The asymptotic volume of diagonal subpolytopes of symmetric stochastic matrices

TL;DR

This paper investigates the asymptotic volume of diagonal subpolytopes of symmetric stochastic matrices, extending enumeration techniques to matrices with specified diagonals and analyzing correction factors.

Contribution

It extends asymptotic enumeration methods to symmetric stochastic matrices with fixed diagonals, providing new volume estimates and correction factors.

Findings

Asymptotic enumeration of symmetric matrices with zero diagonal and varying row sums

Derivation of a third order correction factor for volume approximation

Extension of methods from Birkhoff polytope to diagonal subpolytopes

Abstract

The asymptotic volume of the polytope of symmetric stochastic matrices can be determined by asymptotic enumeration techniques as in the case of the Birkhoff polytope. These methods can be extended to polytopes of symmetric stochastic matrices with given diagonal, if this diagonal varies not too wildly. To this end, the asymptotic number of symmetric matrices with natural entries, zero diagonal and varying row sums is determined and a third order correction factor to this is examined.