KPZ universality describes a scaling behaviour that differs from the central limit theorem by the size of the fluctuations (cube-root instead of square-root) and the limiting distribution. Instead of the Gaussian, the Tracy-Widom distributions from random matrix theory appear in the limit. It is a long standing conjecture that the KPZ universality class contains a large group of models, including particle systems and polymer models.

A key model in the KPZ universality is last-passage percolation (LPP), i.e. the distribution of maximal weights across directed paths in a random environment. In this talk I will discuss several variants of the LPP model with independent exponential weights. In particular, I will show how probabilistic coupling, originating from work by Cator and Groeneboom on Hammersley’s process and the Poisson LPP, an be used to derive optimal-order upper and lower bounds on the central moment for these two variants of exponential LPP. That is, letting $v$ be the LPP end-point we obtain bounds proportional to $\|v\|^{p/2}$ (CLT scaling) when $v$ is close to the axis and to $\|v\|^{p/3}$ (KPZ regime) otherwise. These bounds are also uniform over vertices taking values in these regions.

The talk is based on joint work with Elnur Emrah and Nicos Georgiou.