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Methods Of Modern Mathematical Physics

Motivated by the eigenvalue problems for ordinary and partial differential operators, we shall develop the spectral theory for linear operators in Hilbert spaces. Here we transform the unbounded differential operators into singular integral operators which are completely continuous. With his study of integral equations D. Hilbert, together with his students E. Schmidt, I. Schur, and H. Weyl, opened a new era for the analysis.

Linear Operators in Hilbert Spaces

Large deviations is concerned with the study of rare events and of small probabilities. Let Xi,  ≤ i ≤ n, be independent identically distributed (i.i.d.) real random variables with expectationm, and Xn = (X + . . . +Xn)/n their empirical mean. e law of large numbers shows that, for any Borel set A ⊂ R not containing m in its closure, P(Xn ∈ A) →  as n → ∞, but does not tell us how fast the probability vanishes. Large deviations theory gives us the rate of decay, which is exponential in n. Cramer’s theorem states that,

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Large deviations and applications

Fixed energy universality of Dyson Brownian motion

An Introduction to Random Matrices: Introduction

We prove that the (non-symmetric) adjacency matrix of a uniform random d-regular directed graph on n vertices is asymptotically almost surely invertible, assuming $$\min (d,n-d)\ge C\log ^2n$$
 for a sufficiently large constant $$C>0$$
 . The proof makes use of a coupling of random regular digraphs formed by “shuffling” the neighborhood of a pair of vertices, as well as concentration results for the distribution of edges, proved in Cook (Random Struct Algorithms. arXiv:1410.5595
 , 2014). We also apply our general approach to prove asymptotically almost surely invertibility of Hadamard products $$\varSigma {{\mathrm{\circ }}}\varXi $$
 , where $$\varXi $$
 is a matrix of iid uniform $$\pm 1$$
 signs, and $$\varSigma $$
 is a 0/1 matrix whose associated digraph satisfies certain “expansion” properties.

On the singularity of adjacency matrices for random regular digraphs

Large deviations of subgraph counts for sparse Erdős–Rényi graphs

Let $\lambda$ be the second largest eigenvalue in absolute value of a uniform random $d$-regular graph on $n$ vertices. It was famously conjectured by Alon and proved by Friedman that if $d$ is fixed independent of $n$, then $\lambda=2\sqrt{d-1} +o(1)$ with high probability. In the present work we show that $\lambda=O(\sqrt{d})$ continues to hold with high probability as long as $d=O(n^{2/3})$, making progress towards a conjecture of Vu that the bound holds for all $1\le d\le n/2$. 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 $d=o(n^{1/2})$. We are able to go beyond this barrier by proving concentration of measure results directly for the uniform distribution on $d$-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.

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Size biased couplings and the spectral gap for random regular graphs

For each $n$, let $A_n=(\sigma _{ij})$ be an $n\times n$ deterministic matrix and let $X_n=(X_{ij})$ be an $n\times n$ random matrix with i.i.d. centered entries of unit variance. We study the asymptotic behavior of the empirical spectral distribution $\mu _n^Y$ of the rescaled entry-wise product \[ Y_n = \left (\frac 1{\sqrt{n} } \sigma _{ij}X_{ij}\right ). \] For our main result we provide a deterministic sequence of probability measures $\mu _n$, each described by a family of Master Equations, such that the difference $\mu ^Y_n - \mu _n$ converges weakly in probability to the zero measure. A key feature of our results is to allow some of the entries $\sigma _{ij}$ to vanish, provided that the standard deviation profiles $A_n$ satisfy a certain quantitative irreducibility property. An important step is to obtain quantitative bounds on the solutions to an associate system of Schwinger–Dyson equations, which we accomplish in the general sparse setting using a novel graphical bootstrap argument.

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Non-Hermitian random matrices with a variance profile (I): deterministic equivalents and limiting ESDs

We prove that the (non-symmetric) adjacency matrix of a uniform random $d$-regular directed graph on $n$ vertices is asymptotically almost surely invertible, assuming $\min(d,n-d)\ge C\log^2n$ for a sufficiently large constant $C>0$. The proof makes use of a coupling of random regular digraphs formed by "shuffling" the neighborhood of a pair of vertices, as well as concentration results for the distribution of edges recently obtained by the author (arXiv:1410.5595). We also apply our general approach to prove a.a.s.\ invertibility of Hadamard products $\Sigma\circ \Xi$, where $\Xi$ is a matrix of iid uniform $\pm1$ signs, and $\Sigma$ is a 0/1 matrix whose associated digraph satisfies certain "expansion" properties.

Nicholas A. Cook

Papers

On the singularity of adjacency matrices for random regular digraphs

Large deviations of subgraph counts for sparse Erdős–Rényi graphs

Size biased couplings and the spectral gap for random regular graphs

Non-Hermitian random matrices with a variance profile (I): deterministic equivalents and limiting ESDs

On the singularity of adjacency matrices for random regular digraphs