A central limit theorem for point-to-line last passage percolation

We derive the thermodynamic limit of a discrete probabilistic model inspired by statistical

mechanics. Suppose there are independent exponentially distributed times attached to all

vertices of the 2-dimensional integer lattice. Starting from a given vertex, perform a path

made of N steps taken as follows: wait for the random time attached to the vertex where you

are, then move either to the rightwards or to the downwards neighbouring vertex. The

longest waiting time over all N-step paths is called point-to-line last passage percolation, and

is strongly connected to an interacting particle system called totally asymmetric simple

exclusion process (TASEP). We show that the model is exactly solvable, by expressing the

distribution of the last passage time in terms of an integral of symplectic Schur functions. The

rich algebraic structure behind this formula permits deriving a central limit theorem (very

different from the classical Gaussian one!), characterised by cube root of N fluctuations and

a limiting distribution from random matrix theory. Techniques to obtain asymptotics include:

infinite-dimensional determinants, contour integrals, and steepest descent method.

Based on arXiv:1703.07337 and arXiv:1711.05120, joint work with Nikos Zygouras