The Exact Rank of Sparse Random Graphs

Abstract

Two landmark results in combinatorial random matrix theory, due to Komlós and Costello-Tao-Vu, show that discrete random matrices and symmetric discrete random matrices are typically nonsingular. In particular, in the language of graph theory, when p is a fixed constant, the biadjacency matrix of a random Erdős-Rényi bipartite graph G(n, n, p) and the adjacency matrix of an Erdős-Rényi random graph G(n, p) are both nonsingular with high probability. However, very sparse random graphs (i.e., where p is allowed to decay rapidly with n) are typically singular, due to the presence of "local" dependencies such as isolated vertices and pairs of degree-1 vertices with the same neighbour.In this paper we give a combinatorial description of the rank of a sparse random graph G(n, n, c/n) or G(n, c/n) in terms of such local dependencies, for all constants c = e (and we present some evidence that the situation is very different for c = e). This gives an essentially complete answer to a question raised by Vu at the 2014 International Congress of Mathematicians.As applications of our main theorem and its proof, we also determine the asymptotic singularity probability of the 2-core of a sparse random graph, we show that the rank of a sparse random graph is extremely well-approximated by its matching number, and we deduce a central limit theorem for the rank of G(n, c/n). Corollary 1.3. Let G ∼ G(n, c/n) for a constant c < 1 or c > e, let G ∼ G(n, c/n), and let X be the rank of the adjacency matrix of G. Then (X − EX)/ √ Var X d → N (0, 1).Actually, in an upcoming paper together with Goldschmidt and Kreačić [38], we are able to close the gap between 1 and e in Corollary 1.3. Specifically, the regime c ≤ e is rather different in nature than the regime c > e, and when G ∼ G(n, c/n) for c ≤ e, we are able to give a unified proof that the rank and matching number of G both satisfy a central limit theorem (without going through Theorem 1.2(A1)). Remark.[60] and [49] provide explicit formulas for the asymptotic values of EX and Var X, though these are a bit too complicated to describe here. It is worth remarking that the asymptotic formula for Var X is the single place in this paper where there is a material difference between the "binomial" model of Erdős-Rényi random graphs (where each edge is present with probability p independently) and the "uniform" model of Erdős-Rényi random graphs (where we choose a random subset of exactly m edges, for say m = ⌊p n 2 ⌋). Indeed, the variance of the matching number (and therefore the variance of the rank) differs by a constant factor between these two settings; see [49,60] for details. For all the other results in the paper (which are all stated for the binomial model), one can make trivial changes to the proofs to obtain exactly the same result in the uniform model.Remark. We believe that a central limit theorem does not hold for the rank of G(n, n, c/n); see Section 1.6. 1.4. Exactly characterising the rank. In this subsection we finally state our main theorem, giving an exact combinatorial ...