Delocalization and Continuous Spectrum for Ultrametric Random Operators

Soosten, Per von; Warzel, Simone

doi:10.1007/s00023-019-00809-z

Cited by 6 publications

(8 citation statements)

References 43 publications

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“…Using the above relation (136) we see that this problem corresponds to a rank-n perturbation of a N × N GUE matrix. It is known that as N → +∞, there are in that case n successive and distinct BBP transitions as θ = x/N is increased from 0 (see [90, Theorem 2.1 and Section 3.1] with the correspondence σ → √ N x and θ i → xa i ).…”

Section: B Few Lines and Few Columnar Defectsmentioning

confidence: 97%

“…From the above property (136) we see that if M is fixed and N becomes large (i.e. few lines, many columns) one is probing the edge of the spectrum of the matrix M(x).…”

Section: A Relation To Rmtmentioning

confidence: 99%

“…The problem studied here is closely related to the socalled generalized Rosenzweig-Porter (GRP) model recently studied in physics [130][131][132][133] and mathematics [71,[134][135][136]. The GRP model is a cousin of the Anderson models on the Bethe lattice and on the random regular graph, themselves studied [137] as simpler settings for investigating the many-body localization transition [138].…”

Section: The Generalized Rosenzweig-porter Modelmentioning

confidence: 99%

“…It is thus not surprising that for γ < 1 at large N , which corresponds to τ 1, the spectrum of H is a semicircle, while for γ > 1, i.e. τ 1, is it exactly ρ(a) [131,[134][135][136]. However the transition in the local spectral correlations of H between the Wigner-Dyson statistics and the Poisson statistics takes place at a different value, γ = 2.…”

Section: The Generalized Rosenzweig-porter Modelmentioning

confidence: 99%

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Tilted elastic lines with columnar and point disorder, non-Hermitian quantum mechanics and spiked random matrices: pinning and localization

Krajenbrink,

Doussal,

O'Connell

2020

Preprint

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We revisit the problem of an elastic line (such as a vortex line in a superconductor) subject to both columnar disorder and point disorder in dimension d = 1 + 1. Upon applying a transverse field, a delocalization transition is expected, beyond which the line is tilted macroscopically. We investigate this transition in the fixed tilt angle ensemble and within a "one-way" model where backward jumps are neglected. From recent results about directed polymers in the mathematics literature, and their connections to random matrix theory, we find that for a single line and a single strong defect this transition in presence of point disorder coincides with the Baik-Ben Arous-Péché (BBP) transition for the appearance of outliers in the spectrum of a perturbed random matrix in the Gaussian Unitary Ensemble. This transition is conveniently described in the polymer picture by a variational calculation. In the delocalized phase, the ground state energy exhibits Tracy-Widom fluctuations. In the localized phase we show, using the variational calculation, that the fluctuations of the occupation length along the columnar defect are described by fKPZ, a distribution which appears ubiquitously in the Kardar-Parisi-Zhang universality class. We then consider a smooth density of columnar defect energies. Depending on how this density vanishes at its lower edge we find either (i) a delocalized phase only (ii) a localized phase with a delocalization transition. We analyze this transition which is an infinite-rank extension of the BBP transition. The fluctuations of the ground state energy of a single elastic line in the localized phase (for fixed columnar defect energies) are described by a Fredholm determinant based on a new kernel, closely related to the kernel describing the largest real eigenvalues of the real Ginibre ensemble. The case of many columns and many non-intersecting lines, relevant for the study of the Bose glass phase, is also analyzed. The ground state energy is obtained using free probability and the Burgers equation. Connections with recent results on the generalized Rosenzweig-Porter model suggest that the localization of many polymers occurs gradually upon increasing their lengths.

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Section: B Few Lines and Few Columnar Defectsmentioning

confidence: 97%

“…From the above property (136) we see that if M is fixed and N becomes large (i.e. few lines, many columns) one is probing the edge of the spectrum of the matrix M(x).…”

Section: A Relation To Rmtmentioning

confidence: 99%

Section: The Generalized Rosenzweig-porter Modelmentioning

confidence: 99%

Section: The Generalized Rosenzweig-porter Modelmentioning

confidence: 99%

See 2 more Smart Citations

Tilted elastic lines with columnar and point disorder, non-Hermitian quantum mechanics and spiked random matrices: pinning and localization

Krajenbrink,

Doussal,

O'Connell

2020

Preprint

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show abstract

“…This approach can be used to derive the validity of the semicircle law on global scales Im z = O(1) in the infinitevolume limit [2]. More recently, the works [20,21] showed that the SDE (1.3) can also be analyzed on local scales by considering the evolution along the random characteristic…”

Section: Introductionmentioning

confidence: 99%