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31
On the geometry of metric measure spaces
 II, Acta Math
"... We introduce and analyze lower (’Ricci’) curvature bounds Curv(M, d,m) ≥ K for metric measure spaces (M, d,m). Our definition is based on convexity properties of the relative entropy Ent(.m) regarded as a function on the L2Wasserstein space of probability measures on the metric space (M, d). Amo ..."
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Cited by 116 (4 self)
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We introduce and analyze lower (’Ricci’) curvature bounds Curv(M, d,m) ≥ K for metric measure spaces (M, d,m). Our definition is based on convexity properties of the relative entropy Ent(.m) regarded as a function on the L2Wasserstein space of probability measures on the metric space (M, d). Among others, we show that Curv(M, d,m) ≥ K implies estimates for the volume growth of concentric balls. For Riemannian manifolds, Curv(M, d,m) ≥ K if and only if RicM (ξ, ξ) ≥ K · ξ2 for all ξ ∈ TM. The crucial point is that our lower curvature bounds are stable under an appropriate notion of Dconvergence of metric measure spaces. We define a complete and separable metric D on the family of all isomorphism classes of normalized metric measure spaces. The metric D has a natural interpretation, based on the concept of optimal mass transportation. We also prove that the family of normalized metric measure spaces with doubling constant ≤ C is closed under Dconvergence. Moreover, the family of normalized metric measure spaces with doubling constant ≤ C and radius ≤ R is compact under Dconvergence. 1
The BrunnMinkowski inequality
 Bull. Amer. Math. Soc. (N.S
, 2002
"... Abstract. In 1978, Osserman [124] wrote an extensive survey on the isoperimetric inequality. The BrunnMinkowski inequality can be proved in a page, yet quickly yields the classical isoperimetric inequality for important classes of subsets of R n, and deserves to be better known. This guide explains ..."
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Cited by 109 (5 self)
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Abstract. In 1978, Osserman [124] wrote an extensive survey on the isoperimetric inequality. The BrunnMinkowski inequality can be proved in a page, yet quickly yields the classical isoperimetric inequality for important classes of subsets of R n, and deserves to be better known. This guide explains the relationship between the BrunnMinkowski inequality and other inequalities in geometry and analysis, and some applications. 1.
Metatheory and Reflection in Theorem Proving: A Survey and Critique
, 1995
"... One way to ensure correctness of the inference performed by computer theorem provers is to force all proofs to be done step by step in a simple, more or less traditional, deductive system. Using techniques pioneered in Edinburgh LCF, this can be made palatable. However, some believe such an appro ..."
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Cited by 66 (2 self)
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One way to ensure correctness of the inference performed by computer theorem provers is to force all proofs to be done step by step in a simple, more or less traditional, deductive system. Using techniques pioneered in Edinburgh LCF, this can be made palatable. However, some believe such an approach will never be efficient enough for large, complex proofs. One alternative, commonly called reflection, is to analyze proofs using a second layer of logic, a metalogic, and so justify abbreviating or simplifying proofs, making the kinds of shortcuts humans often do or appealing to specialized decision algorithms. In this paper we contrast the fullyexpansive LCF approach with the use of reflection. We put forward arguments to suggest that the inadequacy of the LCF approach has not been adequately demonstrated, and neither has the practical utility of reflection (notwithstanding its undoubted intellectual interest). The LCF system with which we are most concerned is the HOL proof ...
Asymptotic enumeration of spanning trees
 Combin. Probab. Comput
, 2005
"... Note: Theorem numbers differ from the published version. Abstract. We give new general formulas for the asymptotics of the number of spanning trees of a large graph. A special case answers a question of McKay (1983) for regular graphs. The general answer involves a quantity for infinite graphs that ..."
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Cited by 36 (6 self)
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Note: Theorem numbers differ from the published version. Abstract. We give new general formulas for the asymptotics of the number of spanning trees of a large graph. A special case answers a question of McKay (1983) for regular graphs. The general answer involves a quantity for infinite graphs that we call “tree entropy”, which we show is a logarithm of a normalized determinant of the graph Laplacian for infinite graphs. Tree entropy is also expressed using random walks. We relate tree entropy to the metric entropy of the uniform spanning forest process on quasitransitive amenable graphs, extending a result of Burton and Pemantle (1993). §1. Introduction. Methods of enumeration of spanning trees in a finite graph G and relations to various areas of mathematics and physics have been investigated for more than 150 years. The number of spanning trees is often called the complexity of the graph, denoted here by τ(G). The best known formula for the complexity, proved in every basic text on graph
A CLT for a band matrix model
 Probab. Theory Relat. Fields
, 2005
"... Abstract. A law of large numbers and a central limit theorem are derived for linear statistics of random symmetric matrices whose onorabove diagonal entries are independent, but neither necessarily identically distributed, nor necessarily all of the same variance. The derivation is based on system ..."
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Cited by 15 (0 self)
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Abstract. A law of large numbers and a central limit theorem are derived for linear statistics of random symmetric matrices whose onorabove diagonal entries are independent, but neither necessarily identically distributed, nor necessarily all of the same variance. The derivation is based on systematic combinatorial enumeration, study of generating functions, and concentration inequalities of the Poincaré type. Special cases treated, with an explicit evaluation of limiting variances, are generalized Wigner and Wishart matrices. 1.
ON STEIN’S METHOD FOR MULTIVARIATE NORMAL APPROXIMATION
, 2009
"... The purpose of this paper is to synthesize the approaches taken by ChatterjeeMeckes and ReinertRöllin in adapting Stein’s method of exchangeable pairs for multivariate normal approximation. The more general linear regression condition of ReinertRöllin allows for wider applicability of the metho ..."
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The purpose of this paper is to synthesize the approaches taken by ChatterjeeMeckes and ReinertRöllin in adapting Stein’s method of exchangeable pairs for multivariate normal approximation. The more general linear regression condition of ReinertRöllin allows for wider applicability of the method, while the method of bounding the solution of the Stein normal approximation theorems are proved, one for use when the underlying symmetries of the random variables are discrete, and one for use in contexts in which continuous symmetry groups are present. The application to runs on the line from ReinertRöllin is reworked to demonstrate the improvement in convergence rates, and a new application to joint value distributions of eigenfunctions of the LaplaceBeltrami operator on a compact Riemannian manifold is presented.
Entropy and the fourth moment phenomenon
, 2013
"... We develop a new method for bounding the relative entropy of a random vector in terms of its Stein factors. Our approach is based on a novel representation for the score function of smoothly perturbed random variables, as well as on the de Bruijn’s identity of information theory. When applied to seq ..."
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Cited by 3 (3 self)
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We develop a new method for bounding the relative entropy of a random vector in terms of its Stein factors. Our approach is based on a novel representation for the score function of smoothly perturbed random variables, as well as on the de Bruijn’s identity of information theory. When applied to sequences of functionals of a general Gaussian field, our results can be combined with the CarberyWright inequality in order to yield multidimensional entropic rates of convergence that coincide, up to a logarithmic factor, with those achievable in smooth distances (such as the 1Wasserstein distance). In particular, our findings settle the open problem of proving a quantitative version of the multidimensional fourth moment theorem for random vectors having chaotic components, with explicit rates of convergence in total variation that are independent of the order of the associated Wiener chaoses. The results proved in the present paper are outside the scope of other existing techniques, such as for instance the multidimensional Stein’s
Uniform Large and Moderate Deviations for Functional Empirical Processes.
, 1995
"... For fX i g i1 a sequence of i.i.d. random variables taking values in a Polish space \Sigma with distribution ¯, we obtain large and moderate deviation principles for the processes fn \Gamma1 P [nt] i=1 ffi X i ; t 0g n1 and fn \Gamma1=2 P [nt] i=1 (ffi X i \Gamma ¯); t 0g n1 , respectively. G ..."
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For fX i g i1 a sequence of i.i.d. random variables taking values in a Polish space \Sigma with distribution ¯, we obtain large and moderate deviation principles for the processes fn \Gamma1 P [nt] i=1 ffi X i ; t 0g n1 and fn \Gamma1=2 P [nt] i=1 (ffi X i \Gamma ¯); t 0g n1 , respectively. Given a class of bounded functions F on \Sigma, we then consider the above processes as taking values in the Banach space of bounded functionals over F and obtain the corresponding (uniform over F ), large and moderate deviation principles. Among the corollaries considered are functional laws of the iterated logarithm. 1 Introduction. We recall that, given a sequence fX i g i1 of random variables taking values in a measure space and a class of bounded measurable functions F , one may view the empirical measure L n = n \Gamma1 P n i=1 ffi X i as a bounded functional over F ; that is, as an element of l 1 (F ). Viewed Partially supported by NSF DMS9209712 and DMS9403553 grants, and by a...
ASYMPTOTIC ANALYSIS OF A CLASS OF OPTIMAL LOCATION PROBLEMS
"... Abstract. Given a density function f on an compact subset of R d, we look at the problem of finding the best approximation of f by discrete measures ν = P ciδxi in the sense of the pWasserstein distance, subject to size constraints of the form P h(ci) ≤ α where h is a given weight function. This i ..."
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Abstract. Given a density function f on an compact subset of R d, we look at the problem of finding the best approximation of f by discrete measures ν = P ciδxi in the sense of the pWasserstein distance, subject to size constraints of the form P h(ci) ≤ α where h is a given weight function. This is an important problem with applications in economic planning of locations, in information theory and in shape optimization problems. The efficiency of the approximation can be measured by studying the rate at which the minimal distance tends to zero as α tends to infinity. In this paper, we introduce a rescaled distance which depends on a small parameter and establish a representation formula for its limit as a function of the local statistics for the distribution of the ci’s. The asymptotic problem for large α can be then treated in the case of quite general entropy functions h. Keywords: MongeKantorovich transportation, Wasserstein distance, entropy, optimal partitions, rate of approximation, asymptotics, Γconvergence, functionals on measures, integral representation. 1.
The optimal fourth moment theorem
, 2013
"... We compute the exact rates of convergence in total variation associated with the ‘fourth moment theorem ’ by Nualart and Peccati (2005), stating that a sequence of random variables living in a fixed Wiener chaos verifies a central limit theorem (CLT) if and only if the sequence of the corresponding ..."
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We compute the exact rates of convergence in total variation associated with the ‘fourth moment theorem ’ by Nualart and Peccati (2005), stating that a sequence of random variables living in a fixed Wiener chaos verifies a central limit theorem (CLT) if and only if the sequence of the corresponding fourth cumulants converges to zero. We also provide an explicit illustration based on the BreuerMajor CLT for Gaussiansubordinated random sequences.