A Koteljanskii inequality for permanents

Abstract

We prove a permanental analogue of Koteljanskii's inequality. If $A$ is an inverse $M$-matrix that becomes symmetric after a positive diagonal similarity, then $\mathrm{per}(A_{S\cup T})\mathrm{per}(A_{S\cap T})\ge \mathrm{per}(A_S)\mathrm{per}(A_T)$ for all $S,T\subseteq [n]$, where $A_S$ is the principal submatrix indexed by $S$. The proof expresses permanents as moments of a complex Gaussian vector and uses Ginibre's correlation inequality. Finally, an explicit counterexample shows that symmetry cannot be dropped.

9 pages