Nicholas A. Cook

Size biased couplings and the spectral gap for random regular graphs

Let λλ be the second largest eigenvalue in absolute value of a uniform random dd-regular graph on nn vertices. It was famously conjectured by Alon and proved by Friedman that if dd is fixed independent of nn, then λ=2d−1−−−−√+o(1)λ=2d−1+o(1) with high probability. In the present work, we show that λ=O(d−−√)λ=O(d) continues to hold with high probability as long as d=O(n2/3)d=O(n2/3), making progress toward a conjecture of Vu that the bound holds for all 1≤d≤n/21≤d≤n/2. Prior to this work the best result was obtained by Broder, Frieze, Suen and Upfal (1999) using the configuration model, which hits a barrier at d=o(n1/2)d=o(n1/2). We are able to go beyond this barrier by proving concentration of measure results directly for the uniform distribution on dd-regular graphs. These come as consequences of advances we make in the theory of concentration by size biased couplings. Specifically, we obtain Bennett-type tail estimates for random variables admitting certain unbounded size biased couplings.

Circular law for the sum of random permutation matrices

Let

The circular law for random regular digraphs with random edge weights

P_n^1,\dots, P_n^d

On the singularity of adjacency matrices for random regular digraphs

be

Discrepancy properties for random regular digraphs

n\times n

Lower bounds for the smallest singular value of structured random matrices

permutation matrices drawn independently and uniformly at random, and set

The circular law for random regular digraphs

S_n^d:=\sum_{\ell=1}^d P_n^\ell

Limiting spectral distribution for non-Hermitian random matrices with a variance profile

. We show that if

The circular law for signed random regular digraphs

\log^{12}n/(\log \log n)^{4} \le d=O(n)

Spectral properties of non-Hermitian random matrices

, then the empirical spectral distribution of