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Airy process with wanderers, KPZ fluctuations, and a deformation of the Tracy-Widom GOE distribution. (English. French summary) Zbl 1498.60037

Summary: 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 [A. Bloemendal and B. Virág, Probab. Theory Relat. Fields 156, No. 3–4, 795–825 (2013; Zbl 1356.60014); Ann. Probab. 44, No. 4, 2726–2769 (2016; Zbl 1396.60004)] 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.

MSC:

60B20 Random matrices (probabilistic aspects)
60J65 Brownian motion
60K35 Interacting random processes; statistical mechanics type models; percolation theory
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