Modèles de mouvements browniens, processus critiques et matrices aléatoires
(2010)
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- Authors
- Supervisors
- van Moerbeke, Pierre
- Abstract
- (en) A few years ago, Aptekarev, Bleher and Kuijlaars have demonstrated, using an earlier result due to Karlin and McGregor, that the random matrix theory, especially that of the Gaussian ensemble of Hermitian matrices, deformed by an external potential, enables us to study the dynamical evolution of n non-intersecting, one-dimensional Brownian motions, subjected to specific constraints with regard to their points of departure and arrival. In this work, we have established that, if these movements all leave the same origin at the initial time, and are sent on p targets, then the probability that all paths pass through a "window" consisting of a finite union of intervals at an intermediate time t, satisfies a non-linear partial differential equation (PDE) in t and in the boundary points of the spatial intervals. This very general PDE, which takes the form of a determinant of order (p+1), is mainly based on the fact that, by deforming the probability with the introduction of several infinite sets of auxiliary variables, it becomes a "tau function" satisfying a whole hierarchy of integrable equations. Then, after suitable changes of scale in space and time, we have studied the critical processes obtained by taking various limits to the model above, where the total number n of Brownian particles tends to infinity. We have successively considered the Airy, r-Airy (or Airy "with r outliers") and Pearcey processes, and also the Pearcey process with intermediate targets (or "inliers"), and that in the case of one or more different times.
- Affiliations
Citations
Delepine, J. (2010). Modèles de mouvements browniens, processus critiques et matrices aléatoires. https://hdl.handle.net/2078.5/130781