Nonintersecting Brownian motions on the unit circle

We consider an ensemble of $n$ nonintersecting Brownian particles on the unit circle with diffusion parameter $n^{-1/2}$, which are conditioned to begin at the same point and to return to that point after time $T$, but otherwise not to intersect. There is a critical value of $T$ which separates the subcritical case, in which it is vanishingly unlikely that the particles wrap around the circle, and the supercritical case, in which particles may wrap around the circle. In this paper, we show that in the subcritical and critical cases the probability that the total winding number is zero is almost surely 1 as $n\to\infty$, and in the supercritical case that the distribution of the total winding number converges to the discrete normal distribution. We also give a streamlined approach to identifying the Pearcey and tacnode processes in scaling limits. The formula of the tacnode correlation kernel is new and involves a solution to a Lax system for the Painlev\'{e} II equation of size 2 $\times$ 2. The proofs are based on the determinantal structure of the ensemble, asymptotic results for the related system of discrete Gaussian orthogonal polynomials, and a formulation of the correlation kernel in terms of a double contour integral.Comment: Published at http://dx.doi.org/10.1214/14-AOP998 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Asymptotics of free fermions in a quadratic well at finite temperature and the Moshe-Neuberger-Shapiro random matrix model

We derive the local statistics of the canonical ensemble of free fermions in a quadratic potential well at finite temperature, as the particle number approaches infinity. This free fermion model is equivalent to a random matrix model proposed by Moshe, Neuberger and Shapiro. Limiting behaviors obtained before for the grand canonical ensemble are observed in the canonical ensemble: We have at the edge the phase transition from the Tracy--Widom distribution to the Gumbel distribution via the Kardar-Parisi-Zhang (KPZ) crossover distribution, and in the bulk the phase transition from the sine point process to the Poisson point process. A similarity between this model and a class of models in the KPZ universality class is explained. We also derive the multi-time correlation functions and the multi-time gap probability formulas for the free fermions along the imaginary time.Comment: 46 pages, 2 figure

Exact solution of the six-vertex model with domain wall boundary conditions. Ferroelectric phase

This is a continuation of the paper [4] of Bleher and Fokin, in which the large $n$ asymptotics is obtained for the partition function $Z_n$ of the six-vertex model with domain wall boundary conditions in the disordered phase. In the present paper we obtain the large $n$ asymptotics of $Z_n$ in the ferroelectric phase. We prove that for any \ep>0, as $n\to\infty$, Z_n=CG^nF^{n^2}[1+O(e^{-n^{1-\ep}})], and we find the exact value of the constants $C,G$ and $F$. The proof is based on the large $n$ asymptotics for the underlying discrete orthogonal polynomials and on the Toda equation for the tau-function.Comment: 22 pages, 7 figure

Riemann-Hilbert Approach to the Six-Vertex Model

The six-vertex model, or the square ice model, with domain wall boundary conditions (DWBC) has been introduced and solved for finite $n$ by Korepin and Izergin. The solution is based on the Yang-Baxter equations and it represents the free energy in terms of an $n\times n$ Hankel determinant. Paul Zinn-Justin observed that the Izergin-Korepin formula can be re-expressed in terms of the partition function of a random matrix model with a nonpolynomial interaction. We use this observation to obtain the large $n$ asymptotics of the six-vertex model with DWBC. The solution is based on the Riemann-Hilbert approach. In this paper we review asymptotic results obtained in different regions of the phase diagram.Comment: 15 pages, 4 figures, Submitted to the MSRI volume on "Random Matrix Theory, Interacting Particle Systems and Integrable Systems". arXiv admin note: text overlap with arXiv:math-ph/051003

Uniform Asymptotics for Discrete Orthogonal Polynomials with Respect to Varying Exponential Weights on a Regular Infinite Lattice

We consider the large-$N$ asymptotics of a system of discrete orthogonal polynomials on an infinite regular lattice of mesh $\frac{1}{N}$, with weight $e^{-NV(x)}$, where $V(x)$ is a real analytic function with sufficient growth at infinity. The proof is based on formulation of an interpolation problem for discrete orthogonal polynomials, which can be converted to a Riemann-Hilbert problem, and steepest descent analysis of this Riemann-Hilbert problem.Comment: 32 pages, 4 figures; corrected versio

Two Lax systems for the Painlev\'e II equation, and two related kernels in random matrix theory

We consider two Lax systems for the homogeneous Painlev\'{e} II equation: one of size $2\times 2$ studied by Flaschka and Newell in the early 1980's, and one of size $4\times 4$ introduced by Delvaux-Kuijlaars-Zhang and Duits-Geudens in the early 2010's. We prove that solutions to the $4\times 4$ system can be derived from those to the $2\times 2$ system via an integral transform, and consequently relate the Stokes multipliers for the two systems. As corollaries we are able to express two kernels for determinantal processes as contour integrals involving the Flaschka-Newell Lax system: the tacnode kernel arising in models of nonintersecting paths, and a critical kernel arising in a two-matrix model.Comment: 46 pages, 20 figure

Six-vertex model with partial domain wall boundary conditions: ferroelectric phase

We obtain an asymptotic formula for the partition function of the six-vertex model with partial domain wall boundary conditions in the ferroelectric phase region. The proof is based on a formula for the partition function involving the determinant of a matrix of mixed Vandermonde/Hankel type. This determinant can be expressed in terms of a system of discrete orthogonal polynomials, which can then be evaluated asymptotically by comparison with the Meixner polynomials.Comment: 32 pages, 6 figures, minor changes in version

Nonintersecting Brownian bridges on the unit circle with drift

Nonintersecting Brownian bridges on the unit circle form a determinantal stochastic process exhibiting random matrix statistics for large numbers of walkers. We investigate the effect of adding a drift term to walkers on the circle conditioned to start and end at the same position. For each return time $T<\pi^2$ we show that if the absolute value of the drift is less than a critical value then the expected total winding number is asymptotically zero. In addition, we compute the asymptotic distribution of total winding numbers in the double-scaling regime in which the expected total winding is finite. The method of proof is Riemann--Hilbert analysis of a certain family of discrete orthogonal polynomials with varying complex exponential weights. This is the first asymptotic analysis of such a class of polynomials. We determine asymptotic formulas and demonstrate the emergence of a second band of zeros by a mechanism not previously seen for discrete orthogonal polynomials with real weights.Comment: 40 pages, 11 figure

The k-tacnode process

The tacnode process is a universal behavior arising in nonintersecting particle systems and tiling problems. For Dyson Brownian bridges, the tacnode process describes the grazing collision of two packets of walkers. We consider such a Dyson sea on the unit circle with drift. For any integer k, we show that an appropriate double scaling of the drift and return time leads to a generalization of the tacnode process in which k particles are expected to wrap around the circle. We derive winding number probabilities and an expression for the correlation kernel in terms of functions related to the generalized Hastings-McLeod solutions to the inhomogeneous Painleve-II equation. The method of proof is asymptotic analysis of discrete orthogonal polynomials with a complex weight.Comment: 38 pages, 8 figure

The Fourier extension method and discrete orthogonal polynomials on an arc of the circle

The Fourier extension method, also known as the Fourier continuation method, is a method for approximating non-periodic functions on an interval using truncated Fourier series with period larger than the interval on which the function is defined. When the function being approximated is known at only finitely many points, the approximation is constructed as a projection based on this discrete set of points. In this paper we address the issue of estimating the absolute error in the approximation. The error can be expressed in terms of a system of discrete orthogonal polynomials on an arc of the unit circle, and these polynomials are then evaluated asymptotically using Riemann--Hilbert methods.Comment: 47 pages, 3 figure