Related papers: A determinantal formula for the GOE Tracy-Widom di…
We undertake an analysis of Fredholm determinants arising from kernels whose defining functions satisfy a Schr\"odinger type equation. When this defining function is the Airy one, the evaluation of the corresponding Fredholm determinant…
The Hermite polynomials are ubiquitous but can be difficult to work with due to their unwieldy definition in terms of derivatives. To remedy this, we showcase an underappreciated Gaussian integral formula for the Hermite polynomials, which…
In a recent study of large non-null sample covariance matrices, a new sequence of functions generalizing the GUE Tracy-Widom distribution of random matrix theory was obtained. This paper derives Painlev\'e formulas of these functions and…
In earlier work the authors obtained integral formulas for probabilities for a single particle in the asymmetric simple exclusion process. Here formulas are obtained for joint probabilities for several particles. In the case of a single…
While originally discovered in the context of the Gaussian Unitary Ensemble, the Tracy-Widom distribution also rules the height fluctuations of growth processes. This suggests that there might be other nonequilibrium processes in which the…
We derive an asymptotic expansion for the distribution of a compound sum of independent random variables, all having the same light-tailed subexponential distribution. The examples of a Poisson and geometric number of summands serve as an…
We consider the totally asymmetric simple exclusion process with initial conditions generating a shock. The fluctuations of particle positions are asymptotically governed by the randomness around the two characteristic lines joining at the…
We study current fluctuations of a two-species asymmetric exclusion process, known as the Arndt-Heinzel-Rittenberg model. For a step-Bernoulli initial condition with finite number of particles, we provide an explicit multiple integral…
Uncovering genuine relationships between a response variable of interest and a large collection of covariates is a fundamental and practically important problem. In the context of Gaussian linear models, both the Bayesian and non-Bayesian…
In the present paper, we derive a renormalization formula "\`a la Hardy-Littlewood" for the Gaussian exponential sums with an exact formula for the remainder term. We use this formula to describe the typical growth of the Gaussian…
This paper presents the asymptotic distributions of a general likelihood-based test statistic, derived using results of Wilks and Wald. The general form of the test statistic incorporates the test statistics and associated asymptotic…
We establish some asymptotic expansions for infinite weighted convolutions of distributions having light subexponential tails. Examples are presented, some showing that in order to obtain an expansion with two significant terms, one needs…
This is a survey of Harold Widom's work in random matrices. We start with his pioneering papers on the sine-kernel determinant, continue with his and Craig Tracy's groundbreaking results concerning the distribution functions of random…
By an extension of the Bethe ansatz method used by Gwa and Spohn, we obtain an exact expression for the large deviation function of the time averaged current for the fully asymmetric exclusion process in a ring containing $N$ sites and $p$…
Within the framework of likelihood-based statistical tests for high energy physics measurements, we derive generalized expressions for estimating the statistical significance of discovery using the asymptotic approximations of Wilks and…
As a solvable and broadly applicable model system, the totally asymmetric exclusion process enjoys iconic status in the theory of non-equilibrium phase transitions. Here, we focus on the time dependence of the total number of particles on a…
In random matrix theory (RMT), the Tracy-Widom (TW) distribution describes the behavior of the largest eigenvalue. We consider here two models in which TW undergoes transformations. In the first one disorder is introduced in the Gaussian…
Gaussian Quadrature is a well known technique for numerical integration. Recently Gaussian quadrature with respect to discrete measures corresponding to finite sums have found some new interest. In this paper we apply these ideas to…
This is an expanded version of a series of lectures delivered by the second author in June, 2009. It describes the results of three of the authors' papers on ASEP, from the derivation of exact formulas for configuration probabilities,…
We study a linear-fractional Bienaym\'e-Galton-Watson process with a general type space. The corresponding tree contour process is described by an alternating random walk with the downward jumps having a geometric distribution. This leads…