A phase transition for non-intersecting Brownian motions, and the Painleve II equation

Details A phase transition for non-intersecting Brownian motions, and the Painleve II equation A phase transition for non-intersecting Brownian motions, and the Painleve II equation

2008-09-05

arxiv.org/abs/0809.1000v1

Mathematics

Complex Variables

75 pages, 13 figures

Scientific paper

We consider n non-intersecting Brownian motions with two fixed starting positions and two fixed ending positions in the large n limit. We show that in case of 'large separation' between the endpoints, the particles are asymptotically distributed in two separate groups, with no interaction between them, as one would intuitively expect. We give a rigorous proof using the Riemann-Hilbert formalism. In the case of 'critical separation' between the endpoints we are led to a model Riemann-Hilbert problem associated to the Hastings-McLeod solution of the Painleve II equation. We show that the Painleve II equation also appears in the large n asymptotics of the recurrence coefficients of the multiple Hermite polynomials that are associated with the Riemann-Hilbert problem.

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