Non-intersecting Brownian motions leaving from and going to several points

PHYSICA D-NONLINEAR PHENOMENA

卷:241

Article

Adler, Mark²  van Moerbeke, Pierre^1,2  Vanderstichelen, Didier¹ 

[1] Catholic Univ Louvain, Dept Math, B-1348 Louvain, Belgium

[2] Brandeis Univ, Dept Math, Waltham, MA 02454 USA

关键词: Non-intersecting Brownian motions;    Multi-component KP equation;    Virasoro constraints;   

DOI  :  10.1016/j.physd.2011.10.019

来源: Elsevier

【 摘 要 】

Consider n non-intersecting Brownian motions on R, depending on time t is an element of [0, I], with m(i) particles forced to leave from a(i) at time t = 0, 1 <= i <= q, and n(j) particles forced to end up at Pi at time t = 1, 1 <= j <= p. For arbitrary p and q, it is not known if the distribution of the positions of the non-intersecting Brownian particles at a given time 0 < t < 1, is the same as the joint distribution of the eigenvalues of a matrix ensemble. This paper proves the existence, for general p and q, of a partial differential equation (PDE) satisfied by the log of the probability to find all the particles in a disjoint union of intervals E = boolean OR(r)(i-1) vertical bar C2i-1, C-2i vertical bar subset of R at a given time 0 < t < 1. The variables are the coordinates of the starting and ending points of the particles, and the boundary points of the set E. 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 p = q = 2 will be discussed in the last section. (C) 2012 Published by Elsevier B.V.

【 授权许可】

Free