A note on the singularity probability of random directed \(d\)-regular graphs (Q6612308)

The singularity problem in combinatorial random matrix theory states that if a square matrix \(A_n\) of size \(n\) is ``sufficiently random'', then \(A_n\) is non-singular asymptotically almost surely as \(n\) tends to infinity, in other words, \(p_n\), the probability of \(A_n\) being singular, tends to zero. In this paper, the authors show that the singular probability of the adjacency matrix of a random \(d\)-regular graph on \(n\) vertices, where \(d\) is fixed and \(n \rightarrow \infty\), is bounded by \(n^{-1/3+o(1)}\). \N\NThis improves a recent bound by \textit{J. Huang} [Duke Math. J. 170, No. 18, 3977--4032 (2021; Zbl 1512.15003)] and their method is based on the study of the singularity problem modulo a prime developed in Huang [loc. cit.] (and also partially in [\textit{A. Mészáros}, Trans. Am. Math. Soc. 373, No. 9, 6529--6594 (2020; Zbl 1475.05156); \textit{H. H. Nguyen} and \textit{M. M. Wood}, ``Cokernels of adjacency matrices of random $r$-regular graphs'', Preprint, \url{arXiv:1806.10068}]), together with an inverse-type result on the decay of the characteristic function. The latter is related to the inverse Kneser problem in combinatorics.