Cutoff on all Ramanujan graphs

Lubetzky, Eyal; Peres, Yuval

doi:10.1007/s00039-016-0382-7

Cited by 56 publications

(77 citation statements)

References 33 publications

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“…See the discussion in Section 8 for full details. This work is similar in spirit to the results of [20], and shows the general connection between the common distance and cutoff phenomena in quotients of symmetric spaces (infinite regular trees and the hyperbolic plane in these cases) and temperedness of representations (or the Ramanujan conjecture).…”

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confidence: 78%

Cutoff on hyperbolic surfaces

Golubev

Kamber

2019

Geom Dedicata

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In this paper we study the common distance between points and the behavior of a constant length step discrete random walk on finite area hyperbolic surfaces. We show that if the second smallest eigenvalue of the Laplacian is at least 1/4, then the distances on the surface are highly concentrated around the minimal possible value, and that the discrete random walk exhibits cutoff. This extends the results of Lubetzky and Peres ([20]) from the setting of Ramanujan graphs to the setting of hyperbolic surfaces. By utilizing density theorems of exceptional eigenvalues from [27], we are able to show that the results apply to congruence subgroups of SL 2 (Z) and other arithmetic lattices, without relying on the well known conjecture of Selberg ([28]). Conceptually, we show the close relation between the cutoff phenomenon and temperedness of representations of algebraic groups over local fields, partly answering a question of Diaconis ([7]), who asked under what general phenomena cutoff exists.

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confidence: 78%

Cutoff on hyperbolic surfaces

Golubev

Kamber

2019

Geom Dedicata

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“…The idea of considering a non-backtracking quantum variance first appeared in [7]. Note that the idea of replacing simple random walks by non-backtracking ones is also useful to solve many other problems [29,21,17,1,42].…”

Section: Discussion Of Assumptionsmentioning

confidence: 99%

Recent results of quantum ergodicity on graphs and further investigation

Anantharaman¹,

Sabri²

2019

Annales De La Faculté Des Sciences De Toulouse : Mathématiques

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“…It is a fairly wellunderstood theme that large, d-regular Ramanujan graphs locally resemble the dregular tree in that they contain few short cycles. For an illustration of such results, see [1,4,9,11,12] and references therein. This relation is not as well understood for sparse, irregular graphs.…”

Section: Introductionmentioning

confidence: 97%

On the local geometry of graphs in terms of their spectra

Huang

Rahman

2019

European Journal of Combinatorics

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In this paper, we consider the relation between the spectrum and the number of short cycles in large graphs. Suppose G1, G2, G3, . . . is a sequence of finite and connected graphs that share a common universal cover T and such that the proportion of eigenvalues of Gn that lie within the support of the spectrum of T tends to 1 in the large n limit. We prove such a sequence of graphs is asymptotically locally tree-like. This is deduced by way of an analogous spectral rigidity theorem proved for certain infinite sophic graphs. We present additional results and questions in this spirit.

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Cutoff on all Ramanujan graphs

Cited by 56 publications

References 33 publications

Cutoff on hyperbolic surfaces

Cutoff on hyperbolic surfaces

Recent results of quantum ergodicity on graphs and further investigation

On the local geometry of graphs in terms of their spectra

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