It is now believed that the limiting distribution function of the largest eigenvalue in the three classic random matrix models GOE, GUE and GSE describe new universal limit laws for a wide variety of processes arising in mathematical physics and interacting particle systems. These distribution functions, expressed in terms of a certain Painlev e II function, are described and their occurences surveyed.

We consider the diagonal susceptibility of the isotropic 2D Ising model for temperatures below the critical temperature. For a parameter k related to temperature and the interaction constant, we extend the diagonal susceptibility to complex k inside the unit disc, and prove the conjecture that the unit circle is a natural boundary.

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.

In this paper we obtain general integral formulas for probabilities in the asymmetric simple exclusion process (ASEP) on the integer lattice with nearest neighbor hopping rates p to the right and q=1-p to the left. For the most part we consider an N-particle system but for certain of these formulas we can take the limit as N goes to infinity. First we obtain, for the N-particle system, a formula for the probability of a configuration at time t, given the initial configuration. For this we use Bethe Ansatz ideas to solve the master equation, extending a result of Schuetz for the case N=2. The main results of the paper, derived from this, are integral formulas for the probability, for given initial configuration, that the m'th left-most particle is at x at time t. In one of these formulas we can take the limit as N goes to infinity, and it gives the probability for an infinite system where the initial configuration is bounded on one side. For the special case of the totally asymmetric simple exclusion process (TASEP) our formulas reduce to the known ones.

In earlier work, the authors obtained formulas for the probability in the asymmetric simple exclusion process that the mth particle from the left is at site x at time t. They were expressed in general as sums of multiple integrals and, for the case of step initial condition, as an integral involving a Fredholm determinant. In the present work, these results are generalized to the case where the mth particle is the left-most one in a contiguous block of L particles. The earlier work depended in a crucial way on two combinatorial identities, and the present work begins with a generalization of these identities to general L.

This paper first surveys the connection of integrable systems of the Painleve type to various distribution functions appearing in Wigner-Dyson random matrix theory. A short discussion is then given of the appearance of these same distributions in other areas of mathematics.

We consider the distributions of the lengths of the longest weakly increasing and strongly decreasing subsequences in words of length N from an alphabet of k letters. We find Toeplitz determinant representations for the exponential generating functions (on N) of these distribution functions and show that they are expressible in terms of solutions of Painlev e V equations. We show further that in the weakly increasing case the generating function gives the distribution of the smallest eigenvalue in the k x k Laguerre random matrix ensemble and that the distribution itself has, after centering and normalizing, an N -> infinity limit which is equal to the distribution function for the largest eigenvalue in the Gaussian Unitary Ensemble of k x k hermitian matrices of trace zero.

For the asymmetric simple exclusion process η t on the integer lattice with two-sided Bernoulli initial condition, we derive exact formulas for the following quantities: (1) ℙ(η t(x)=1), the probability that site x is occupied at time t; (2) the correlation function ℙ(η t(x)=1,η 0(0)=1); (3) the distribution function for Q t, the total flux across 0 at time t, and its exponential generating function.

In this article we derive, using standard methods of Toeplitz theory, an asymptotic formula for certain large minors of Toeplitz matrices. D. Bump and P. Diaconis obtained the same asymptotics using representation theory, with an answer having a different form.

We give an exact expression for the distribution of the position X(t) of a single second class particle in the asymmetric simple exclusion process (ASEP) where initially the second class particle is located at the origin and the first class particles occupy the sites {1,2,...}.