Limiting distributions of last passage percolation models in half space

Part 1: Last passage percolation models in half-space is a model in the Kardar--Parisi--Zhang (KPZ) universality class of stochastic growth models. Here, half-space means that the random variables are restricted to the sites (i,j) with i<=j. Starting from geometric weights, two natural limit can be made: (a) limit to exponential random variables, which is tightly related with the totally asymmetric simple exclusion process with a reservoir at the origin, and (b) the limit to Poisson points, which can be identified with the so-called polynuclear growth model. In this part we give an overview of the limiting distributions and explain how the derivation of the limiting distribution within the determinantal framework is obtained.

Part 2: We consider a polynuclear growth model in half space with two external sources generated by Poisson point processes on the diagonal, x=0, and at the boundary x=t. Under certain values of the boundary parameters, this setting corresponds to a stationary initial conditions. The goal is characterise the large time limit distribution of the height function. We will review the strategy developed to study the model in the full space setting (Baik--Rains ’00), which relies on algebraic and orthogonal polynomials identities, and Riemann--Hilbert techniques, and which led to a limit distribution formulated in terms of the solution to Painlevé II equation; then we will underline the differences with respect to the half space case. By universality the distribution obtained should coincide with the ones derived by Betea--Ferrari--Occelli '20 for a last passage percolation model with exponential weights, and by Barraquand--Krajenbrink--Le Doussal '22 for the stationary KPZ equation on the half line. Based on joint work with M. Cafasso, D. Ofner, H. Walsh.