Spectral measure for uniform $d$-regular digraphs
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866909711179186176 |
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| author | Adhikari, Arka Dembo, Amir |
| author_facet | Adhikari, Arka Dembo, Amir |
| contents | 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 $μ_d$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_14132 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Spectral measure for uniform $d$-regular digraphs Adhikari, Arka Dembo, Amir Probability Combinatorics 46L53, 60B10, 60B20, 05C50, 05C20 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 $μ_d$ 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. |
| title | Spectral measure for uniform $d$-regular digraphs |
| topic | Probability Combinatorics 46L53, 60B10, 60B20, 05C50, 05C20 |
| url | https://arxiv.org/abs/2310.14132 |