Nick Simm

Last passage percolation and q-ensembles

Consider the integer lattice Z^{2} and assign independent random weights \omega_{i,j} to each site. The last passage time G_{n,m} is the maximum sum of the weights along up-right paths connecting (1,1) to a fixed target site (n,m). A central problem in last passage percolation is to determine the distribution of G_{n,m}. For certain special weight distributions, this problem has intimate connections to random matrix theory and the KPZ universality class.

I will discuss recent work on an inhomogeneous model that we show is related to a q-deformed random matrix ensemble. When the inhomogeneity is weak enough, we establish Tracy–Widom fluctuations for G_{n,m}, as n and m tend to infinity. For stronger inhomogeneity, we obtain a different limiting distribution, expressed as a Fredholm determinant involving the Bessel kernel.

This is joint work with Sung-Soo Byun and Yeong-Gwang Jung.