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We consider n non-intersecting Brownian motion paths with p prescribedstarting 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 pointprocess, which in the case p=1 is equivalent to the eigenvalue distribution ofa random matrix from the Gaussian unitary ensemble with external source. Forgeneral 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 nlimit by a vector equilibrium problem with an interaction matrix that is basedon a bipartite planar graph. Our proof is based on a steepest descent analysisof an associated (p+q) by (p+q) matrix valued Riemann-Hilbert problem whosesolution is built out of multiple orthogonal polynomials. A new feature of thesteepest descent analysis is a systematic opening of a large number of globallenses.

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