Consecutive level spacings in the chiral Gaussian unitary ensemble: from the hard and soft edge to the bulk

We study heavy-tailed Hermitian random matrices that are unitarily invariant. The invariance implies that the eigenvalue and eigenvector statistics are decoupled. The motivating question has been whether a freely stable random matrix has stable eigenvalue statistics for the largest eigenvalues in the tail. We investigate this question through the use of both numerical and analytical means, the latter of which makes use of the supersymmetry method. A surprising behaviour is uncovered in that a freely stable random matrix does not necessarily yield stable statistics and if it does then it might exhibit Poisson or Poisson-like statistics. The Poisson statistics have been already observed for heavy-tailed Wigner matrices. We conclude with two conjectures on this peculiar behaviour.

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“…which is pretty close to the universal bulk distribution [52] as well as to the level spacing distributions at the hard edge [53].…”

Section: Figuresupporting

confidence: 68%

Local tail statistics of heavy-tailed random matrix ensembles with unitary invariance

Kieburg¹,

Monteleone²

2021

J. Phys. A: Math. Theor.

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“…We also want to point out that for traditional β-ensembles, starting with the well known Wigner surmise, various different notions of spacings have been introduced and studied in great detail. For more about this we refer to [1,2,15,19,20,22,27,28,54,61,64] and the references there.…”

mentioning

confidence: 99%

Gaussian beta ensembles: the perfect freezing transition and its characterization in terms of Beurling-Landau densities

Ameur¹,

Marceca²,

Romero³

2022

Preprint

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The Gaussian β-ensemble is a real n-point configuration {x j } n 1 picked randomly with respect to the Boltzmann factor e − β 2 Hn ,It is well known that the point process {x j } n 1 tends to follow the semicircle law σ(x) = 1 2π (4 − x 2 ) + in certain average senses.A Fekete configuration (minimizer of H n ) is spread out in a much more uniform way in the interval [−2, 2] with respect to the regularization σ n (x) = max{σ(x), n − 1 3 } of the semicircle law. In particular, Fekete configurations are "equidistributed" with respect to σ n (x), in a certain technical sense of Beurling-Landau densities.We consider the problem of characterizing sequences β n of inverse temperatures, which guarantee almost sure equidistribution as n → ∞. We find that a necessary and sufficient condition is that β n grows at least logarithmically in n:We call this growth rate the perfect freezing regime. Along the way, we give several further results on the distribution of particles when β n log n, for example on minimal spacing and discrepancies, and that with high probability a random sample solves certain sampling and interpolation problems for weighted polynomials. (In this context, Fekete sets correspond to β ≡ ∞.)The condition β n log n was introduced in earlier works due to some of the authors in the context of two-dimensional Coulomb gas ensembles, where it is shown to be sufficient for equidistribution. Interestingly, although the technical implementation requires some considerable modifications, the strategy from dimension two adapts well to prove sufficiency also for one-dimensional Gaussian ensembles. On a technical level, we use estimates for weighted polynomials due to Levin, Lubinsky, Gustavsson and others. The other direction (necessity) involves estimates due to Ledoux and Rider on the distribution of particles which fall near or outside of the boundary.

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The resurgence of the plateau in supersymmetric $$ \mathcal{N} $$ = 1 Jackiw-Teitelboim gravity

Griguolo,

Papalini,

Russo

et al. 2024

J. High Energ. Phys.

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Significant progresses have been made recently in understanding the spectral form factor of Jackiw-Teitelboim gravity, particularly at late times where non-perturbative effects are expected to play a dominant role. By focusing on a peculiar regime of large time and fixed temperature, called τ-scaling limit, it was found that it is possible to analytically investigate the late-time plateau directly through the gravitational genus expansion. We extend this analysis to the supersymmetric $$ \mathcal{N} $$ N = 1 generalization of the bosonic theory, revealing an interesting structure. First, we notice that in the τ-scaling limit the perturbative sum over genera truncates after a single term, which solely accounts for the ramp behaviour. Instead a non-perturbative completion, responsible for the plateau, is encoded into an exact formula coming from the properties of the chiral gaussian ensemble, governing the spectral properties of the supersymmetric theory. We are able to recover the non-perturbative contributions by slightly deforming the genus of the involved surfaces and using resurgence theory. We derive a closed-form analytical expression for the late-time plateau and a trans-series expansion that captures the ramp-plateau transition.

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Consecutive level spacings in the chiral Gaussian unitary ensemble: from the hard and soft edge to the bulk

Cited by 4 publications

References 47 publications

Local tail statistics of heavy-tailed random matrix ensembles with unitary invariance

Local tail statistics of heavy-tailed random matrix ensembles with unitary invariance

Gaussian beta ensembles: the perfect freezing transition and its characterization in terms of Beurling-Landau densities

The resurgence of the plateau in supersymmetric $$ \mathcal{N} $$ = 1 Jackiw-Teitelboim gravity

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