Limiting Root Distribution of Random Log-concave Polynomials

Abstract

Log-concave polynomials arise naturally in combinatorics, geometry, and probability. We introduce a natural family of random log-concave polynomials and study their limiting root distributions. Our starting point is the \emph{uniform model}, in which the coefficient vector $(a_0,\dots ,a_n)$ is chosen uniformly from $[0,1{]}^{n+1}$ subject to the log-concavity constraint. More generally, for $\alpha \ge 0$, we define \[ P_{n,α}(z)=\sum_{i=0}^n a_i^{n^α}z^i. \] Thus, $\alpha =0$ corresponds to the uniform model. For each fixed $0\le \alpha <1$, we prove that the empirical root measure converges almost surely to the uniform probability measure on the unit circle. For the critical case $\alpha =1$, which we call the {\emph beta model}, the empirical root measure instead converges in probability to a rotationally invariant probability measure that is absolutely continuous with respect to planar Lebesgue measure.

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