Local Semicircle Law for Random Regular Graphs

Bauerschmidt, Roland; Knowles, Antti; Yau, Horng‐Tzer

doi:10.1002/cpa.21709

Cited by 80 publications

(126 citation statements)

References 56 publications

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“…In particular, G N,d does not exhibit a connectivity crossover and remains connected down to d = 3. A local law for the random regular graph G N,d for d (log N ) 4 was proved in [4] and for fixed but large d in [3].…”

Section: Introductionmentioning

confidence: 99%

Local law and complete eigenvector delocalization for supercritical Erdős–Rényi graphs

He¹,

Knowles²,

Marcozzi³

2019

Ann. Probab.

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We prove a local law for the adjacency matrix of the Erdős-Rényi graph G (N, p) in the supercritical regime pN C log N where G(N, p) has with high probability no isolated vertices. In the same regime, we also prove the complete delocalization of the eigenvectors. Both results are false in the complementary subcritical regime. Our result improves the corresponding results from [11] by extending them all the way down to the critical scale pN = O(log N ).A key ingredient of our proof is a new family of multilinear large deviation estimates for sparse random vectors, which carefully balance mixed 2 and ∞ norms of the coefficients with combinatorial factors, allowing us to prove strong enough concentration down to the critical scale pN = O(log N ). These estimates are of independent interest and we expect them to be more generally useful in the analysis of very sparse random matrices.

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Section: Introductionmentioning

confidence: 99%

Local law and complete eigenvector delocalization for supercritical Erdős–Rényi graphs

He¹,

Knowles²,

Marcozzi³

2019

Ann. Probab.

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“…As was demonstrated in a recent series of papers, adding some randomness may allow to settle the problem completely. For instance almost sure optimal ℓ ∞ -bounds and quantum unique ergodicity for various models of random matrices and random graphs, such as Wigner matrices, sparse Erdös-Rényi graphs, random regular graphs of slowly increasing or bounded degrees were obtained in [29,30,22,28,13,14,15]. The invariance of the probability distribution under certain elementary transformations plays an important role.…”

Section: Introductionmentioning

confidence: 99%

Quantum ergodicity on graphs: From spectral to spatial delocalization

Anantharaman

Sabri

2019

Ann. of Math. (2)

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We prove a quantum-ergodicity theorem on large graphs, for eigenfunctions of Schrödinger operators in a very general setting. We consider a sequence of finite graphs endowed with discrete Schrödinger operators, assumed to have a local weak limit. We assume that our graphs have few short loops, in other words that the limit model is a random rooted tree endowed with a random discrete Schrödinger operator. We show that absolutely continuous spectrum for the infinite model, reinforced by a good control of the moments of the Green function, imply "quantum ergodicity", a form of spatial delocalization for eigenfunctions of the finite graphs approximating the tree. This roughly says that the eigenfunctions become equidistributed in phase space. Our result applies in particular to graphs converging to the Anderson model on a regular tree, in the régime of extended states studied by Klein and Aizenman-Warzel.2010 Mathematics Subject Classification. Primary 58J51. Secondary 60B20, 81Q10.

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“…Moreover, since H is related to A by a (complex) rank one perturbation, the spectral properties of the ITE can be related to the complex eigenvalues of random tournaments [13]. However, to the best of our knowledge, there are no such results for the RITE, although linear statistics [37,38], local semicircle estimates [39,40] and local universality results [41] have been obtained for random regular graphs using switching methods. Theorem 2.1 (Convergence for ITE).…”

Section: Definitions and Resultsmentioning

confidence: 98%

A random walk approach to linear statistics in random tournament ensembles

Joyner¹,

Smilansky²

2018

Electron. J. Probab.

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We investigate the linear statistics of random matrices with purely imaginary Bernoulli entries of the form Hpq = Hqp = ±i, that are either independently distributed or exhibit global correlations imposed by the condition q Hpq = 0. These are related to ensembles of so-called random tournaments and random regular tournaments respectively. Specifically, we construct a random walk within the space of matrices and show that the induced motion of the first k traces in a Chebyshev basis converges to a suitable Ornstein-Uhlenbeck process. Coupling this with Stein's method allows us to compute the rate of convergence to a Gaussian distribution in the limit of large matrix dimension.

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Local Semicircle Law for Random Regular Graphs

Cited by 80 publications

References 56 publications

Local law and complete eigenvector delocalization for supercritical Erdős–Rényi graphs

Local law and complete eigenvector delocalization for supercritical Erdős–Rényi graphs

Quantum ergodicity on graphs: From spectral to spatial delocalization

A random walk approach to linear statistics in random tournament ensembles

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