Ngoc Mai Tran (University of Texas at Austin)
Title: Zeros of random tropical polynomials, random polytopes and stick-breaking
Abstract:
For $i = 0, 1, \ldots, n$, let $C_i$ be independent and identically distributed according to some distribution $F$ with support $(0,\infty)$. The number of zeros of random tropical polynomials $\Tf_n(x) = \min_{i=1,\ldots,n}(C_i + ix)$, $R_n$ is also the number of faces of the lower convex hull of the $n+1$ random points $(i,C_i)$. We show that $R_n$ satisfies a central limit theorem when $F$ has polynomial decay near $0$. Specifically, if $F$ near $0$ behaves like $gamma(a,1)$ distribution for some $a > 0$, then $R_n$ has the same asymptotics as the number of renewals on the interval $[0,\log(n)/a]$ of a renewal process with inter-arrival distribution $-\log(Beta(1+a,1))$. Our proof draws on connections between random partitions, renewal theory and random polytopes. In particular, we obtain generalizations and simple proofs of the central limit theorem for the number of vertices of the convex hull of $n$ uniform random points in a square. Our work leads to many open problems in stochastic tropical geometry, the study of linear functionals and intersections of random tropical varieties.