Perfect Matchings in Inhomogeneous Random Bipartite Graphs in Random Environment

TL;DR

This paper investigates inhomogeneous random bipartite graphs in random environments, providing precise asymptotic estimates for the expected number of perfect matchings and introducing an efficient iterative approximation method.

Contribution

It extends classical Erdős-Rényi graph models to inhomogeneous cases in random environments and offers a fast converging iterative approach for asymptotic analysis.

Findings

Expected number of perfect matchings follows a precise quenched asymptotic

An iterative process approximates the expected number with exponential convergence

Extension of Erdős-Rényi models to inhomogeneous, random environment settings

Abstract

In this note we study inhomogeneous random bipartite graphs in random environment. These graphs can be thought of as an extension of the classical Erd\"os-R\'enyi random graphs in a random environment. We show that the expected number of perfect matchings obeys a precise quenched asymptotic and that it can be approximated using an iterative process that converges exponentially fast.