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

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.