A Graph-based Equilibrium Problem for the Limiting Distribution of Nonintersecting Brownian Motions at Low Temperature

We consider n nonintersecting Brownian motion paths with p prescribed starting positions at time t=0 and q prescribed ending positions at time t=1. The positions of the paths at any intermediate time are a determinantal point process, which in the case p=1 is equivalent to the eigenvalue distribution of a random matrix from the Gaussian unitary ensemble with external source. For general p and q, we show that if a temperature parameter is sufficiently small, then the distribution of the Brownian paths is characterized in the large n limit by a vector equilibrium problem with an interaction matrix that is based on a bipartite planar graph. Our proof is based on a steepest descent analysis of an associated (p+q)×(p+q) matrix-valued Riemann–Hilbert problem whose solution is built out of multiple orthogonal polynomials. A new feature of the steepest descent analysis is a systematic opening of a large number of global lenses.     

Universality of the double scaling limit in random matrix models

We study unitary random matrix ensembles in the critical case where the limiting mean eigenvalue density vanishes quadratically at an interior point of the support. We establish universality of the limits of the eigenvalue correlation kernel at such a critical point in a double scaling limit. The limiting kernels are constructed out of functions associated with the second Painlevé equation. This extends a result of Bleher and Its for the special case of a critical quartic potential. The two main tools we use are equilibrium measures and Riemann‐Hilbert problems. In our treatment of equilibrium measures we allow a negative density near the critical point, which enables us to treat all cases simultaneously. The asymptotic analysis of the Riemann‐Hilbert problem is done with the Deift‐Zhou steepest‐descent analysis. For the construction of a local parametrix at the critical point we introduce a modification of the approach of Baik, Deift, and Johansson so that we are able to satisfy the required jump properties exactly. © 2005 Wiley Periodicals, Inc.     

Dyson's nonintersecting Brownian motions with a few outliers

Consider n nonintersecting Brownian particles on ? (Dyson Brownian motions), all starting from the origin at time t = 0 and forced to return to x = 0 at time t = 1. For large n, the average mean density of particles has its support, for each 0 < t < 1, on the interval ±√2nt(1 ? t). The Airy process ??(τ) is defined as the motion of these nonintersecting Brownian motions for large n but viewed from the curve ?? : y = √2nt(1 ? t) with an appropriate space‐time rescaling. Assume now a finite number r of these particles are forced to a different target point, say a = ρ₀√n/2 > 0. Does it affect the Brownian fluctuations along the curve ?? for large n? In this paper, we show that no new process appears as long as one considers points (y, t) ∈ ?? such that 0 < t < (1 + ρ)^?1, which is the t‐coordinate of the point of tangency of the tangent to the curve passing through (ρ₀√n/2, 1). At this point the fluctuations obey a new statistics, which we call the Airy process with r outliers ??^(r)(τ) (in short, r‐Airy process). The log of the probability that at time τ the cloud does not exceed x is given by the Fredholm determinant of a new kernel (extending the Airy kernel), and it satisfies a nonlinear PDE in x and τ, from which the asymptotic behavior of the process can be deduced for τ → ?∞. This kernel is closely related to one found by Baik, Ben Arous, and Péché in the context of multivariate statistics. © 2008 Wiley Periodicals, Inc.     

Gaussian unitary ensembles with two jump discontinuities,PDEs, and the coupled Painlevé II and IV systems

We consider the Hankel determinant generated by the Gaussian weight with two jump discontinuities. Utilizing the results of Min and Chen [Math. Methods Appl Sci. 2019;42:301‐321] where a second‐order partial differential equation (PDE) was deduced for the log derivative of the Hankel determinant by using the ladder operators adapted to orthogonal polynomials, we derive the coupled Painlevé IV system which was established in Wu and Xu [arXiv: 2002.11240v2] by a study of the Riemann‐Hilbert problem for orthogonal polynomials. Under double scaling, we show that, as , the log derivative of the Hankel determinant in the scaled variables tends to the Hamiltonian of a coupled Painlevé II system and it satisfies a second‐order PDE. In addition, we obtain the asymptotics for the recurrence coefficients of orthogonal polynomials, which are connected with the solutions of the coupled Painlevé II system.     

Riemann‐Hilbert problem and matrix discrete Painlevé II systems

Matrix Szeg? biorthogonal polynomials for quasi‐definite matrices of Hölder continuous weights are studied. A Riemann‐Hilbert problem is uniquely solved in terms of the matrix Szeg? polynomials and its Cauchy transforms. The Riemann‐Hilbert problem is given as an appropriate framework for the discussion of the Szeg? matrix and the associated Szeg? recursion relations for the matrix orthogonal polynomials and its Cauchy transforms. Pearson‐type differential systems characterizing the matrix of weights are studied. These are linear systems of ordinary differential equations that are required to have trivial monodromy. Linear ordinary differential equations for the matrix Szeg? polynomials and its Cauchy transforms are derived. It is shown how these Pearson systems lead to nonlinear difference equations for the Verblunsky matrices and two examples, of Fuchsian and non‐Fuchsian type, are considered. For both cases, a new matrix version of the discrete Painlevé II equation for the Verblunsky matrices is found. Reductions of these matrix discrete Painlevé II systems presenting locality are discussed.     

Asymptotics of Discrete Painlevé V transcendents via the Riemann–Hilbert Approach

We study a system of discrete Painlevé V equations via the Riemann–Hilbert approach. We begin with an isomonodromy problem for dPV, which admits a discrete Riemann–Hilbert problem formulation. The asymptotics of the discrete Riemann–Hilbert problem is derived via the nonlinear steepest descent method of Deift and Zhou. In the analysis, a parametrix is constructed in terms of specific Painlevé V transcendents. As a result, the asymptotics of the dPV transcendents are represented in terms of the PV transcendents. In the special case, our result confirms a conjecture of Borodin, that the difference Schlesinger equations converge to the differential Schlesinger equations at the solution level.     

Painlevé II asymptotics near the leading edge of the oscillatory zone for the Korteweg—de Vries equation in the small‐dispersion limit

In the small‐dispersion limit, solutions to the Korteweg—de Vries equation develop an interval of fast oscillations after a certain time. We obtain a universal asymptotic expansion for the Korteweg—de Vries solution near the leading edge of the oscillatory zone up to second‐order corrections. This expansion involves the Hastings‐McLeod solution of the Painlevé II equation. We prove our results using the Riemann‐Hilbert approach. © 2009 Wiley Periodicals, Inc.     

A general framework for solving Riemann–Hilbert problems numerically

A new, numerical framework for the approximation of solutions to matrix-valued Riemann?CHilbert problems is developed, based on a recent method for the homogeneous Painlevé II Riemann?CHilbert problem. We demonstrate its effectiveness by computing solutions to other Painlevé transcendents. An implementation in Mathematica is made available online.     

Fredholm determinants,Jimbo‐Miwa‐Ueno τ‐functions,and representation theory

The authors show that a wide class of Fredholm determinants arising in the representation theory of “big” groups, such as the infinite‐dimensional unitary group, solve Painlevé equations. Their methods are based on the theory of integrable operators and the theory of Riemann‐Hilbert problems. © 2002 Wiley Periodicals, Inc.     

Painlevé Kernels in Hermitian Matrix Models

After reviewing the Hermitian one-matrix model, we will give a brief introduction to the Hermitian two-matrix model and present a summary of some recent results on the asymptotic behavior of the two-matrix model with a quartic potential. In particular, we will discuss a limiting kernel in the quartic/quadratic case that is constructed out of a 4×4 Riemann–Hilbert problem related to the Painlevé II equation. Also an open problem will be presented.     

Tanaka Formula for Multidimensional Brownian Motions

We study the Tanaka formula for multidimensional Brownian motions in the framework of generalized Wiener functionals. More precisely, we show that the submartingale U(B _t –x) is decomposed in the sence of generalized Wiener functionals into the sum of a martingale and the Brownian local time, U being twice of the kernel of Newtonian potential and B _t the multidimensional Brownian motion. We also discuss on an aspect of the Tanaka formula for multidimensional Brownian motions as the Doob–Meyer decomposition.     

PDEs for the Gaussian ensemble with external source and the Pearcey distribution

The present paper studies a Gaussian Hermitian random matrix ensemble with external source, given by a fixed diagonal matrix with two eigenvalues ±a. As a first result, the probability that the eigenvalues of the ensemble belong to an interval E satisfies a fourth‐order PDE with quartic nonlinearity; the variables are the eigenvalue a and the boundary of E. This equation enables one to find a PDE for the Pearcey distribution. The latter describes the statistics of the eigenvalues near the closure of a gap, i.e., when the support of the equilibrium measure for large‐size random matrices has a gap that can be made to close. The Gaussian Hermitian random matrix ensemble with external source, described above, has this feature. The Pearcey distribution is shown to satisfy a fourth‐order PDE with cubic nonlinearity. This also gives the PDE for the transition probability of the Pearcey process, a limiting process associated with nonintersecting Brownian motions on ℝ. © 2006 Wiley Periodicals, Inc.     

Genealogy of extremal particles of branching Brownian motion

Branching Brownian motion describes a system of particles that diffuse in space and split into offspring according to a certain random mechanism. By virtue of the groundbreaking work by M. Bramson on the convergence of solutions of the Fisher‐KPP equation to traveling waves, the law of the rightmost particle in the limit of large times is rather well understood. In this work, we address the full statistics of the extremal particles (first‐, second‐, third‐largest, etc.). In particular, we prove that in the large t‐limit, such particles descend with overwhelming probability from ancestors having split either within a distance of order 1 from time 0, or within a distance of order 1 from time t. The approach relies on characterizing, up to a certain level of precision, the paths of the extremal particles. As a byproduct, a heuristic picture of branching Brownian motion “at the edge” emerges, which sheds light on the still unknown limiting extremal process. © 2011 Wiley Periodicals, Inc.     

On the Reproducing Kernel Hilbert Spaces Associated with the Fractional and Bi-Fractional Brownian Motions

We present decompositions of various positive kernels as integrals or sums of positive kernels. Within this framework we study the reproducing kernel Hilbert spaces associated with the fractional and bi-fractional Brownian motions. As a tool, we define a new function of two complex variables, which is a natural generalization of the classical Gamma function for the setting we consider. D. Alpay thanks the Earl Katz family for endowing the chair which supports his research.     

Gibbs Measures on Mutually Interacting Brownian Paths under Singularities

We are interested in the analysis of Gibbs measures defined on two independent Brownian paths in ?^d interacting through a mutual self‐attraction. This is expressed by the Hamiltonian with two probability measures μ and ν representing the occupation measures of two independent Brownian motions. We will be interested in a class of potentials V that are singular , e.g., Dirac‐ or Coulomb‐type interactions in ?³, or the correlation function of the parabolic Anderson problem with white noise potential. The mutual interaction of the Brownian paths inspires a compactification of the quotient space of orbits of product measures, which is structurally different from the self‐interacting case introduced in [27], owing to the lack of shift‐invariant structure in the mutual interaction. We prove a strong large‐deviation principle for the product measures of two Brownian occupation measures in such a compactification and derive asymptotic path behavior under Gibbs measures on Wiener paths arising from mutually attracting singular interactions. For the spatially smoothened parabolic Anderson model with white noise potential, our analysis allows a direct computation of the annealed Lyapunov exponents, and a strict ordering of them implies the intermittency effect present in the smoothened model. © 2017 Wiley Periodicals, Inc.     

Higher‐order analogues of the Tracy‐Widom distribution and the Painlevé II hierarchy

We study Fredholm determinants related to a family of kernels that describe the edge eigenvalue behavior in unitary random matrix models with critical edge points. The kernels are natural higher‐order analogues of the Airy kernel and are built out of functions associated with the Painlevé I hierarchy. The Fredholm determinants related to those kernels are higher‐order generalizations of the Tracy‐Widom distribution. We give an explicit expression for the determinants in terms of a distinguished smooth solution to the Painlevé II hierarchy. In addition, we compute large gap asymptotics for the Fredholm determinants. © 2009 Wiley Periodicals, Inc.     

Large Deviations for Brownian Intersection Measures

We consider p independent Brownian motions in \input amssym ${\Bbb R}^d$ . We assume that p ≥ 2 and p (d ? 2) < d. Let ?_t denote the intersection measure of the p paths by time t, i.e., the random measure on \input amssym ${\Bbb R}^d$ that assigns to any measurable set \input amssym $A \subset {\Bbb R}^d$ the amount of intersection local time of the motions spent in A by time t. Earlier results of X. Chen derived the logarithmic asymptotics of the upper tails of the total mass \input amssym $\ell _t \left({{\Bbb R}^d } \right)$ as t → ∞. In this paper, we derive a large‐deviation principle for the normalized intersection measure t^?p?_t on the set of positive measures on some open bounded set \input amssym $B \subset {\Bbb R}^d$ as t → ∞ before exiting B. The rate function is explicit and gives some rigorous meaning, in this asymptotic regime, to the understanding that the intersection measure is the pointwise product of the densities of the normalized occupation times measures of the p motions. Our proof makes the classical Donsker‐Varadhan principle for the latter applicable to the intersection measure. A second version of our principle is proved for the motions observed until the individual exit times from B, conditional on a large total mass in some compact set . This extends earlier studies on the intersection measure by König and Mörters. © 2012 Wiley Periodicals, Inc.     

Double scaling limit in the random matrix model: The Riemann‐Hilbert approach

We prove the existence of the double scaling limit in the unitary matrix model with quartic interaction, and we show that the correlation functions in the double scaling limit are expressed in terms of the integrable kernel determined by the ψ function for the Hastings‐McLeod solution to the Painlevé II equation. The proof is based on the Riemann‐Hilbert approach, and the central point of the proof is an analysis of analytic semiclassical asymptotics for the ψ function at the critical point in the presence of four coalescing turning points. © 2003 Wiley Periodicals, Inc.     

Painlevé VI,Painlevé III,and the Hankel determinant associated with a degenerate Jacobi unitary ensemble

This paper studies the Hankel determinant generated by a perturbed Jacobi weight, which is closely related to the largest and smallest eigenvalue distribution of the degenerate Jacobi unitary ensemble. By using the ladder operator approach for the orthogonal polynomials, we find that the logarithmic derivative of the Hankel determinant satisfies a nonlinear second-order differential equation, which turns out to be the Jimbo–Miwa–Okamoto σ-form of the Painlevé VI equation by a translation transformation. We also show that, after a suitable double scaling, the differential equation is reduced to the Jimbo–Miwa–Okamoto σ-form of the Painlevé III. In the end, we obtain the asymptotic behavior of the Hankel determinant as t→1⁻ and t→0⁺ in two important cases, respectively.