Spectral measure for uniform $d$-regular digraphs
| Autor: | Adhikari, Arka, Dembo, Amir |
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| Rok vydání: | 2023 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | Consider the matrix $A_{\mathcal{G}}$ chosen uniformly at random from the finite set of all $N$-dimensional matrices of zero main-diagonal and binary entries, having each row and column of $A_{\mathcal{G}}$ sum to $d$. That is, the adjacency matrix for the uniformly random $d$-regular simple digraph $\mathcal{G}$. Fixing $d \ge 3$, it has long been conjectured that as $N \to \infty$ the corresponding empirical eigenvalue distributions converge weakly, in probability, to an explicit non-random limit, %measure $\mu_d$ on $\mathbb{C}$, which is given by the Brown measure of the free sum of $d$ Haar unitary operators. We reduce this conjecture to bounding the decay in $N$ of the probability that the minimal singular value of the shifted matrix $A(w) = A_{\mathcal{G}} - w I$ is very small. While the latter remains a challenging task, the required bound is comparable to the recently established control on the singularity of $A_{\mathcal{G}}$. The reduction is achieved here by sharp estimates on the behavior at large $N$, near the real line, of the Green's function (aka resolvent) of the Hermitization of $A(w)$, which is of independent interest. Comment: 65 pages |
| Databáze: | arXiv |
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