On asymptotic behavior of Bell polynomials and concentration of vertex degree of large random graphs

Khorunzhiy, O.

doi:10.48550/arxiv.1904.01339

Cited by 4 publications

(11 citation statements)

References 27 publications

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“…Although (3.23) is not exactly the Borel transform of (3.22), we expect it will have the same convergence properties. Using theorem 2.1 of [20] one can check that (3.23) converges to an entire function of λ.…”

Section: A Splitting Formulamentioning

confidence: 99%

“…. , b n } for A, so θ(a i b j ) = δ ij , and such that a i b i = s. A good example of this structure is provided by the cohomology of an arbitrary manifold, 20 A = H * (M ; κ). Then if dimM = m we can take N k to be the subspace of forms of degree ≥ m − k. Then we take a 1 = 1, .…”

Section: Some Remarks On the General Non-semisimple Casementioning

confidence: 99%

“…for fixed λ, B n (λ) grows, roughly speaking, like n! [20] so the series is at best asymptotic. 19 In particular, rather surprisingly, (Ψ ∨ HH , Ψ HH ) is not the vacuum to vacuum to vacuum amplitude!…”

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confidence: 99%

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Comments on Summing over bordisms in TQFT

Banerjee¹,

Moore²

2022

Preprint

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Section: A Splitting Formulamentioning

confidence: 99%

Section: Some Remarks On the General Non-semisimple Casementioning

confidence: 99%

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Comments on Summing over bordisms in TQFT

Banerjee¹,

Moore²

2022

Preprint

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“…Here R k = B k (2) for infinite 1D system, R k = B k for semi-infinite 1D system with a boundary, and R k = C P k (q −1 0 ) for D ≥ 2. Using the asymptotic behavior of Bell polynomials [31]…”

Section: B Constraints On Power Spectrummentioning

confidence: 99%

Euclidean operator growth and quantum chaos

Avdoshkin,

Dymarsky

2019

Preprint

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We consider growth of local operators under Euclidean time evolution in lattice systems with local interactions. We derive rigorous bounds on the operator norm growth and then proceed to establish an analog of the Lieb-Robinson bound for the spatial growth. In contrast to the Minkowski case when ballistic spreading of operators is universal, in the Euclidean case spatial growth is systemdependent and indicates if the system is integrable or chaotic. In the integrable case, the Euclidean spatial growth is at most polynomial. In the chaotic case, it is the fastest possible: exponential in 1D, while in higher dimensions and on Bethe lattices local operators can reach spatial infinity in finite Euclidean time. We use bounds on the Euclidean growth to establish constraints on individual matrix elements and operator power spectrum. We show that one-dimensional systems are special with the power spectrum always being superexponentially suppressed at large frequencies. Finally, we relate the bound on the Euclidean growth to the bound on the growth of Lanczos coefficients. To that end, we develop a path integral formalism for the weighted Dyck paths and evaluate it using saddle point approximation. Using a conjectural connection between the growth of the Lanczos coefficients and the Lyapunov exponent controlling the growth of OTOCs, we propose an improved bound on chaos valid at all temperatures.

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“…In paper [14], we considered the maximal vertex degree of random graphs d max = max i=1,...,n d i in the limit when n and p tend to infinity simultaneously. We have studied the concentration properties of d max with the help of high moments of random variables d i .…”

Section: Introductionmentioning

confidence: 99%

On asymptotic properties of high moments of compound Poisson distribution

Khorunzhiy

2020

Preprint

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We study asymptotic properties of the moments M k (λ) of compound Poisson random variable N j=1 W j . These moments can be regarded as a weighted version of Bell polynomials. We consider a limiting transition when the number of the moment k infinitely increases at the same time as the mean value λ of the Poisson random variable N tends to infinity and find an explicit expression for the rate function in dependence of the ratio k/λ. This rate function is expressed in terms of the exponential generating function of the moments of W j . We illustrate the general theorem by three particular cases corresponding to the normal, gamma and Bernoulli distributions of the weights W j . We apply our results to the study of the concentration properties of weighted vertex degree of large random graphs. Finally, as a by-product of the main theorem, we determine an asymptotic behavior of the number of even partitions in comparison with the asymptotic properties of the Bell numbers.

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On asymptotic behavior of Bell polynomials and concentration of vertex degree of large random graphs

Cited by 4 publications

References 27 publications

Comments on Summing over bordisms in TQFT

Comments on Summing over bordisms in TQFT

Euclidean operator growth and quantum chaos

On asymptotic properties of high moments of compound Poisson distribution

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