  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) by (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.
