Log-concavity in one-dimensional Coulomb gases and related ensembles

TL;DR

This paper establishes log-concavity properties for various probabilistic models related to Young diagrams, last passage percolation, beta ensembles, and eigenvalue distributions, confirming several conjectures and revealing new structural insights.

Contribution

It proves log-concavity for top row lengths in Young diagrams, passage times in percolation, beta ensemble maxima, and eigenvalues, advancing understanding of these distributions' structure.

Findings

Log-concavity of Young diagram top rows under Poissonized Plancherel measure.

Log-concavity of passage times in geometric last passage percolation.

Log-concavity of Tracy-Widom distributions for all β > 0.

Abstract

We prove log-concavity of the lengths of the top rows of Young diagrams under Poissonized Plancherel measure. This is the first known positive result towards a 2008 conjecture of Chen that the length of the top row of a Young diagram under the Plancherel measure is log-concave. This is done by showing that the ordered elements of several discrete ensembles have log-concave distributions. In particular, we show the log-concavity of passage times in last passage percolation with geometric weights, using their connection to Meixner ensembles. In the continuous setting, distributions of the maximal elements of beta ensembles with convex potentials on the real line are shown to be log-concave. As a result, log-concavity of the Tracy-Widom distributions for all parameters $\beta>0$ follows, confirming a folklore conjecture that was partially proved by Deift for $\beta=2$. Furthermore, we also obtain log-concavity and positive association for the joint distribution of the $k$ smallest eigenvalues of the stochastic Airy operator. Our methods also show the log-concavity of the Airy-2 process and the Airy distribution. A log-concave distribution with full-dimensional support must have density, a fact that was apparently not known for some of these examples.