A graph-based equilibrium problem for the limiting distribution of non-intersecting Brownian motions at low temperature

We consider n non-intersecting Brownian motion paths with p prescribed starting positions at time t = 0 and q prescribed ending positions at time t = 1. The positions of the paths at any intermediate time are a determinantal point process, which in the case p = 1 is equivalent to the eigenvalue distribution of a random matrix from the Gaussian unitary ensemble with external source. For general p and q, we show that if a temperature parameter is sufficiently small, then the distribution of the Brownian paths is characterized in the large n limit by a vector equilibrium problem with an interaction matrix that is based on a bipartite planar graph. Our proof is based on a steepest descent analysis of an associated (p+ q)× (p+ q) matrix valued Riemann-Hilbert problem whose solution is built out of multiple orthogonal polynomials. A new feature of the steepest descent analysis is a systematic opening of a large number of global lenses.