### 27.11.2018 16:30 Nicos Georgiou (University of Sussex) :Last passage times in discontinuous environments.2.01.10 (Parkring 11, 85748 Garching-Hochbrück)

We are studying a last passage percolation model on the two dimensional lattice, where the environment is a field of independent random exponential weights with different parameters. Each variable is associated with a lattice vertex and its parameter is selected according to a discretization of lower semi-continuous parameter function that may admit discontinuities on a set of curves. We prove a law of large numbers for the sequence of last passage times, defined as the maximum sum of weights which a directed path can collect from (0, 0) to a target point (Nx, Ny) as N tends to infinity and the mesh of the discretisation of the parameter function tends to 0 as 1/N. The LLN is cast in the form of a variational formula, optimised over a given set of macroscopic paths. Properties of maximizers to the variational formula above are investigated in two models where the parameter function allows for analytical tractability. This is joint work with Federico Ciech.