Universality of Mesoscopic Fluctuations for Orthogonal Polynomial Ensembles

Breuer, Jonathan; Duits, Maurice

doi:10.1007/s00220-015-2514-6

Cited by 36 publications

(40 citation statements)

References 37 publications

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“…It turns out that the appearance of the H 1/2 -Gaussian noise is remarkably universal in the mesoscopic limit of one dimensional ensembles with random matrix type repulsion and this problem has attracted renewed interest in the last couple of years. For instance, the analogue of Soshnikov's CLT (1.9) was obtained for the GUE [28], more general invariant ensembles [12,46], Wigner matrices [23,47,34], β-ensembles [10,2] and for zeros of the Riemann zeta function [11,63]. It is likely the counterpart of Theorem 1.3 continues to hold for these models as well.…”

Section: Background and Results For The Cuementioning

confidence: 90%

Subcritical Multiplicative Chaos for Regularized Counting Statistics from Random Matrix Theory

Lambert

Ostrovsky

Simm

2018

Commun. Math. Phys.

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For an N × N Haar distributed random unitary matrix UN , we consider the random field defined by counting the number of eigenvalues of UN in a mesoscopic arc centered at the point u on the unit circle. We prove that after regularizing at a small scale ǫN > 0, the renormalized exponential of this field converges as N → ∞ to a Gaussian multiplicative chaos measure in the whole subcritical phase. We discuss implications of this result for obtaining a lower bound on the maximum of the field. We also show that the moments of the total mass converge to a Selberg-like integral and by taking a further limit as the size of the arc diverges, we establish part of the conjectures in [55]. By an analogous construction, we prove that the multiplicative chaos measure coming from the sine process has the same distribution, which strongly suggests that this limiting object should be universal. Our approach to the L 1 -phase is based on a generalization of the construction in Berestycki [5] to random fields which are only asymptotically Gaussian. In particular, our method could have applications to other random fields coming from either random matrix theory or a different context.

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Section: Background and Results For The Cuementioning

confidence: 90%

Subcritical Multiplicative Chaos for Regularized Counting Statistics from Random Matrix Theory

Lambert

Ostrovsky

Simm

2018

Commun. Math. Phys.

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“…for any function f ∈ L ∞ ([−1, 1]). The CLT (4.2) holds for more general potentials and for other orthogonal polynomial ensembles as well, [38, section 11.3] or [10,14,35] and it is known that the one-cut condition, i.e. the assumption that the support of the equilibrium measure is connected, is necessary.…”

Section: Just Like For a Poisson Point Process With Intensity Functionmentioning

confidence: 99%

Incomplete Determinantal Processes: From Random Matrix to Poisson Statistics

Lambert

2019

J Stat Phys

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We study linear statistics of a class of determinantal processes which interpolate between Poisson and GUE/Ginibre statistics in dimension 1 or 2. These processes are obtained by performing an independent Bernoulli percolation on the particle configuration of a Coulomb gas with logarithmic interaction confined in a general potential. We show that, depending on the expected number of deleted particles, there is a universal transition for mesoscopic statistics. Namely, at small scales, the point process still behaves like a Coulomb gas, while, at large scales, it needs to be renormalized because the variance of any linear statistic diverges. The crossover is explicitly characterized as the superposition of a H 1/2 -or H 1 -correlated Gaussian noise and an independent Poisson process. The proof consists in computing the limits of the cumulants of linear statistics using the asymptotics of the correlation kernel of the process.

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“…Interest in mesoscopic linear statistics has surged in recent years. Results in this field of study were obtained in a variety of settings, for Gaussian random matrices [9,12], and for invariant ensembles [11,15]. In many cases the results were shown at all scales α ∈ (0; 1), often with the use of distribution specific properties.…”

Section: Introductionmentioning

confidence: 99%