Peaks of cylindric plane partitions

TL;DR

This paper investigates the asymptotic behavior of the maximum parts of cylindric plane partitions, revealing connections to random matrix theory, last passage percolation, and the KPZ equation, with results interpolating between Gumbel and Tracy--Widom distributions.

Contribution

It introduces a novel analysis of the peak distribution of cylindric plane partitions, linking it to finite temperature analogues of classical random matrix distributions and exploring new interpolations.

Findings

Distributions governed by finite temperature Bessel and Airy gap probabilities.

Interpolate between Gumbel and Tracy--Widom extremal distributions.

Connections established with last passage percolation and KPZ equation.

Abstract

We study the asymptotic distribution, as the volume parameter goes to 1, of the peak (largest part) of finite- or slowly-growing-width cylindric plane partitions weighted by their trace, seam, and volume. There are two natural asymptotic regimes depending on the trace/seam parameters, and in both cases we obtain asymptotics governed by finite temperature (periodic) analogues of the Bessel and Airy gap probabilities from random matrix theory. In particular, the distributions we obtain interpolate \emph{in more than one way} between two well-known extremal value distributions: the Gumbel distribution of maxima of iid random variables and the Tracy--Widom distribution of maxima of eigenvalues of random Hermitian matrices. We also interpret our results in terms of last passage percolation on a cylinder, which yields to interesting connections to the Kardar--Parisi--Zhang equation.