Moduli spaces of d-connections and difference Painleve equations

We show that difference Painleve equations can be interpreted as isomorphisms of moduli spaces of d-connections on the projective line with given singularity structure. We also derive a new difference equation. It is the most general difference Painleve equation known so far, and it degenerates to both difference Painleve V and classical (differential) Painleve VI equations.Comment: 30 pages (LaTeX

Stochastic higher spin six vertex model and Macdonald measures

We prove an identity that relates the q-Laplace transform of the height function of a (higher spin inhomogeneous) stochastic six vertex model in a quadrant on one side and a multiplicative functional of a Macdonald measure on the other. The identity is used to prove the GUE Tracy-Widom asymptotics for two instances of the stochastic six vertex model via asymptotic analysis of the corresponding Schur measures.National Science Foundation (U.S.) (Grant DMS-1056390)National Science Foundation (U.S.) (Grant DMS-1607901

Generalized Green Functions and current correlations in the TASEP

We study correlation functions of the totally asymmetric simple exclusion process (TASEP) in discrete time with backward sequential update. We prove a determinantal formula for the generalized Green function which describes transitions between positions of particles at different individual time moments. In particular, the generalized Green function defines a probability measure at staircase lines on the space-time plane. The marginals of this measure are the TASEP correlation functions in the space-time region not covered by the standard Green function approach. As an example, we calculate the current correlation function that is the joint probability distribution of times taken by selected particles to travel given distance. An asymptotic analysis shows that current fluctuations converge to the ${Airy}_2$ process.Comment: 46 pages, 3 figure

Asymptotics of a discrete-time particle system near a reflecting boundary

We examine a discrete-time Markovian particle system on the quarter-plane introduced by M. Defosseux. The vertical boundary acts as a reflecting wall. The particle system lies in the Anisotropic Kardar-Parisi-Zhang with a wall universality class. After projecting to a single horizontal level, we take the longtime asymptotics and obtain the discrete Jacobi and symmetric Pearcey kernels. This is achieved by showing that the particle system is identical to a Markov chain arising from representations of the infinite-dimensional orthogonal group. The fixed-time marginals of this Markov chain are known to be determinantal point processes, allowing us to take the limit of the correlation kernel. We also give a simple example which shows that in the multi-level case, the particle system and the Markov chain evolve differently.Comment: 16 pages, Version 2 improves the expositio

Infinite-dimensional diffusions as limits of random walks on partitions

The present paper originated from our previous study of the problem of harmonic analysis on the infinite symmetric group. This problem leads to a family {P_z} of probability measures, the z-measures, which depend on the complex parameter z. The z-measures live on the Thoma simplex, an infinite-dimensional compact space which is a kind of dual object to the infinite symmetric group. The aim of the paper is to introduce stochastic dynamics related to the z-measures. Namely, we construct a family of diffusion processes in the Toma simplex indexed by the same parameter z. Our diffusions are obtained from certain Markov chains on partitions of natural numbers n in a scaling limit as n goes to infinity. These Markov chains arise in a natural way, due to the approximation of the infinite symmetric group by the increasing chain of the finite symmetric groups. Each z-measure P_z serves as a unique invariant distribution for the corresponding diffusion process, and the process is ergodic with respect to P_z. Moreover, P_z is a symmetrizing measure, so that the process is reversible. We describe the spectrum of its generator and compute the associated (pre)Dirichlet form.Comment: AMSTex, 33 pages. Version 2: minor changes, typos corrected, to appear in Prob. Theor. Rel. Field

Anisotropic KPZ growth in 2+1 dimensions: fluctuations and covariance structure

In [arXiv:0804.3035] we studied an interacting particle system which can be also interpreted as a stochastic growth model. This model belongs to the anisotropic KPZ class in 2+1 dimensions. In this paper we present the results that are relevant from the perspective of stochastic growth models, in particular: (a) the surface fluctuations are asymptotically Gaussian on a sqrt(ln(t)) scale and (b) the correlation structure of the surface is asymptotically given by the massless field.Comment: 13 pages, 4 figure