Airy processes with wanderers and new universality classes - Mathematics > ProbabilityReport as inadecuate

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Abstract: Consider $n+m$ nonintersecting Brownian bridges, with $n$ of them leavingfrom 0 at time $t=-1$ and returning to 0 at time $t=1$, while the $m$ remainingones wanderers go from $m$ points $a i$ to $m$ points $b i$. First, we keep$m$ fixed and we scale $a i,b i$ appropriately with $n$. In the large-$n$limit, we obtain a new Airy process with wanderers, in the neighborhood of$\sqrt{2n}$, the approximate location of the rightmost particle in the absenceof wanderers. This new process is governed by an Airy-type kernel, with arational perturbation. Letting the number $m$ of wanderers tend to infinity aswell, leads to two Pearcey processes about two cusps, a closing and an openingcusp, the location of the tips being related by an elliptic curve. Upon tuningthe starting and target points, one can let the two tips of the cusps grow veryclose; this leads to a new process, which might be governed by a kernel,represented as a double integral involving the exponential of a quinticpolynomial in the integration variables.

Author: Mark Adler, Patrik L. Ferrari, Pierre van Moerbeke



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