  Consider non-intersecting Brownian motions on the line leaving from the
origin and forced to two arbitrary points. Letting the number of Brownian
particles tend to infinity, and upon rescaling, there is a point of
bifurcation, where the support of the density of particles goes from one
interval to two intervals. In this paper, we show that at that very point of
bifurcation a cusp appears, near which the Brownian paths fluctuate like the
Pearcey process. This is a universality result within this class of problems.
Tracy and Widom obtained such a result in the symmetric case, when the two
target points are symmetric with regard to the origin. This asymmetry enabled
us to improve considerably a result concerning the non-linear partial
differential equations governing the transition probabilities for the Pearcey
process, obtained by Adler and van Moerbeke.
