Quantum Chaos on Random Cayley Graphs of <i> SL 2[Z/pZ]</i>

Rivin, Igor; Sardari, Naser T.

doi:10.1080/10586458.2017.1403982

Cited by 9 publications

(12 citation statements)

References 24 publications

(29 reference statements)

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“…Remark 1.5. In [Sar18,RS17], we demonstrated some numerical experiments on the diameter of the LPS Ramanujan graphs X p,m , which shows that their diameter is asymptotically (4/3) log p |X p,m |. In our experiments, p is fixed and m is growing.…”

Section: Introductionmentioning

confidence: 89%

Optimal strong approximation for quadratic forms

Sardari¹

2019

Duke Math. J.

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For a non-degenerate integral quadratic form F (x 1 , . . . , x d ) in d ≥ 5 variables, we prove an optimal strong approximation theorem. Let Ω be a fixed compact subset of the affine quadric F (x 1 , . . . , x d ) = 1 over the real numbers. Take a small ball B of radius 0 < r < 1 inside Ω, and an integer m. Further assume that N is a given integer which satisfies N ≫ δ,Ω (r −1 m) 4+δ for any δ > 0. Finally assume that an integral vector (λ 1 , . . . , λ d ) mod m is given. Then we show that there exists an integral solution X = (x 1 , . . . , x d ) of F (X) = N such that x i ≡ λ i mod m and X √ N ∈ B, provided that all the local conditions are satisfied. We also show that 4 is the best possible exponent. Moreover, for a non-degenerate integral quadratic form in 4 variables we prove the same result if N is odd and N ≫ δ,Ω (r −1 m) 6+ǫ . Based on our numerical experiments on the diameter of LPS Ramanujan graphs and the expected square root cancellation in a particular sum that appears in Remark 6.8, we conjecture that the theorem holds for any quadratic form in 4 variables with the optimal exponent 4.

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Section: Introductionmentioning

confidence: 89%

Optimal strong approximation for quadratic forms

Sardari¹

2019

Duke Math. J.

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“…However it is plausible that they are in fact almost Ramanujan, in the sense that for each > 0 with very high probability as p → +∞ all non-trivial eigenvalues are bounded by 2 √ k − 1+ . See [RS17] where an upper bound on the number of exceptional eigenvalues is established and numerics are given. The same could be said of the family of alternating groups Alt(n) (and perhaps even of the full family of all finite simple groups).…”

Section: This Is Still Widely Open the Best Results As Of Now Ismentioning

confidence: 99%

Expansion in finite simple groups of Lie type

Breuillard¹,

Green²,

Guralnick³

et al. 2015

J. Eur. Math. Soc.

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Two short seminal papers of Margulis used Kazhdan's property (T ) to give, on the one hand, explicit constructions of expander graphs, and to prove, on the other hand, the uniqueness of some invariant means on compact simple Lie groups. These papers opened a rich line of research on expansion and spectral gap phenomena in finite and compact simple groups. In this paper we survey the history of this area and point out a number of problems which are still open.

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“…It is known that random d-regular graphs form a sequence of expander graphs, close to being Ramanujan [15]. For possible Cayley graph examples with diameter close to log d´1 n see [42][Section 4.1], who provide a numerical evidence that the diameter of SLp2, Z{pZq, with respect to a random generating set on d generators is close to ln d´1 n, as p Ñ 8.…”

Section: Weak Expansion and Infinite Order Breakpointmentioning

confidence: 99%

Assouad-Nagata dimension and gap for ordered metric spaces

Erschler¹,

Mitrofanov²

2021

Preprint

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We prove that all spaces of finite Assouad-Nagata dimension admit a good order for Travelling Salesman Problem, and provide sufficient conditions under which the converse is true. We formulate a conjectural characterisation of spaces of finite AN -dimension, which would yield a gap statement for the efficiency of orders on metric spaces. Under assumption of doubling, we prove a stronger gap phenomenon about all orders on a given metric space.

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Quantum Chaos on Random Cayley Graphs of SL 2[Z/pZ]

Cited by 9 publications

References 24 publications

Optimal strong approximation for quadratic forms

Optimal strong approximation for quadratic forms

Expansion in finite simple groups of Lie type

Assouad-Nagata dimension and gap for ordered metric spaces

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