A central limit theorem for the matching number of a sparse random graph

TL;DR

The paper establishes central limit theorems for the matching number $\nu(G)$ (and for the rank) in sparse Erdős--Rényi graphs, covering the subcritical and critical regimes where the Karp--Sipser leaf-removal process becomes degenerate. It extends the classical KS81 LLN using a stochastic differential equation–type analysis (Ethier–Kurtz framework), a stopped-leaf-removal scheme, and a reduction to random multigraphs via the configuration model, complemented by coupling and concentration arguments. A key innovation is handling the $c\le e$ phase transition, where core degeneracy prevents a direct Gaussian process limit; the authors show that stopping before degeneracy allows a dominant fluctuation in the stopping time to govern the CLT, with conditional fluctuations of the remainder being negligible. The results also yield a CLT for the rank of the adjacency matrix, and the methods introduce new analytic and probabilistic tools (center-manifold–style stability, Lipschitz concentration, and robust multigraph comparisons) that advance the analysis of sparse random graph parameters with optimisation interpretations.

Abstract

In 1981, Karp and Sipser proved a law of large numbers for the matching number of a sparse Erdős-Rényi random graph, in an influential paper pioneering the so-called differential equation method for analysis of random graph processes. Strengthening this classical result, and answering a question of Aronson, Frieze and Pittel, we prove a central limit theorem in the same setting: the fluctuations in the matching number of a sparse random graph are asymptotically Gaussian. Our new contribution is to prove this central limit theorem in the subcritical and critical regimes, according to a celebrated algorithmic phase transition first observed by Karp and Sipser. Indeed, in the supercritical regime, a central limit theorem has recently been proved in the PhD thesis of Kreačić, using a stochastic generalisation of the differential equation method (comparing the so-called Karp-Sipser process to a system of stochastic differential equations). Our proof builds on these methods, and introduces new techniques to handle certain degeneracies present in the subcritical and critical cases. Curiously, our new techniques lead to a non-constructive result: we are able to characterise the fluctuations of the matching number around its mean, despite these fluctuations being much smaller than the error terms in our best estimates of the mean. We also prove a central limit theorem for the rank of the adjacency matrix of a sparse random graph.

Theorems & Definitions (29)

• Theorem 1.1
• Theorem 1.2
• Remark 1.3
• Theorem 1.4
• Definition 2.1: Karp--Sipser leaf removal
• Theorem 2.2
• Remark 2.3
• Definition 3.1
• Definition 3.2
• Theorem 3.3