Harald Luschgy, Gilles Pages
The sharp asymptotics for the L^2-quantization errors of Gaussian measures on a Hilbert space and, in particular, for Gaussian processes is derived. The condition imposed is regular variation of the eigenvalues.
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In this paper we establish the sharp rate of the optimal dual quantization problem. The notion of dual quantization was recently introduced in the paper [8], where it was shown that, at least in an Euclidean setting, dual quantizers are based on a Delaunay triangulation, the dual counterpart of the Voronoi tessellation on which "regular" quantization relies. Moreover, this new approach shares an intrinsic stationarity property, which makes it very valuable for numerical applications. We establish in this paper the counterpart for dual quantization of the celebrated Zador theorem, which describ…
We describe quantization designs which lead to asymptotically and order optimal functional quantizers. Regular variation of the eigenvalues of the covariance operator plays a crucial role to achieve these rates. For the development of a constructive quantization scheme we rely on the knowledge of the eigenvectors of the covariance operator in order to transform the problem into a finite dimensional quantization problem of normal distributions. Furthermore we derive a high-resolution formula for the $L^2$-quantization errors of Riemann-Liouville processes.
Sharp functional quantization and empirical Wasserstein rates for Itô processes
该研究针对Itô过程经验路径律，建立了(log N)^{-1/2}的Sharp Wasserstein收敛速率，采用自适应随机时间区间划分方法，其结果可应用于路径依赖SDE及McKean-Vlasov粒子系统相关估计。
We derive a high-resolution formula for the $L^2$-quantization errors of Riemann-Liouville processes and the sharp Kolmogorov entropy asymptotics for related Sobolev balls. We describe a quantization procedure which leads to asymptotically optimal functional quantizers. Regular variation of the eigenvalues of the covariance operator plays a crucial role.
We consider the optimal quantization problem with Rényi-$α$-entropy constraints for centered Gaussian measures on a separable Banach space. For $α= \infty$ we can compute the optimal quantization error by a moment on a ball. For $α\in {}]1,\infty]$ and large entropy bound we derive sharp asymptotics for the optimal quantization error in terms of the small ball probability of the Gaussian measure. We apply our results to several classes of Gaussian measures. The asymptotical order of the optimal quantization error for $α> 1$ is different from the well-known cases $α= 0$ and $α= 1$.
We offer in this paper the non-asymptotical bilateral sharp exponential estimates for tail of maximum distribution of {\it discontinuous} random fields. Our consideration based on the theory of Prokhorov-Skorokhod spaces of random fields and on the theory of multivariate Banach spaces of random variables with exponential decreasing tails of distributions.
In this work, we consider critical planar site percolation on the triangular lattice and derive sharp estimates on the asymptotics of the probability of half-plane $j$-arm events for $j \geq 1$ and planar (polychromatic) $j$-arm events for $j>1$. These estimates greatly improve previous results and in particular answer (a large part of) a question of Schramm (ICM Proc., 2006).
The current research of fractional Sturm-Liouville boundary value problems focuses on the qualitative theory and numerical methods, and much progress has been recently achieved in both directions. The objective of this paper is to explore a different route, namely, construction of explicit asymptotic approximations for the solutions. As a study case, we consider a problem with left and right Riemann-Liouville derivatives, for which our analysis yields asymptotically sharp estimates for the sequence of eigenvalues and eigenfunctions.
We give sharp estimate for the free energy of directed polymers in random environment in dimension 1+1. This estimate was known for a Gaussian environment, we extend it to the case where the law of the environment is infinitely divisible.
We consider continuous and discrete (1+1)-dimensional wetting models which undergo a localization/delocalization phase transition. Using a simple approach based on Renewal Theory we determine the precise asymptotic behavior of the partition function, from which we obtain the scaling limits of the models and an explicit construction of the infinite volume measure (thermodynamic limit) in all regimes, including the critical one.
The entropy, the spectral radius and the drift are important numerical quantities associated to random walks on countable groups. We prove sharp inequalities relating those quantities for walks with a finite second moment, improving upon previous results of Avez, Varopoulos, Carne, Ledrappier. We also deduce inequalities between these quantities and the volume growth of the group. Finally, we show that the equality case in our inequality is rather rigid.
In the present paper, we investigate the collapsed phase of the interacting partially-directed self-avoiding walk (IPDSAW) that was introduced in Zwanzig and Lauritzen (1968). We provide sharp asymptotics of the partition function inside the collapsed phase, proving rigorously a conjecture formulated in Guttmann (2015) and Owczarek et al. (1993). As a by-product of our result, we obtain that, inside the collapsed phase, a typical IPDSAW trajectory is made of a unique macroscopic bead, consisting of a concatenation of long vertical stretches of alternating signs, outside which only finitely man…
We consider a class of parabolic semi-linear stochastic partial differential equations driven by space-time white noise on a compact space interval. Our aim is to obtain precise asymptotics of the transition times between metastable states. A version of the so-called Eyring-Kramers Formula is proven in an infinite dimensional setting. The proof is based on a spatial finite difference discretization of the stochastic partial differential equation. The expected transition time is computed for the finite dimensional approximation and controlled uniformly in the dimension.
The study of finite approximations of probability measures has a long history. In (Xu and Berger, 2017), the authors focus on constrained finite approximations and, in particular, uniform ones in dimension $d=1$. The present paper gives an elementary construction of a uniform decomposition of probability measures in dimension $d\geq 1$. This decomposition is then used to give upper-bounds on the rate of convergence of the optimal uniform approximation error. These bounds appear to be the generalization of the ones obtained in (Xu and Berger, 2017) and to be sharp for generic probability measur…
Convergence Rates for Regularized Optimal Transport via Quantization
本文研究散度正则化最优传输在正则化参数趋于零时的收敛速率，提出基于量化与鞅耦合的新方法，适用于非紧致边际，并获得了熵正则化2-Wasserstein距离的尖锐首阶项。
For an ergodic Brownian diffusion with invariant measure $ν$, we consider a sequence of empirical distributions ($ν$n) n$\ge$1 associated with an approximation scheme with decreasing time step ($γ$n) n$\ge$1 along an adapted regular enough class of test functions f such that f --$ν$(f) is a coboundary of the infinitesimal generator A. Denote by $σ$ the diffusion coefficient and $Φ$ the solution of the Poisson equation A$Φ$ = f -- $ν$(f). When the square norm of |$σ$ * $Φ$| 2 lies in the same coboundary class as f , we establish sharp non-asymptotic concentration bounds for suitable normalizati…
Sharp large deviation results of Bahadur-Ranga Rao type are provided for the $q$-norm of random vectors distributed on the $\ell _{p}^{n}$-ball ${\mathbb{B}}^{n}_{p}$ according to the cone probability measure or the uniform distribution for $1 \le q<p < \infty $, thereby furthering previous large deviation results by Kabluchko, Prochno and Thäle in the same setting. These results are then applied to deduce sharp asymptotics for intersection volumes of different $\ell _{p}^{n}$-balls in the spirit of Schechtman and Schmuckenschläger, and for the length of the projection of an $\ell _{p}^{n}$-ba…
Optimal Quantization for Nonuniform Densities on Spherical Curves
本文研究了球面曲线上的非均匀密度最优量化问题，推导了质心条件和点密度公式，并应用于von Mises分布，为几何概率和量化理论提供了新方法。
We describe the asymptotic properties of the edge-triangle exponential random graph model as the natural parameters diverge along straight lines. We show that as we continuously vary the slopes of these lines, a typical graph drawn from this model exhibits quantized behavior, jumping from one complete multipartite graph to another, and the jumps happen precisely at the normal lines of a polyhedral set with infinitely many facets. As a result, we provide a complete description of all asymptotic extremal behaviors of the model.
Sharp Minimax Gram-Determinant Asymptotics for Labelled Unit Decompositions of the Identity and Recent Developments on Makai's Conjecture
该研究证明了González Merino和Schymura的猜想4.8在足够大维度下不成立，结合随机框架确定了向量数与维度比大于1时的Sharp指数行列式保证，为Makai反向等径猜想相关研究提供了关键进展。
For a family of random-cluster models with cluster weights $q\geq 1$, we prove that the probability that $0$ is connected to $x$ is asymptotically equal to $\tfrac{1}{q}χ(β)^{2}βJ_{0,x}$. The method developed in this article can be applied to any spin model for which there exists a random-cluster representation which is one-monotonic.
We prove that the correction to exponential decay of the truncated two points function in the homogeneous positive field Ising model is $c\|x\|^{-(d-1)/2}$. The proof is based on the development in the random current representation of a "modern" Ornstein-Zernike theory, as developed by Campanino, Ioffe and Velenik.
We provide O(1) asymptotics for the average number of deep minima of the (p,k) spiked tensor model. We also derive an explicit formula for the limiting ground state energy on the N-dimensional sphere, similar to the work of Jagannath-Lopatto-Miolane. Moreover, when the signal to noise ratio is large enough, the expected number of deep minima is asymptotically finite as N tends to infinity and we determine its limit as the signal-to-noise ratio diverges.
The Wasserstein distance between two probability measures on a metric space is a measure of closeness with applications in statistics, probability, and machine learning. In this work, we consider the fundamental question of how quickly the empirical measure obtained from $n$ independent samples from $μ$ approaches $μ$ in the Wasserstein distance of any order. We prove sharp asymptotic and finite-sample results for this rate of convergence for general measures on general compact metric spaces. Our finite-sample results show the existence of multi-scale behavior, where measures can exhibit radic…
Geometric quantization results for semi-positive line bundles on a Riemann surface
本文探讨了Riemann面上半正线丛Bergman核展开的相关结果，包括Tian逼近定理、Toeplitz算子渐近性、随机截面零点及全纯挠渐近性。
In this paper we study the metastable behavior of one of the simplest disordered spin system, the random field Curie-Weiss model. We will show how the potential theoretic approach can be used to prove sharp estimates on capacities and metastable exit times also in the case when the distribution of the random field is continuous. Previous work was restricted to the case when the random field takes only finitely many values, which allowed the reduction to a finite dimensional problem using lumping techniques. Here we produce the first genuine sharp estimates in a context where entropy is importa…
Random Matrices, Intrinsic Freeness, and Sharp Non-Asymptotic Inequalities
该研究介绍随机矩阵理论中的内蕴自由性现象，提出用其推导一般随机矩阵谱的新易用精确非渐近界，并展示高维统计推断中的应用，为2026年国际数学家大会讲座配套内容。
The purpose of this paper is to provide a sharp analysis on the asymptotic behavior of the Durbin-Watson statistic. We focus our attention on the first-order autoregressive process where the driven noise is also given by a first-order autoregressive process. We establish the almost sure convergence and the asymptotic normality for both the least squares estimator of the unknown parameter of the autoregressive process as well as for the serial correlation estimator associated to the driven noise. In addition, the almost sure rates of convergence of our estimates are also provided. It allows us …
We recently introduced a robust approach to the derivation of sharp asymptotic formula for correlation functions of statistical mechanics models in the high-temperature regime. We describe its application to the nonperturbative proof of Ornstein-Zernike asymptotics of 2-point functions for self-avoiding walks, Bernoulli percolation and ferromagnetic Ising models. We then extend the proof, in the Ising case, to arbitrary odd-odd correlation functions. We discuss the fluctuations of connection paths (invariance principle), and relate the variance of the limiting process to the geometry of the eq…
Asymptotics of Constrained Quantization for Compactly Supported Measures
本文研究闭集约束下紧支撑概率测度的高速率量化渐近性，借助度量投影、回拉不等式与加权扰动建立匹配的上下界，证明 Ahlfors 正则维数即约束量化维数。
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