Lifshitz tails and long-time decay in random systems with arbitrary disorder

Luck, J. M.; Nieuwenhuizen, Th. M.

doi:10.1007/bf01016401

Cited by 24 publications

(12 citation statements)

References 22 publications

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“…where b is again an arbitrary positive parameter, the Laplace transform f The similarity between the two problems is now patent by comparing (6.12) and (6. 19). The probability densities of the variable V = 1/R and of the Kesten variable Z are related to each other by the equation…”

Section: Discussionmentioning

confidence: 99%

A record-driven growth process

Godrèche¹,

Luck²

2008

J. Stat. Mech.

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We introduce a novel stochastic growth process, the Record-driven growth process, which originates from the analysis of a class of growing networks in a universal limiting regime. Nodes are added one by one to a network, each node possessing a quality. The new incoming node connects to the preexisting node with best quality, that is, with record value for the quality. The emergent structure is that of a growing network, where groups are formed around record nodes (nodes endowed with the best intrinsic qualities). Special emphasis is put on the statistics of leaders (nodes whose degrees are the largest). The asymptotic probability for a node to be a leader is equal to the Golomb-Dickman constant ω = 0.624 329 . . ., which arises in problems of combinatorical nature. This outcome solves the problem of the determination of the record breaking rate for the sequence of correlated inter-record intervals. The process exhibits temporal self-similarity in the late-time regime. Connections with the statistics of the cycles of random permutations, the statistical properties of randomly broken intervals, and the Kesten variable are given.

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Section: Discussionmentioning

confidence: 99%

A record-driven growth process

Godrèche¹,

Luck²

2008

J. Stat. Mech.

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“…The expression (A.4) signals the appearance of the Lifshitz tail in the density of states. A more precise analysis shows that the tail (A.4) is modulated by a periodic function [54]. Equation (A.3) displays some interesting features.…”

Section: Jhep04(2021)080mentioning

confidence: 95%

“…The spectral behavior of the Anderson model for the distribution (A.2) up to some minor details is identical to the problem encountered in the study of harmonic chains with binary distribution of random masses. The last problem, which goes back to F. Dyson, has been investigated in [52,53] and then thoroughly discussed in [54]. They considered a harmonic chain of masses, which can take two values, m and M > m as implied by (A.2).…”

Section: Jhep04(2021)080mentioning

confidence: 99%

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Lifshitz tails at spectral edge and holography with a finite cutoff

Gorsky

Nechaev

Valov

2021

J. High Energ. Phys.

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We propose the holographic description of the Lifshitz tail typical for one-particle spectral density of bounded disordered system in D = 1 space. To this aim the “polymer representation” of the Jackiw-Teitelboim (JT) 2D dilaton gravity at a finite cutoff is used and the corresponding partition function is considered as the weighted sum over paths of fixed length in an external magnetic field. We identify the regime of small loops, responsible for emergence of a Lifshitz tail in the Gaussian disorder, and relate the strength of disorder to the boundary value of the dilaton. The geometry corresponding to the Poisson disorder in the boundary theory involves random paths fluctuating in the vicinity of the hard impenetrable cut-off disc in a 2D plane. It is shown that the ensemble of “stretched” paths evading the disc possesses the Kardar-Parisi-Zhang (KPZ) scaling for fluctuations, which is the key property that ensures the dual description of the Lifshitz tail in the spectral density for the Poisson disorder.

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“…(ii) How does disorder enter the estimate? More precisely, can one generalize existing results showing logarithmic corrections for random uniform disorder as compared to binary disorder [18,19]? (iii) Can these results be extended to a full interval instead of being asymptotic near the lower bound of the spectrum?…”

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confidence: 96%

Sharp estimates for the integrated density of states in Anderson tight-binding models

Desforges,

Mayboroda,

Zhang

et al. 2020

Preprint

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Recent work [1] has proved the existence of bounds from above and below for the Integrated Density of States (IDOS) of the Schrödinger operator throughout the spectrum, called the Landscape Law. These bounds involve dimensional constants whose optimal values are yet to be determined. Here, we investigate the accuracy of the Landscape Law in 1D and 2D tight-binding Anderson models, with binary or uniform random distributions. We show, in particular, that in 1D, the IDOS can be approximated with high accuracy through a single formula involving a remarkably simple multiplicative energy shift. In 2D, the same idea applies but the prefactor has to be changed between the bottom and top parts of the spectrum.

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Lifshitz tails and long-time decay in random systems with arbitrary disorder

Cited by 24 publications

References 22 publications

A record-driven growth process

A record-driven growth process

Lifshitz tails at spectral edge and holography with a finite cutoff

Sharp estimates for the integrated density of states in Anderson tight-binding models

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