ ( µ j ) u ℓµ ( µ j ) σ D − k,ℓ = πm m X µ D − =1 u kµ ( µ D − ) u ℓµ ( µ D − )(60)10here in the central expression, j ∈ { , ..., D − } .Note that σ jk := σ jk,k for all relevant values of j and k and note that if k = ℓ we may assume without loss ofgenerality that k < ℓ since σ jk,ℓ = σ jℓ,k for all j , k, ℓ . Considering the latter case ( k = ℓ ) first, we thus have thefollowing: σ jk,ℓ = πm P mµ j =1 sin D +1 − j (cid:0) µ j πm (cid:1) , j < k < ℓ πm P mµ j =1 sin D − j (cid:0) µ j πm (cid:1) cos (cid:0) µ j πm (cid:1) , j = k < ℓ πm P mµ j =1 sin D − − j (cid:0) µ j πm (cid:1) , k < j < ℓ πm P mµ j =1 sin D − − j (cid:0) µ j πm (cid:1) cos (cid:0) µ j πm (cid:1) , k < j = ℓ < D − πm P mµ j =1 sin D − − j (cid:0) µ j πm (cid:1) , k < ℓ < j < D − π, k < ℓ < j = D − πm P mµ D − =1 cos (cid:0) µ D − πm (cid:1) , k < j = ℓ = D − , πm P mµ D − =1 sin (cid:0) µ D − πm (cid:1) , k < j = D − < ℓ = D πm P mµ D − =1 cos (cid:0) µ D − πm (cid:1) , j = k = D − < ℓ = D (61)Since sin( θ ) , cos( θ ) ∈ [ − ,
1] for all θ we have σ jk,ℓ = O (1) for all σ jk,ℓ , k < ℓ , trivially; moreover the cases with j = D − j ≤ ℓ all vanish trivially since m X µ D − =1 cos (cid:16) µ D − πm (cid:17) = m X µ D − =1 (cid:18) cos (cid:16) µ D − πm (cid:17) + cos (cid:18) ( µ D − + m ) πm (cid:19)(cid:19) = 0 (62a) m X µ D − =1 sin (cid:16) µ D − πm (cid:17) = m X µ D − =1 (cid:18) sin (cid:16) µ D − πm (cid:17) + sin (cid:18) ( µ D − + m ) πm (cid:19)(cid:19) = 0 (62b)and σ ( k, ℓ ) = 0 for k = ℓ .On the other hand, note that for k = ℓ we have σ jk = πm P mµ j =1 sin D +1 − j (cid:0) µ j πm (cid:1) , j < min( k, D − πm P mµ j =1 sin D − − j (cid:0) µ j πm (cid:1) cos (cid:0) µ j πm (cid:1) , j = k < D − πm P mµ j =1 sin D − − j (cid:0) µ j πm (cid:1) , k < j < D − π, k < j = D − πm P mµ D − =1 cos (cid:0) µ D − πm (cid:1) , j = k = D − πm P mµ D − =1 sin (cid:0) µ D − πm (cid:1) , j = D − < k = D (63)To deduce the vaues of these expressions we note that each expression is the Riemann sum of its summand. Inparticular, for any smooth function f : [ a, b ] → R we have: (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z ba d xf ( x ) − m m X k =1 f (cid:18) a + k ( b − a ) m (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = O ( m − ) . (64)Thus σ jk = R π d ϕ j sin D +1 − j ( ϕ j ) ± O ( m − ) , j < min( k, D − R π d ϕ j sin D − − j ( ϕ j ) cos ( ϕ j ) ± O ( m − ) , j = k < D − R π d ϕ j sin D − − j ( ϕ j ) ± O ( m − ) , k < j < D − π, k < j = D − R π d ϕ D − cos ( ϕ D − ) ± O ( m − ) , j = k = D − R π d ϕ D − sin ( ϕ D − ) ± O ( m − ) , j = D − < k = D (65)Recall the identities: Z ba d θ sin p θ = 1 p (cid:0) sin p − b cos b − sin p − a cos a (cid:1) + p − p Z ba d θ sin p − θ (66a) Z ba d θ sin p θ cos θ = − p + 2 (sin p +1 b cos b − sin p +1 a cos a ) + 1 p + 2 Z ba d θ sin p θ, (66b)which hold for all integers p > θ =1 − sin θ into the integrand. It turns out that the latter identity also follows for p = 0, which can immediately11e verified by integrating cos θ = (1 + cos 2 θ ) /
2. Substituting these identities into 65 then gives σ jk = D − jD +1 − j R π d ϕ j sin D − − j ( ϕ j ) ± O ( m − ) , j < min( k, D − D +1 − j R π d ϕ j sin D − − j ( ϕ j ) ± O ( m − ) , j = k < D − R π d ϕ j sin D − − j ( ϕ j ) ± O ( m − ) , k < j < D − π, k < j = D − π, j = k = D − π, j = D − < k = D (67)Thus for k < D − σ ( k ) = 1 D − k k Y j =1 D − jD + 1 − j (cid:18)Z π d ϕ j sin D − − j ( ϕ j ) ± O ( m − ) (cid:19) × D − Y j = k +1 (cid:18)Z π d ϕ j sin D − − j ( ϕ j ) ± O ( m − ) (cid:19) Z π d ϕ D − = 1 D D − Y j =1 (cid:18)Z π d ϕ j sin D − − j ( ϕ j ) (cid:19) Z π d ϕ D − ± O ( m − )= 1 D Z S D − d vol S D − ± O ( m − )= vol( S D − ) D ± O ( m − )= ω D ± O ( m − ) . (68)Similarly, for k ∈ { D − , D } we have σ ( k ) = 12 D − Y j =1 D − jD + 1 − j (cid:18)Z π d ϕ j sin D − − j ( ϕ j ) ± O ( m − ) (cid:19) Z π d ϕ D − = 1 D D − Y j =1 (cid:18)Z π d ϕ j sin D − − j ( ϕ j ) (cid:19) Z π d ϕ D − ± O ( m − )= ω D ± O ( m − ) (69)Hence Tr d U ( T ) = 1 ω D X k = ℓ T ( e k , e ℓ ) σ ( k, ℓ ) + 1 ω D D X k =1 T ( e k , e k ) σ ( k ) = tr( T ) ± O ( m − ) . (70)Hence | Tr d U ( T ) − tr( T ) | = O ( m − ) and (cid:12)(cid:12) tr d U ( T ) − tr( T ) (cid:12)(cid:12) = O (max( σ, n − / ( D − ))as required. B.2 Approximating a Uniform Grid with an ε -Net For a metric space (
X, ρ X ), any point x ∈ X and any numbers r, ℓ > r ≪ ℓ , let S Xℓ,r ( x ) := B Xℓ + r ( x ) /B Xℓ − r ( x ) . (71)Now let M be a Riemannian D -manifold, let A ⊆ M be an ε -net in M . For some point a ∈ A , let n := | A a | where A a := A ∩ S M ℓ,r ( a ). Then given m = ⌊ ( n/ /D − ⌋ , let µ be a multi-index as in the previous section andlet U = { u µ } ⊆ T a M be a uniform grid of points specified via equation 52 as previously. Let ˜ U = { u µ t µ } where for each µ , t µ ∈ ( l + r, l − r ). Clearly ˜ U ⊆ S R D ℓ,r (0) ⊆ T a M ; the distance between any two points of ˜ U isat least O (( ℓ − r ) m − ). Now let U a := exp a ( ˜ U ); since exp a preserves radial geodesics we have U a ⊆ S M ℓ,r ( a ) =exp a ( S R D ℓ,r (0)). Since A is an ε -net in M , every point of U a lies within a distance ε of some point of A a up tosome negligible boundary errors; moreover, exp a restricted to a ball of radius R has distortion O ( R ) (equation37), and the distance between any two points of U a is at least R := O (cid:18) ( ℓ − r ) m − ( ℓ + r ) (cid:19) = O (cid:18) ℓm − ℓ (cid:19) . (72)12hus if ε ≪ R we can ensure that any point of A a that lies within a distance ε of a point of U a does so uniquely;thus we have an injection ι : U a ֒ → A a such that ρ M ( b, ι ( b )) < ε for all b ∈ U a .Now suppose that we have a bilinear form T : T a M × T a M → R and a smooth partial function κ on M × M such that for all b ∈ S M ℓ,r ( a ) we have | κ ( a, b ) − T ( V, V ) | = O ( σ ) where V is the unit velocity of the uniquegeodesic connecting a and b . Now let d : S D − → R be as in the preceding section; given tr a M ( T ) as specifiedin equation 14, we have:tr a M ( T ) = 2 π D − ω D n X b ∈ A a d (cid:18) exp − a ( b ) || exp − a ( b ) || (cid:19) κ ( a, b )= 2 π D − ω D n X b ∈ ι ( U a ) d (cid:18) exp − a ( b ) || exp − a ( b ) || (cid:19) κ ( a, b ) + 2 π D − ω D n X b ∈ A a /ι ( U a ) d (cid:18) exp − a ( b ) || exp − a ( b ) || (cid:19) κ ( a, b )= 2 π D − ω D n X b ∈ ι ( U a ) d (cid:18) exp − a ( b ) || exp − a ( b ) || (cid:19) κ ( a, b ) + O (( n − m D − ) /n )= 2 π D − ω D n X b ∈ ι ( U a ) d (cid:18) exp − a ( b ) || exp − a ( b ) || (cid:19) κ ( a, b ) + O ( m − ) (73)where we define κ : S R D ℓ,r (0) → R via κ ( b ) = κ ( a, exp a ( b )). For each b ∈ ι ( U a ) there is a unique c ∈ U a such that ι ( c ) = b which we denote by ι − ( b ) := c . For each b ∈ ι ( U a ) we may then Taylor expand the summand about ι − ( b ); to first-order this introduces a term proportional to ( b − ι ( b )), and since ρ M ( β, ι ( b )) < ε we havetr M ( T ) = 2 π D − ω D n X b ∈ U a (cid:18) d (cid:18) exp − a ( b ) || exp − a ( b ) || (cid:19) κ ( a, b ) + O ( ε ) (cid:19) + O ( m − )= 2 π D − ω D n X b ∈ U a d (cid:18) exp − a ( b ) || exp − a ( b ) || (cid:19) κ ( a, b ) + O ( m − (1 + ε ))= 2 π D − ω D n X µ d ( u µ ) κ ( u µ ) + O ( m − ) (74)where we define κ : U → R via κ ( u µ ) := κ ( a, exp a ( u µ t µ )) for all µ . Comparing with equation 44, we see thattr M ( T ) = tr d U ( T ) + O ( m − ) (75)and hence | tr M ( T ) − tr( T ) | = O (cid:16) max (cid:16) σ, n − D − (cid:17)(cid:17) as required. C Proof of Theorem 3
As stated in appendix A, the constraints 19 ensure that the inequality 7 is satisfied; this means in particularthat (cid:12)(cid:12)(cid:12)(cid:12) D + 2) κ δG ( u, v ) δ − D + 2) κ δ M ( ι ( u ) , ι ( v )) δ (cid:12)(cid:12)(cid:12)(cid:12) = O ( δ ( δ + ℓ )) (76)so by equation 3 we have (cid:12)(cid:12)(cid:12)(cid:12) D + 2) κ δG ( u, v ) δ − Ric ι ( u ) ( V, V ) (cid:12)(cid:12)(cid:12)(cid:12) = O ( δ ( δ + ℓ )) . (77)Noting that ι ( G ) is a O ( ε )-net in M , we see by lemma 2 and the fact that R ( u ) = tr(Ric u ), we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) R ( u ) − π D − ( D + 2) ω D δ n ( u ) X b ∈ S Gℓ,r ( u ) κ δG ( u, v ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = O (cid:16) max (cid:16) δ ( δ + ℓ ) , n ( u ) − D − (cid:17)(cid:17) (78)13or each u ∈ G , where n ( u ) = | S Gℓ,r ( u ) ∩ G | , as long as ε ≪ ℓn − / ( D − − ℓ ; note that S Gℓ,r ( u ) ∩ G = S M ℓ,r ( u ) ∩ ι ( G ),but since dis( ι ) = O ( ε ) this introduces an error of order O ( ε ) ≪ O ( δ ( δ + ℓ )). Then since f : R → R is smooth,for each u ∈ G , we may Taylor expand f about R ( u ) and apply lemma 1 to find that |A f dis ( G ; δ, ℓ, r ) − A f ( M ) | = O (cid:16) max (cid:16) ε, δ ( δ + ℓ ) , n − D − (cid:17)(cid:17) (79)where n = inf u ∈ G ( n ( u )). By the constraints 19, ε ≪ δ ℓ ≪ δ ( δ + ℓ ); also, since S v ∈ S Gℓ,r ( u ) B M ε ( ι ( v )) covers S M ℓ,r ( ι ( u )) up to some negligible boundary defects, we have n = O vol( S M ℓ,r ( a ))vol( B M ε ( b ) ! = O (cid:18) ω D ( ℓ + r ) D − ω D ( ℓ − r ) D ω D ε D (cid:19) = O (cid:18) rℓ D − ε D (cid:19) = O (cid:16) N − ( c +( D − b ) (cid:17) . (80)Thus n is large—and hence the error 20 small—as long as 1 > c + ( D − b ; this is guaranteed by the constraint19 since c < /D and ( D − b + 1 /D <
1. Substituting in the definitions 17 then gives the desired result.It remains to verify the trace constraint 16: ε ≪ ℓn − D − − ℓ . Suppose that ℓ ≪ ℓn − / ( D − ; then it is sufficient to show that ε ≪ ℓn − / ( D − . This is equivalent to N − D − D ≪ N − (2( D − b + c − (81)i.e. 2( D − b + c − < − D . (82)But since c + ( D − b <
1, we have 2( D − b + c − < ( D − b (83)and it is sufficient for 1 D + ( D − b < , (84)which holds by assumption (see constraint 19). Also ℓ ≪ ℓn − / ( D − iff N − b ( D − ≪ N − (2( D − b + c − (85)i.e. if c − ( D − b < . (86)But since c − ( D − b < c + ( D − b < − /D < < aD ; then1 D + ( D − b < aD + ( D − b < aD + Db. (87)Also 3 aD + bD < < aD implies that bD < aD , i.e. b < a . Let ǫ > a = c = 1 + ǫ D < D , (88)where the right-hand inequality follows as long as ǫ <
3. Clearly1 < aD = 1 + ǫ. (89)Also 3 aD = 3 D + 3 ǫD > D > ε >
0. Thus it is sufficient to verify that 3 aD + bD <
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