Gaussian Waves on the Regular Tree

Elon, Yehonatan

doi:10.48550/arxiv.0907.5065

Cited by 9 publications

(22 citation statements)

References 18 publications

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“…The random wave model is based on the conjecture that the distribution of a typical adjacency eigenvector of a graph G ∈ G(n, d) graph, with an eigenvalue λ ∈ σ(T d ) converges locally to that of GS d (λ). This hypothesis, was supported by various numerical tests, and found a partial formal justification in [15]. The random waves hypothesis will be the basis for the analysis of the morphology of level sets in (n, d) graphs, which will be carried out in this paper.…”

Section: The Random Waves Model For the Adjacency Eigenvectors Of G ∈...mentioning

confidence: 56%

“…The described dominance of short range correlations in the process GS d (λ), is utilized in [15], to bound rigorously the effect of long range correlation in T ω (α), resulting in the identification of T ω (α) as a quasi Bernoulli process and in the establishment of As was already stated above, the random wave conjecture is valid locally. In the present context, however it applies globally.…”

Section: The Distribution Of Level Sets In Gs D (λ)mentioning

confidence: 99%

“…The main tool in the present study is the random wave model [1,15]. It is based on the observation that for the d regular tree T d , the distribution of a typical eigenvector can be approximated by a real Gaussian process GS d (λ).…”

Section: The Random Waves Model For the Adjacency Eigenvectors Of G ∈...mentioning

confidence: 99%

“…As the proof of the theorem is rather technical, we refer the interested reader to [15] where a complete and detailed proof of the theorem can be found. Here, we shall provide the main line of the proof, skipping much of the technical aspects.…”

Section: The Distribution Of Level Sets In Gs D (λ)mentioning

confidence: 99%

“…The numerically deduced critical level α c (λ, d) as a function of the eigenvalue and the degree of the graph summarizes these computation. The next chapter is dedicated to a systematic construction of a Gaussian random waves model on the d-regular tree [15]. For this process, we are able to prove that a percolation transition occurs.…”

Section: Introductionmentioning

confidence: 99%

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Percolating level sets of the adjacency eigenvectors ofd-regular graphs

Elon¹,

Smilansky²

2010

J. Phys. A: Math. Theor.

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One of the most surprising discoveries in quantum chaos was that nodal domains of eigenfunctions of quantum-chaotic billiards and maps in the semi-classical limit display critical percolation. Here we extend these studies to the level sets of the adjacency eigenvectors of d-regular graphs. Numerical computations show that the statistics of the largest level sets (the maximal connected components of the graph for which the eigenvector exceeds a prescribed value) depend critically on the level. The critical level is a function of the eigenvalue and the degree d. To explain the observed behavior we study a random Gaussian waves ensemble over the d-regular tree. For this model, we prove the existence of a critical threshold. Using the local tree property of d-regular graphs, and assuming the (local) applicability of the random waves model, we can compute the critical percolation level and reproduce the numerical simulations.These results support the random-waves model for random regular graphs, suggested in [1] and provides an extension to Bogomolny's percolation model [2] for two-dimensional chaotic billiards.

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Section: The Random Waves Model For the Adjacency Eigenvectors Of G ∈...mentioning

confidence: 56%

Section: The Distribution Of Level Sets In Gs D (λ)mentioning

confidence: 99%

Section: The Random Waves Model For the Adjacency Eigenvectors Of G ∈...mentioning

confidence: 99%

Section: The Distribution Of Level Sets In Gs D (λ)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Percolating level sets of the adjacency eigenvectors ofd-regular graphs

Elon¹,

Smilansky²

2010

J. Phys. A: Math. Theor.

Self Cite

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Spectrum of random d‐regular graphs up to the edge

Huang,

Yau

2023

Comm Pure Appl Math

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Consider the normalized adjacency matrices of random d‐regular graphs on N vertices with fixed degree . We prove that, with probability for any , the following two properties hold as provided that : (i) The eigenvalues are close to the classical eigenvalue locations given by the Kesten–McKay distribution. In particular, the extremal eigenvalues are concentrated with polynomial error bound in N, that is, . (ii) All eigenvectors of random d‐regular graphs are completely delocalized.

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On large‐girth regular graphs and random processes on trees

Backhausz

Szegedy

2018

Random Struct Algorithms

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We study various classes of random processes defined on the regular tree T d that are invariant under the automorphism group of T d . The most important ones are factor of i.i.d. processes (randomized local algorithms), branching Markov chains and a new class that we call typical processes. Using Glauber dynamics on processes we give a sufficient condition for a branching Markov chain to be factor of i.i.d. K E Y W O R D Sfactor of i.i.d.

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Gaussian Waves on the Regular Tree

Cited by 9 publications

References 18 publications

Percolating level sets of the adjacency eigenvectors ofd-regular graphs

Percolating level sets of the adjacency eigenvectors ofd-regular graphs

Spectrum of random d‐regular graphs up to the edge

On large‐girth regular graphs and random processes on trees

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