A central limit theorem for the matching number of a sparse random graph
arxiv(2024)
摘要
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.
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