arXiv subjects

Sourav Chatterjee

Publications and source records attributed to Sourav Chatterjee.

226 records · Page 13Linked to original sources

arXiv2007

Estimation in spin glasses: A first step

The Sherrington--Kirkpatrick model of spin glasses, the Hopfield model of neural networks and the Ising spin glass are all models of binary data belonging to the one-parameter exponential family with quadratic sufficient statistic. Under bare minimal conditions, we establish the $\sqrt{N}$-consistency of the maximum pseudolikelihood estimate of the natural parameter in this family, even at critical temperatures. Since very little is known about the low and critical temperature regimes of these extremely difficult models, the proof requires several new ideas. The author's version of Stein's method is a particularly useful tool. We aim to introduce these techniques into the realm of mathematical statistics through an example and present some open questions.

math.PR↗

arXiv2007

Effects of stellar collisions on star cluster evolution and core collapse

We systematically study the effects of collisions on the overall dynamical evolution of dense star clusters using Monte Carlo simulations over many relaxation times. We derive many observable properties of these clusters, including their core radii and the radial distribution of collision products. We also study different aspects of collisions in a cluster taking into account the shorter lifetimes of more massive stars, which has not been studied in detail before. Depending on the lifetimes of the significantly more massive collision products, observable properties of the cluster can be modified qualitatively; for example, even without binaries, core collapse can sometimes be avoided simply because of stellar collisions.

astro-ph↗

arXiv2007

Gravitational allocation to Poisson points

For d>=3, we construct a non-randomized, fair and translation-equivariant allocation of Lebesgue measure to the points of a standard Poisson point process in R^d, defined by allocating to each of the Poisson points its basin of attraction with respect to the flow induced by a gravitational force field exerted by the points of the Poisson process. We prove that this allocation rule is economical in the sense that the "allocation diameter", defined as the diameter X of the basin of attraction containing the origin, is a random variable with a rapidly decaying tail. Specifically, we have the tail bound: P(X > R) < C exp[ -c R(log R)^(alpha_d) ], for all R>2, where: alpha_d = (d-2)/d for d>=4; alpha_3 can be taken as any number <-4/3; and C,c are positive constants that depend on d and alpha_d. This is the first construction of an allocation rule of Lebesgue measure to a Poisson point process with subpolynomial decay of the tail P(X>R).

math.PR↗

arXiv2007

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.

math.PR↗

arXiv2007

Concentration of Haar measures, with an application to random matrices

We show that the mixing times of random walks on compact groups can be used to obtain concentration inequalities for the respective Haar measures. As an application, we derive a concentration inequality for the empirical distribution of eigenvalues of sums of random hermitian matrices, with possible applications in free probability. The advantage over existing techniques is that the new method can deal with functions that are non-Lipschitz or even discontinuous with respect to the usual metrics.

math.PR↗

arXiv2006

Stein's method for concentration inequalities

We introduce a version of Stein's method for proving concentration and moment inequalities in problems with dependence. Simple illustrative examples from combinatorics, physics, and mathematical statistics are provided.

math.PR↗

arXiv2005

Dynamical Interactions of Planetary Systems in Dense Stellar Environments

We study dynamical interactions of star--planet binaries with other single stars. We derive analytical cross sections for all possible outcomes, and confirm them with numerical scattering experiments. We find that a wide mass ratio in the binary introduces a region in parameter space that is inaccessible to comparable-mass systems, in which the nature of the dynamical interaction is fundamentally different from what has traditionally been considered in the literature on binary scattering. We study the properties of the planetary systems that result from the scattering interactions for all regions of parameter space, paying particular attention to the location of the "hard--soft" boundary. The structure of the parameter space turns out to be significantly richer than a simple statement of the location of the "hard--soft" boundary would imply. We consider the implications of our findings, calculating characteristic lifetimes for planetary systems in dense stellar environments, and applying the results to previous analytical studies, as well as past and future observations. Recognizing that the system PSR B1620-26 in the globular cluster M4 lies in the "new" region of parameter space, we perform a detailed analysis quantifying the likelihood of different scenarios in forming the system we see today.

astro-ph↗

arXiv2005

Exchangeable pairs and Poisson approximation

This is a survey paper on Poisson approximation using Stein's 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 collector's 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.

math.PR↗

arXiv2005

An error bound in the Sudakov-Fernique inequality

We obtain an asymptotically sharp error bound in the classical Sudakov-Fernique comparison inequality for finite collections of gaussian random variables. Our proof is short and self-contained, and gives an easy alternative argument for the classical inequality, extended to the case of non-centered processes.

math.PR↗

arXiv2005

A simple invariance theorem

We present a simple extension of Lindeberg's argument for the Central Limit Theorem to get a general invariance result. We apply the technique to prove results from random matrix theory, spin glasses, and maxima of random fields.

math.PR↗