A Limit Theorem for Shifted Schur Measures

To each partition $\lambda$ with distinct parts we assign the probability $Q_\lambda(x) P_\lambda(y)/Z$ where $Q_\lambda$ and $P_\lambda$ are the Schur $Q$-functions and $Z$ is a normalization constant. This measure, which we call the shifted Schur measure, is analogous to the much-studied Schur measure. For the specialization of the first $m$ coordinates of $x$ and the first $n$ coordinates of $y$ equal to $\alpha$ ($0<\alpha<1$) and the rest equal to zero, we derive a limit law for $\lambda_1$ as m,n\ra\infty with $\tau=m/n$ fixed. For the Schur measure the $\alpha$-specialization limit law was derived by Johansson. Our main result implies that the two limit laws are identical.Comment: 35 pages, 2 figures. Version 3 adds a section on the Poisson limit of the shifted Schur measur

Random Unitary Matrices, Permutations and Painleve

This paper is concerned with certain connections between the ensemble of n x n unitary matrices -- specifically the characteristic function of the random variable tr(U) -- and combinatorics -- specifically Ulam's problem concerning the distribution of the length of the longest increasing subsequence in permutation groups -- and the appearance of Painleve functions in the answers to apparently unrelated questions. Among the results is a representation in terms of a Painleve V function for the characteristic function of tr(U) and (using recent results of Baik, Deift and Johansson) an expression in terms of a Painleve II function for the limiting distribution of the length of the longest increasing subsequence in the hyperoctahedral group.Comment: 21 pages, 1 figure. Revised paper simplifies the statement of Theorem 1 and adds some additional reference

Correlation Functions, Cluster Functions and Spacing Distributions for Random Matrices

The usual formulas for the correlation functions in orthogonal and symplectic matrix models express them as quaternion determinants. From this representation one can deduce formulas for spacing probabilities in terms of Fredholm determinants of matrix-valued kernels. The derivations of the various formulas are somewhat involved. In this article we present a direct approach which leads immediately to scalar kernels for unitary ensembles and matrix kernels for the orthogonal and symplectic ensembles, and the representations of the correlation functions, cluster functions and spacing distributions in terms of them.Comment: 22 pages. LaTeX file. Minor correctio

A Distribution Function Arising in Computational Biology

Karlin and Altschul in their statistical analysis for multiple high-scoring segments in molecular sequences introduced a distribution function which gives the probability there are at least r distinct and consistently ordered segment pairs all with score at least x. For long sequences this distribution can be expressed in terms of the distribution of the length of the longest increasing subsequence in a random permutation. Within the past few years, this last quantity has been extensively studied in the mathematics literature. The purpose of these notes is to summarize these new mathematical developments in a form suitable for use in computational biology.Comment: 9 pages, no figures. Revised version makes minor change

On the ground state energy of the delta-function Bose gas

The weak coupling asymptotics, to order $(c/\rho)^2$, of the ground state energy of the delta-function Bose gasmis derived. Here $2c\ge 0$ is the delta-function potential amplitude and $\rho$ the density of the gas in the thermodynamic limit. The analysis uses the electrostatic interpretation of the Lieb-Liniger integral equation.Comment: 18 page

Application of Random Matrix Theory to Multivariate Statistics

This is an expository account of the edge eigenvalue distributions in random matrix theory and their application in multivariate statistics. The emphasis is on the Painlev\'e representations of these distributions

On the Singularities in the Susceptibility Expansion for the Two-Dimensional Ising Model

For temperatures below the critical temperature, the magnetic susceptibility for the two-dimensional isotropic Ising model can be expressed in terms of an infinite series of multiple integrals. With respect to a parameter related to temperature and the interaction constant, the integrals may be extended to functions analytic outside the unit circle. In a groundbreaking paper, B. G. Nickel identified a class of singularities of these integrals on the unit circle. In this note we show that there are no other singularities on the unit circle.Comment: 13 page

Asymptotics of a class of Fredholm determinants

In this expository article we describe the asymptotics of certain Fredholm determinants which provide solutions to the cylindrical Toda equations, and we explain how these asymptotics are derived. The connection with Fredholm determinants arising in the theory of random matrices, and their asymptotics, are also discussed.Comment: 8 pages, LaTeX fil

On ASEP with Periodic Step Bernoulli Initial Condition

We consider the asymmetric simple exclusion process (ASEP) on the integers in which the initial density at a site (the probability that it is occupied) is given by a periodic function on the positive integers. (When the function is constant this is the step Bernoulli initial condition.) Starting with a result in earlier work we find a formula for the probability distribution for a given particle at a given time which is a sum over positive integers k of integrals of order k.Comment: 8 pages. In version 2 the proof of the main result is simplified, and another one is obtained in the proces