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

Understand

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.