Last passage percolation, Schur processes and free fermions
Jérémie Bouttier (IPhT-Saclay and ENS Lyon)
We consider the model of (directed 2D) last passage percolation (LPP), defined as follows. To every site of Z^2 we assign a nonnegative random number called weight. To every directed (north-east) path in Z^2, we assign a weight equal to the sum of of the weights of the sites that it visits. If x,y are two sites such that y is north-east of x, we define the LPP time L(x,y) as the maximal weight of a directed path from x to y.
In his seminal paper arXiv:math/9903134 [math.CO], Johansson considered the situation where the weights are all independent and drawn according to the same geometric or exponential distribution. This case is
intimately connected with the Totally Asymmetric Simple Exclusion Process (TASEP) and the so-called Corner Growth Model. He proved that the fluctuations of L(x,y), when |y-x| gets large, are asymptotically governed by the Tracy-Widom GUE distribution.
Several variants have been considered. In particular, Baik and Rains considered the situation where the weight array has symmetries, for instance a reflection symmetry along the main diagonal i-j=0. This is
equivalent to considering LPP in the half-plane i-j≥0. They have shown that the asymptotic fluctuations of L(x,y), for x and y close to the main diagonal, are now governed by the Tracy-Widom GOE or GSE
What allows to obtain such precise results is that, in these models, there exists an exact formula for the distribution of L(x,y) in the form of a Fredholm determinant or pfaffian. Such property holds for the more general class of Schur processes, which may be studied using free fermions.
After reviewing these results, I will discuss Schur processes with periodic and free boundary conditions. In the LPP picture, this corresponds to imposing some periodicity/symmetry properties on the weight array. We will see that, in suitable asymptotic regimes, the fluctuations of the LPP time are governed by nontrivial deformations of the Tracy-Widom distributions.
This talk is based on joint work with Dan Betea, Peter Nejjar and Mirjana Vuletić.