Universality for the Pearcey process |
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Authors: | Mark Adler Pierre van Moerbeke |
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Institution: | a Department of Mathematics, Brandeis University, Waltham, MA 02454, USA b Université de Louvain, 1348 Louvain-la-Neuve, Belgium c Theory division, CERN, CH-1211 Geneva 23, Switzerland d Department of Mathematics, Université de Louvain, 1348 Louvain-la-Neuve, Belgium |
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Abstract: | 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. |
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Keywords: | Non-intersecting Brownian motions Pearcey distribution Matrix models Random Hermitian ensembles Multi-component KP equation Virasoro constraints |
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