Log-concavity in one-dimensional Coulomb gases and related ensembles
Jnaneshwar Baslingker, Manjunath Krishnapur, Mokshay Madiman
TL;DR
The paper addresses the problem of establishing log-concavity for order statistics across discrete and continuum ensembles arising in combinatorics, PoissonizedPlancherel problems, and random matrix theory. It develops a unified framework for discrete log-concavity using the Halikias-Klartag-Slomka inequality and leverages marginalization properties to prove log-concavity for Meixner-type ensembles and Schur measures, with consequences for Poissonization/depoissonization and connections to Chen's conjecture. In the continuum, it treats beta-Coulomb gas ensembles with convex potentials, proving log-concavity of the joint law and of the ordered coordinates and gaps, and deduces log-concavity and positive association for TW_beta, the stochastic Airy operator's smallest eigenvalues, and for the Airy_2 process and Airy distribution. The work also provides alternative proofs of TW_2 log-concavity on the entire real line and discusses density, entropy, and moment implications of log-concavity. Overall, the results offer a broad, interoperable toolkit for certifying log-concavity across discrete and continuous probabilistic models with deep ties to random partitions and random matrices, advancing conjectures such as Chen's and informing future explorations of higher-order Tracy-Widom laws.
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 $β>0$ follows, confirming a folklore conjecture that was partially proved by Deift for $β=2$. 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.