Concentration of permanent estimators for certain large matrices

Abstract

Let $A_n=(a_{ij})_{i,j=1}^n$ be an $n×n$ positive matrix with entries in $[a,b], 0<a≤b$. Let $X_{n}=(\sqrt{a_{ij}}x_{ij})_{i,j=1}^{n}$ be a random matrix, where $\{x_{ij}\}$ are i.i.d. $N(0,1)$ random variables. We show that for large $n$, $\det (X_{n}^{T}X_{n})$ concentrates sharply at the permanent of $A_n$, in the sense that $n^{-1}\log (\det(X_{n}^{T}X_{n})/\operatorname {per}A_{n})\to_{n\to\infty}0$ in probability.