The Coupling Method for Central Moment Bounds in Exponential Last-Passage Percolation (Janosch Ortmann)

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, Ortmann will discuss several variants of the LPP model with independent exponential weights. In particular, they 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.