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

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 -> oo, 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.