For rational beta>0, the Sine-beta and Airy-beta point processes are the unique solutions of the bulk and edge loop equation hierarchies respectively.
Ramanujan Property and Edge Universality of Random Regular Graphs
7 Pith papers cite this work. Polarity classification is still indexing.
7
Pith papers citing it
abstract
We consider the normalized adjacency matrix of a random $d$-regular graph on $N$ vertices with any fixed degree $d\geq 3$ and denote its eigenvalues as $\lambda_1=d/\sqrt{d-1}\geq \lambda_2\geq\lambda_3\cdots\geq \lambda_N$. We establish the following two results as $N\rightarrow \infty$. (i) With high probability, all eigenvalues are optimally rigid, up to an additional $N^{{\rm o}(1)}$ factor. Specifically, the fluctuations of bulk eigenvalues are bounded by $N^{-1+{\rm o}(1)}$, and the fluctuations of edge eigenvalues are bounded by $N^{-2/3+{\rm o}(1)}$. (ii) Edge universality holds for random $d$-regular graphs. That is, the distributions of $\lambda_2$ and $-\lambda_N$ converge to the Tracy-Widom$_1$ distribution associated with the Gaussian Orthogonal Ensemble. As a consequence, for sufficiently large $N$, approximately $69\%$ of $d$-regular graphs on $N$ vertices are Ramanujan, meaning $\max\{\lambda_2,|\lambda_N|\}\leq 2$.
representative citing papers
Inhomogeneous random matrices have identical universal edge statistics if their variance-profile Markov chains satisfy short-to-long comparability, enabling analysis of band matrices, orbital models, and Hankel profiles in subcritical and critical regimes.
Spielman's Laplacian eigenratio conjecture is disproved for infinitely many d>2 using Ramanujan graphs but verified for d≤2 and regular graphs, with stronger results and resolutions of two other conjectures.
Proves that SO(3) lattice Yang-Mills theory fails Wilson's confinement criterion at strong coupling.
Proves eigenvalue bounds implying that isogeny graphs of supersingular elliptic curves are Ramanujan and studies their spectral distribution, components, automorphisms, and links to modular forms.
Develops a mixed quantization technique for graph vector bundles and applies it to the Alon-Boppana bound, Kesten-McKay law, asymptotic determinant, quantum ergodicity, zero divisor convergence, and Ramanujan vector bundles.
A survey of recent results proving that random hyperbolic surfaces of large genus have near-optimal spectral gaps, after Hide–Magee, Anantharaman–Monk, and Hide–Macera–Thomas.
citing papers explorer
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Loop Equations Characterize Random Matrix Statistics
For rational beta>0, the Sine-beta and Airy-beta point processes are the unique solutions of the bulk and edge loop equation hierarchies respectively.
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Edge Universality for Inhomogeneous Random Matrices II: Markov Chain Comparison and Critical Statistics
Inhomogeneous random matrices have identical universal edge statistics if their variance-profile Markov chains satisfy short-to-long comparability, enabling analysis of band matrices, orbital models, and Hankel profiles in subcritical and critical regimes.
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On Spielman's Laplacian Eigenratio Conjecture and Related Problems
Spielman's Laplacian eigenratio conjecture is disproved for infinitely many d>2 using Ramanujan graphs but verified for d≤2 and regular graphs, with stronger results and resolutions of two other conjectures.
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Deconfinement For $\mathrm{SO}(3)$ Lattice Yang-Mills at Strong Coupling
Proves that SO(3) lattice Yang-Mills theory fails Wilson's confinement criterion at strong coupling.
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Spectral Theory of Isogeny Graphs
Proves eigenvalue bounds implying that isogeny graphs of supersingular elliptic curves are Ramanujan and studies their spectral distribution, components, automorphisms, and links to modular forms.
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Discrete Mixed Quantization
Develops a mixed quantization technique for graph vector bundles and applies it to the Alon-Boppana bound, Kesten-McKay law, asymptotic determinant, quantum ergodicity, zero divisor convergence, and Ramanujan vector bundles.
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Bass notes of random hyperbolic surfaces of large genus
A survey of recent results proving that random hyperbolic surfaces of large genus have near-optimal spectral gaps, after Hide–Magee, Anantharaman–Monk, and Hide–Macera–Thomas.