Sourav Chatterjee

A generalization of the Lindeberg principle

We generalize Lindeberg’s proof of the central limit theorem to an invariance principle for arbitrary smooth functions of independent and weakly dependent random variables. The result is applied to get a similar theorem for smooth functions of exchangeable random variables. This theorem allows us to identify, for the first time, the limiting spectral distributions of Wigner matrices with exchangeable entries.

The large deviation principle for the Erdős-Rényi random graph

What does an Erdos-Renyi graph look like when a rare event happens? This paper answers this question when p is fixed and n tends to infinity by establishing a large deviation principle under an appropriate topology. The formulation and proof of the main result uses the recent development of the theory of graph limits by Lovasz and coauthors and Szemeredis regularity lemma from graph theory. As a basic application of the general principle, we work out large deviations for the number of triangles in G(n,p). Surprisingly, even this simple example yields an interesting double phase transition.

Exchangeable Pairs and Poisson Approximation

This is a survery paper on Poisson approximation using Steins method of exchangeable pairs. We illustrate using Poisson-binomial trials and many variations on three classical problems of combinatorial probability: the matching problem, the coupon collectors problem, and the birthday problem. While many details are new, the results are closely related to a body of work developed by Andrew Barbour, Louis Chen, Richard Arratia, Lou Gordon, Larry Goldstein, and their collaborators. Some comparison with these other approaches is offered.

Nonnormal approximation by Stein’s method of exchangeable pairs with application to the Curie–Weiss model

Let (W,W ′) be an exchangeable pair. Assume that E(W − W ′|W ) = g(W ) + r(W ), where g(W ) is a dominated term while r(W ) is negligible. Let G(t) = ∫ t 0 g(s)ds and define p(t) = c1e −c0G(t), where c0 is a properly chosen constant and c1 = 1/ ∫∞ −∞ p(t)dt . Let Y be a random variable with the probability density function p. In this talk we shall proved that W converges to Y in distribution under certain regular conditions. A Berry-Esseen type bound is also given. An application to the Curie-Weiss model will be discussed. The talk is based on a joint work with Sourav Chatterjee.

Applications of Stein's method for concentration inequalities

Stein’s method for concentration inequalities was introduced to prove concentration of measure in problems involving complex dependencies such as random permutations and Gibbs measures. In this paper, we provide some extensions of the theory and three applications: (1) We obtain a concentration inequality for the magnetization in the Curie–Weiss model at critical temperature (where it obeys a nonstandard normalization and super-Gaussian concentration). (2) We derive exact large deviation asymptotics for the number of triangles in the Erdős–Renyi random graph G(n, p) when p ≥ 0.31. Similar results are derived also for general subgraph counts. (3) We obtain some interesting concentration inequalities for the Ising model on lattices that hold at all temperatures.

The missing log in large deviations for triangle counts

This paper solves the problem of sharp large deviation estimates for the upper tail of the number of triangles in an Erdős-Renyi random graph, by establishing a logarithmic factor in the exponent that was missing till now. It is possible that the method of proof may extend to general subgraph counts.

Superconcentration and related topics

Preface.- 1.Introduction.- 2.Markov semigroups.- 3.Super concentration and chaos.- 4.Multiple valleys.- 5.Talagrands method for proving super concentration.- 6.The spectral method for proving super concentration.- 7.Independent flips.- 8.Extremal fields.- 9.Further applications of hypercontractivity.- 10.The interpolation method for proving chaos.- 11.Variance lower bounds.- 12.Dimensions of level sets.- Appendix A. Gaussian random variables.- Appendix B. Hypercontractivity.- Bibliography.- Indices.

Fluctuations of the Bose–Einstein condensate

This paper gives a rigorous analysis of the fluctuations of the Bose?Einstein condensate for a system of non-interacting bosons in an arbitrary potential, assuming that the system is governed by the canonical ensemble. As a result of the analysis, we are able to tell the order of fluctuations of the condensate fraction as well as its limiting distribution upon proper centering and scaling. This yields interesting results. For example, for a system of n bosons in a 3D harmonic trap near the transition temperature, the order of fluctuations of the condensate fraction is n?1/2 and the limiting distribution is normal, whereas for the 3D uniform Bose gas, the order of fluctuations is n?1/3 and the limiting distribution is an explicit non-normal distribution. For a 2D harmonic trap, the order of fluctuations is n?1/2(log?n)1/2, which is larger than n?1/2 but the limiting distribution is still normal. All of these results come as easy consequences of a general theorem.

Consistent estimates of deformed isotropic Gaussian random fields on the plane

This paper proves fixed domain asymptotic results for estimating a smooth invertible transformation f: R 2 -R 2 when observing the deformed random field Z o f on a dense grid in a bounded, simply connected domain Ω, where Z is assumed to be an isotropic Gaussian random field on R 2 is constructed on a simply connected domain U, such that Ū Ω and is defined using kernel smoothed quadratic variations, Bergman projections and results from quasiconformal theory. We show, under mild assumptions on the random field Z and the deformation f, that R θ f + c uniformly on compact subsets of U with probability one as the grid spacing goes to zero, where R θ is an unidentifiable rotation and c is an unidentifiable translation.

The sample size required in importance sampling

The goal of importance sampling is to estimate the expected value of a given function with respect to a probability measure