Airy process with wanderers, KPZ fluctuations, and a deformation of the Tracy--Widom GOE distribution

Abstract

We study the distribution of the supremum of the Airy process with $m$ wanderers minus a parabola, or equivalently the limit of the rescaled maximal height of a system of $N$ non-intersecting Brownian bridges as $N\to \infty$, where the first $N-m$ paths start and end at the origin and the remaining $m$ go between arbitrary positions. The distribution provides a $2m$-parameter deformation of the Tracy--Widom GOE distribution, which is recovered in the limit corresponding to all Brownian paths starting and ending at the origin. We provide several descriptions of this distribution function: (i) A Fredholm determinant formula; (ii) A formula in terms of Painlevé II functions; (iii) A representation as a marginal of the KPZ fixed point with initial data given as the top path in a stationary system of reflected Brownian motions with drift; (iv) A characterization as the solution of a version of the Bloemendal--Virag PDE (arXiv:1011.1877, arXiv:1109.3704) for spiked Tracy--Widom distributions; (v) A representation as a solution of the KdV equation. We also discuss connections with a model of last passage percolation with boundary sources.

arXiv comment34 pages, 1 figure

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• Airy kernel determinant solutions to the KdV equation and integro-differential Painlevé equations
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• Uniform tail asymptotics for Airy kernel determinant solutions to KdV and for the narrow wedge solution to KPZ