Non-uniform Kahn-Kalai, spread, variants, and applications

Abstract

Building on B.Park and Vondrak's recent generalization of the J.Park-Pham Theorem (formerly known as Kahn-Kalai conjecture) to non-uniform probability measures, this paper introduces the notion of "spread" for the non-uniform setting. This provides a framework to establish 1-statements for subgraph containment in inhomogeneous random graphs with or without a set of forced edges. Using this approach, we derived conditions for the emergence of perfect matchings in the Stochastic Block Model and the Chung-Lu model, and verified that these conditions are in general not tight, but they capture thresholds across a broad range of regimes. Finally, we bridge this non-uniform framework with $\mathcal{G}(n,\textbf{d})$, utilizing a coupling argument to demonstrate thresholds for perfect matchings in $\mathcal{G}(n,\textbf{d})$ for a broad range of degree sequences $\textbf{d}$.