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Abstract: In 1962, Dyson (1962) introduced dynamics in random matrix models, in particular into GUE (also for and 4), by letting the entries evolve according to independent Ornstein–Uhlenbeck processes. Dyson shows the spectral points of the matrix evolve according to non-intersecting Brownian motions. The present paper shows that the interlacing spectra of two consecutive principal minors form a Markov process (diffusion) as well. This diffusion consists of two sets of Dyson non-intersecting Brownian motions, with a specific interaction respecting the interlacing. This is revealed in the form of the generator, the transition probability and the invariant measure, which are provided here; this is done in all cases: . It is also shown that the spectra of three consecutive minors ceases to be Markovian for . [Copyright &y& Elsevier]

Abstract: Consider non-intersecting Brownian motions on , depending on time , with particles forced to leave from at time , , and particles forced to end up at at time , . For arbitrary and , it is not known if the distribution of the positions of the non-intersecting Brownian particles at a given time , is the same as the joint distribution of the eigenvalues of a matrix ensemble. This paper proves the existence, for general and , of a partial differential equation (PDE) satisfied by the log of the probability to find all the particles in a disjoint union of intervals at a given time . The variables are the coordinates of the starting and ending points of the particles, and the boundary points of the set . The proof of the existence of such a PDE, using Virasoro constraints and the multicomponent KP hierarchy, is based on the method of elimination of the unwanted partials; that this is possible is a miracle! Unfortunately we were unable to find its explicit expression. The case will be discussed in the last section. [Copyright &y& Elsevier]

Abstract: Consider non-intersecting Brownian motions on the real line, starting from the origin at , with particles forced to reach p distinct target points at time , with . This can be viewed as a diffusion process in a sector of . This work shows that the transition probability, that is the probability for the particles to pass through windows at times , satisfies, in a new set of variables, a non-linear PDE which can be expressed as a near-Wronskian; that is a determinant of a matrix of size , with each row being a derivative of the previous, except for the last column. It is an interesting open question to understand those equations from a more probabilistic point of view. As an application of these equations, let the number of particles forced to the extreme points and tend to infinity; keep the number of particles forced to intermediate points fixed (inliers), but let the target points themselves go to infinity according to a proper scale. A new critical process appears at the point of bifurcation, where the bulk of the particles forced to depart from those going to . These statistical fluctuations near that point of bifurcation are specified by a kernel, which is a rational perturbation of the Pearcey kernel. This work also shows that such equations are an essential tool in obtaining certain asymptotic results. Finally, the paper contains a conjecture. [Copyright &y& Elsevier]

Abstract: Consider non-intersecting Brownian motions on the line leaving from the origin and forced to two arbitrary points. Letting the number of Brownian particles tend to infinity, and upon rescaling, there is a point of bifurcation, where the support of the density of particles goes from one interval to two intervals. In this paper, we show that at that very point of bifurcation a cusp appears, near which the Brownian paths fluctuate like the Pearcey process. This is a universality result within this class of problems. Tracy and Widom obtained such a result in the symmetric case, when the two target points are symmetric with regard to the origin. This asymmetry enabled us to improve considerably a result concerning the non-linear partial differential equations governing the transition probabilities for the Pearcey process, obtained by Adler and van Moerbeke. [Copyright &y& Elsevier]

Questions on random matrices and non-intersecting Brownian motions have led to the study of moment matrices with regard to several weights. The main result of this paper is to show that the determinants of such moment matrices satisfy, upon adding one set of “time” deformations for each weight, the multi-component KP-hierarchy: these determinants are thus “tau-functions” for these integrable hierarchies. The tau-functions, so obtained, with appropriate shifts of the time-parameters (forward and backwards) will be expressed in terms of multiple orthogonal polynomials for these weights and their Cauchy transforms. The main result is a vast generalization of a known fact about infinitesimal deformations of orthogonal polynomials: it concerns an identity between the orthogonality of polynomials on the real line, the bilinear identity in KP theory and a generating functional for the full KP theory. An additional fact not discussed in this paper is that these τ-functions satisfy Virasoro constraints with respect to these time parameters. As one of the many examples worked out in this paper, we consider N non-intersecting Brownian motions in $${\mathbb R}$$ leaving from the origin, with n i particles forced to reach p distinct target points b i at time t = 1; of course, $${\sum_{i=1}^p n_i = N}$$ . We give a PDE, in terms of the boundary points of the interval E, for the probability that the Brownian particles all pass through an interval E at time 0 < t < 1. It is given by the determinant of a ( p + 1) × ( p + 1) matrix, which is nearly a wronskian. This theory is also applied to biorthogonal polynomials and orthogonal polynomials on the circle. [ABSTRACT FROM AUTHOR]