Singularity of random symmetric matrices revisited

Abstract

Where a licence is displayed above, please note the terms and conditions of the licence govern your use of this document.When citing, please reference the published version. Take down policyWhile the University of Birmingham exercises care and attention in making items available there are rare occasions when an item has been uploaded in error or has been deemed to be commercially or otherwise sensitive.

Random symmetric matrices: rank distribution and irreducibility of the characteristic polynomial

“…It was noted in [ 8 ] that given the techniques in [ 3, 8 ], Theorem 1·1 (with the weaker probability bound ) can be deduced from the following universality statement: the probability that is invertible over , for , is essentially the same as for an symmetric matrix whose entries on and above the diagonal are sampled from the uniform distribution on . Despite the intensive efforts to study the singularity probability of symmetric Rademacher matrices ([ 4, 6, 7, 10, 14, 20, 24 ] and especially the recent breakthrough [ 5 ] which confirms the long-standing conjecture that the singularity probability of symmetric Rademacher matrices is exponentially small), a result of this precision has remained elusive. While for , such a result is known due to work of Maples [ 18 ], the bound on p is too restrictive to imply Theorem 1·1 (even with the weaker probability ).…”

The least singular value of a random symmetric matrix

“…We now turn to discuss our second approximate negative dependence result, which deals with the intersection of two different small ball events. This was originally proved in our paper [4], but is put to a different use here. This result tells us that the events…”

“…He went on to conjecture that 𝜀 should replace 𝜀 1/8−𝜀 as the correct order of magnitude, and that 𝑒 −𝑐𝑛 should replace 𝑒 −𝑛 𝑐 . After Vershynin, a series of works [3,5,16,17,19] made progress on singularity probability (i.e., the 𝜀 = 0 case of Vershynin's conjecture), and we, in [4], ultimately showed that the singularity probability is exponentially small, when 𝐴 𝑖, 𝑗 is uniform in {−1, 1}:…”

“…To prove that these quasi-randomness properties hold with probability 1 − 𝑒 −𝑐𝑛 is a difficult task and depends fundamentally on the ideas in our previous paper [4]. Since we don't want these ideas to distract from the new ideas in this paper, we have opted to carry out the details in the Appendix.…”

“…In particular, the proof of Theorem 4.3 is similar to the proof of the main theorem in [4], with only a few tweaks and additions required to make the adaptation go through. In several places, we need only update the constants and will be satisfied in pointing the interested reader to [4] for more detail. Elsewhere, more significant adaptations are required, and we outline these changes in full detail.…”