Recent progress in combinatorial random matrix theory

Abstract

We are going to discuss recent progress on many problems in random matrix theory of a combinatorial nature, including several breakthroughs that solve long standing famous conjectures.

The rank of sparse random matrices

“…It therefore comes as no surprise that the subject has played a central role in probabilistic combinatorics since the early days [27,28,29,30]. The current state of affairs is that the theory of dense random matrices is significantly more advanced than that of sparse ones with a bounded average number of non-zero entries per row or column [46,47]. This is in part because concentration techniques apply more easily in the dense case.…”

“…Zero-one matrices over the rationals. Apart from matrices over finite fields, the rational rank of sparse random {0, 1}-matrices has received a great deal of attention [46,47]. The random graph G naturally induces a {0, 1}matrix, namely the m × n-biadjacency matrix B = B(G).…”

On the permanent of a random symmetric matrix

“…and the symmetric case. It is currently only known that Pr(per M n = a) and Pr(per A n = a) are bounded by n −c for some constant c. It would be very interesting to prove bounds of the form n −ω (1) , where ω(1) is any function going to infinity with n. Actually, Vu conjectured (see [40,Conjecture 6.12]) that Pr(per A n = 0) is of the form ω(1) −n (in contrast to the situation for the determinant, where we have Pr(det A n = 0) = 2 −n+o(n) ). It seems reasonable to conjecture that in fact all probabilities of the form Pr(per M n = a) or Pr(per A n = a) are upper-bounded by n −cn , for some constant c. However, essentially all known tools for studying permanents of random matrices also apply to determinants, so significant new ideas would be required to prove such strong results.…”

“…The statement of Theorem 1.1 has been conjectured by Vu in 2009 (see [39]). He also mentioned the conjecture in a recent survey [40,Conjecture 6.11], and described it as "the still missing piece of the picture" regarding determinants and permanents of random discrete matrices.…”

Rank deficiency of random matrices

“…The above simple lower bound shows that the decay of f (k) can be at most 2 −k ; it has been suggested (cf. [18,Section 4], 'It is tempting to conjecture...') that this rate of decay is essentially sharp, i.e. that…”