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 → ∞, 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 Painleve 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 (Probab. Theory Related Fields 156 (2013) 795.825; Ann. Probab. 44 (2016) 2726.2769) 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.