We investigate the limit behaviour of the spectral measures of matrices following the Gibbs measure for the Ising model on random graphs, Potts model on random graphs, matrices coupled in a chain model or induced QCD model. For most of these models, we prove that the spectral measures converge almost surely and describe their limit via solutions to an Euler equation for isentropic flow with negative pressure $p(ρ)=-3^{-1}π^2 ρ^3$.
We present the product formula for complex multiple Wiener-Ito integrals. As applications, we show the Ustunel-Zakai independent criterion, the Nourdin-Rosinski asymptotic moment-independent criterion and joint convergence criterion for complex multiple Wiener-Ito integrals.
In this paper, we prove that in small parameter regions, arbitrary unitary matrix integrals converge in the large $N$ limit and match their formal expansion. Secondly we give a combinatorial model for our matrix integral asymptotics and investigate examples related to free probability and the HCIZ integral. Our convergence result also leads us to new results of smoothness of microstates. We finally generalize our approach to integrals over the othogonal group.
In this article we study the Dyson Bessel process, which describes the evolution of singular values of rectangular matrix Brownian motions, and prove a large deviation principle for its empirical particle density. We then use it to obtain the asymptotics of the so-called rectangular spherical integrals as $m,n$ go to infinity while $m/n$ converges.
We study a Edgeworth-type refinement of the central limit theorem for the discretizacion error of Itô integrals. Towards this end, we introduce a new approach, based on the anticipating Itô formula. This alternative technique allows us to compute explicitly the terms of the corresponding expansion formula.
For suitable families of locally infinitely divisible Markov processes $\{ξ^{ε}_t\}_{0\leq t\leq T}$ with frequent small jumps depending on a small parameter $ε>0,$ precise asymptotics for large deviations of integral forms $\mathbb{E}^ε[\exp\{ε^{-1}F(ξ^ε)\}]$ are proved for smooth functionals $F.$ The main ingredient of the proof in this paper is a recent result regarding the asymptotic expansions of the expectations $\mathbb{E}^ε[G(ξ^ε)\}]$ for smooth $G.$ Several connections between these large deviation asymptotics and partial integro-differential equations are included as well.
In this short note we announce the construction of scale invariant non-Gaussian generalized stochastic processes over three dimensional p-adic space. The construction includes that of the associated squared field and our result shows this squared field has a dynamically generated anomalous dimension which rigorously confirms a prediction made more than forty years ago, in an essentially identical situation, by Kenneth G. Wilson. We also prove a mild form of universality for the model under consideration.
We characterize the asymptotic independence between blocks consisting of multiple Wiener-Itô integrals. As a consequence of this characterization, we derive the celebrated fourth moment theorem of Nualart and Peccati, its multidimensional extension and other related results on the multivariate convergence of multiple Wiener-Itô integrals, that involve Gaussian and non Gaussian limits. We give applications to the study of the asymptotic behavior of functions of short and long-range dependent stationary Gaussian time series and establish the asymptotic independence for discrete non-Gaussian chao…
We consider Riemann sum approximations of stochastic integrals with respect to the fractional Browian motion of index $H\geq \frac12$. We show the convergence of these schemes at first and second order. The processes obtained in the limit in the second case are stochastic integrals with respect to the Rosenblatt process if $H >\frac34$ and the standard Brownian motion otherwise. These results are obtained under the assumption that the integrand is a `controlled' process. We provide many examples of such processes, in particular fractional semimartingales and multiple Wiener-Itô integrals
In an abstract Wiener space setting, we constract a rigorous mathematical model of the one-loop approximation of the perturbative Chern-Simons integral, and derive its explicit asymptotic expansion for stochastic Wilson lines.
We consider a renewal-like recursion and prove that the solution is polynomially decaying asymptotically under suitable conditions. We prove similar results for the corresponding integral equation. In both cases coefficients and functions are of more general form than in the classic cases.
We consider the spherical integral of real symmetric or Hermitian matrices when the rank of one matrix is one. We prove the existence of the full asymptotic expansions of these spherical integrals and derive the first and the second term in the asymptotic expansion. Using asymptotic expression of the spherical integral, we derive the asymptotic freeness of Wigner matrices with (deterministic) Hermitian matrices.
We investigate generalizations of the Cramér theorem. This theorem asserts that a Gaussian random variable can be decomposed into the sum of independent random variables if and only if they are Gaussian. We prove asymptotic counterparts of such decomposition results for multiple Wiener integrals and prove that similar results are true for the (asymptotic) decomposition of the semicircular distribution into free multiple Wigner integrals.
In this note, we study the asymptotics of a spherical integral that is a multiplicative counterpart to the well-known Harish-Chandra Itzykson Zuber integral. This counterpart can also be expressed in terms the Heckman-Opdam hypergeometric function. When the argument of this spherical integral is of finite support and of order $N$, these asymptotics involve a modified version of the $S$-transform of the limit measure of the matrix argument and its largest eigenvalue. To prove our main result, we are leveraging a technique of successive conditionning. In particular we prove in a "mathematically …
The aim of the presented research is to give a rigorous mathematical approach to Feynman path integrals based on strong (pathwise) approximations based on simple random walks.
Rigorous Asymptotics for First-Order Algorithms Through the Dynamical Cavity Method
Statistical Physics of Computation Laboratory, École polytechnique fédérale de Lausanne (EPFL)（计算统计物理实验室，瑞士联邦理工学院（EPFL））；Sloan School of Management, Operations Research Center and Institute of Data, Systems and Society (IDSS), MIT（斯隆管理学院，运筹学中心和数据、系统与社会研究所（IDSS），麻省理工学院）；CSAIL and LIDS, MIT（计算机科学与人工智能实验室（CSAIL）和麻省理工学院数据科学研究所（LIDS））
We study the behavior of high-dimensional robust regression estimators in the asymptotic regime where $p/n$ tends to a finite non-zero limit. More specifically, we study ridge-regularized estimators, i.e $\widehatβ=\text{argmin}_{β\in \mathbb{R}^p} \frac{1}{n}\sum_{i=1}^n ρ(\varepsilon_i-X_i' β)+\fracτ{2}\lVertβ\rVert^2$. In a recently published paper, we had developed with collaborators probabilistic heuristics to understand the asymptotic behavior of $\widehatβ$. We give here a rigorous proof, properly justifying all the arguments we had given in that paper. Our proof is based on the probabi…
In this monograph, we prove an asymptotic approximation for integrals of probability densities over sets in finite dimensional euclidean space, which are far away from the origin (asymptotic sets). We use this approximation to investigate tails of quadratic forms of random vectors, supremum of random linear forms among others. Applications to the study of finite size random matrices, finite sample statistics of autoregressive processes, and supremum of some stochastic processes.
We rigorously derive the Ornstein-Zernike asymptotics of the pair-correlation functions for finite-range Ising ferromagnets in any dimensions and at any temperature above critical.
We use the steepest descents method to study the integral kernel of a family of normal random matrix ensembles with eigenvalue distribution P_{N}(z_{1},...,z_{N}) = Z_{N}^{-1} e^{-NΣ_{i=1}^{N}V_α(z_{i})} Π_{1\leqi<j\leqN}|z_{i}-z_{j}|^{2} where V_α(z)=|z|^α, z \in C and α\in ]0,\infty[. Asymptotic analysis with error estimates are obtained. A corollary of this expansion is a scaling limit for the n-point function in terms of the integral kernel for the classical Segal--Bargmann space.
We explore a well-known integral representation of the logarithmic function, and demonstrate its usefulness in obtaining compact, easily-computable exact formulas for quantities that involve expectations and higher moments of the logarithm of a positive random variable (or the logarithm of a sum of positive random variables). The integral representation of the logarithm is proved useful in a variety of information-theoretic applications, including universal lossless data compression, entropy and differential entropy evaluations, and the calculation of the ergodic capacity of the single-input, …
The expected number of false inflection points of kernel smoothers is evaluated. To obtain the small noise limit, we use a reformulation of the Leadbetter-Cryer integral for the expected number of zero crossings of a differentiable Gaussian process.
In this article we provide the complete proof of the result announced in arXiv:1210.7717 about the construction of scale invariant non-Gaussian generalized stochastic processes over three dimensional p-adic space. The construction includes that of the associated squared field and our result shows this squared field has a dynamically generated anomalous dimension which rigorously confirms a prediction made more than forty years ago, in an essentially identical situation, by K. G. Wilson. We also prove a mild form of universality for the model under consideration. Our main innovation is that our…
