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Sasha Sodin

Publications and source records attributed to Sasha Sodin.

At least 37 records · Page 2Linked to original sources
arXiv2018

Real eigenvalues in the non-Hermitian Anderson model

The eigenvalues of the Hatano--Nelson non-Hermitian Anderson matrices, in the spectral regions in which the Lyapunov exponent exceeds the non-Hermiticity parameter, are shown to be real and exponentially close to the Hermitian eigenvalues. This complements previous results, according to which the eigenvalues in the spectral regions in which the non-Hermiticity parameter exceeds the Lyapunov exponent are aligned on curves in the complex plane.

math-ph↗
arXiv2017

On the critical points of random matrix characteristic polynomials and of the Riemann $ξ$-function

A one-parameter family of point processes describing the distribution of the critical points of the characteristic polynomial of large random Hermitian matrices on the scale of mean spacing is investigated. Conditionally on the Riemann hypothesis and the multiple correlation conjecture, we show that one of these limiting processes also describes the distribution of the critical points of the Riemann $ξ$-function on the critical line. We prove that each of these processes boasts stronger level repulsion than the sine process describing the limiting statistics of the eigenvalues: the probability to find $k$ critical points in a short interval is comparable to the probability to find $k+1$ eigenvalues there. We also prove a similar property for the critical points and zeros of the Riemann $ξ$-function, conditionally on the Riemann hypothesis but not on the multiple correlation conjecture.

math.PR↗
arXiv2017

Fluctuations of interlacing sequences

In a series of works published in the 1990-s, Kerov put forth various applications of the circle of ideas centred at the Markov moment problem to the limiting shape of random continual diagrams arising in representation theory and spectral theory. We demonstrate on several examples that his approach is also adequate to study the fluctuations about the limiting shape. In the random matrix setting, we compare two continual diagrams: one is constructed from the eigenvalues of the matrix and the critical points of its characteristic polynomial, whereas the second one is constructed from the eigenvalues of the matrix and those of its principal submatrix. The fluctuations of the latter diagram were recently studied by Erdős and Schröder; we discuss the fluctuations of the former, and compare the two limiting processes. For Plancherel random partitions, the Markov correspondence establishes the equivalence between Kerov's central limit theorem for the Young diagram and the Ivanov--Olshanski central limit theorem for the transition measure. We outline a combinatorial proof of the latter, and compare the limiting process with the ones of random matrices.

math.PR↗
arXiv2017

Matrix regularizing effects of Gaussian perturbations

The addition of noise has a regularizing effect on Hermitian matrices. This effect is studied here for $H=A+V$, where $A$ is the base matrix and $V$ is sampled from the GOE or the GUE random matrix ensembles. We bound the mean number of eigenvalues of $H$ in an interval, and present tail bounds for the distribution of the Frobenius and operator norms of $H^{-1}$ and for the distribution of the norm of $H^{-1}$ applied to a fixed vector. The bounds are uniform in $A$ and exceed the actual suprema by no more than multiplicative constants. The probability of multiple eigenvalues in an interval is also estimated.

math.PR↗
arXiv2016

The Tracy--Widom law for some sparse random matrices

Consider the random matrix obtained from the adjacency matrix of a random d-regular graph by multiplying every entry by a random sign. The largest eigenvalue converges, after proper scaling, to the Tracy--Widom distribution.

math-ph↗
arXiv2016

A Limit Theorem for Stochastically Decaying Partitions at the Edge

In this paper, we study the asymptotic behavior of the first, second, and so on rows of stochastically decaying partitions. We establish that, with appropriate scaling in time and length, the sequence of rows converges to the Airy$_2$ line ensemble. This result was first established, in a more general setting, by Borodin and Olshanski, who relied on the determinantal structure of the Poissonized correlation functions. Our argument is based on a different, combinatorial approach, developed by Okounkov. This approach may be useful in other problems in which no determinantal structure is available, and also highlights the similarity between random partitions and random matrices.

math.PR↗
arXiv2016

On the Wegner orbital model

The Wegner orbital model is a class of random operators introduced by Wegner to model the motion of a quantum particle with many internal degrees of freedom (orbitals) in a disordered medium. We consider the case when the matrix potential is Gaussian, and prove three results: localisation at strong disorder, a Wegner-type estimate on the mean density of eigenvalues, and a Minami-type estimate on the probability of having multiple eigenvalues in a short interval. The last two results are proved in the more general setting of deformed block-Gaussian matrices, which includes a class of Gaussian band matrices as a special case. Emphasis is placed on the dependence of the bounds on the number of orbitals. As an additional application, we improve the upper bound on the localisation length for one-dimensional Gaussian band matrices.

math-ph↗
arXiv2015

Semi-classical analysis of non self-adjoint transfer matrices in statistical mechanics. I

We propose a way to study one-dimensional statistical mechanics models with complex-valued action using transfer operators. The argument consists of two steps. First, the contour of integration is deformed so that the associated transfer operator is a perturbation of a normal one. Then the transfer operator is studied using methods of semi-classical analysis. In this paper we concentrate on the second step, the main technical result being a semi-classical estimate for powers of an integral operator which is approximately normal.

math-ph↗
arXiv2015

The trimmed Anderson model at strong disorder: localisation and its breakup

We explore the properties of discrete random Schroedinger operators in which the random part of the potential is supported on a sub-lattice. In particular, we provide new conditions on the sub-lattice under which Anderson localisation happens at strong disorder, and provide examples in which it can be ruled out.

math-ph↗
arXiv2014

A limit theorem at the spectral edge for corners of time-dependent Wigner matrices

For the eigenvalues of principal submatrices of stochastically evolving Wigner matrices, we construct and study the edge scaling limit: a random decreasing sequence of continuous functions of two variables, which at every point has the distribution of the Airy point process. The analysis is based on the methods developed by Soshnikov to study the extreme eigenvalues of a single Wigner matrix.

math-ph↗
arXiv2012

Localisation for non-monotone Schroedinger operators

We study localisation effects of strong disorder on the spectral and dynamical properties of (matrix and scalar) Schroedinger operators with non-monotone random potentials, on the d-dimensional lattice. Our results include dynamical localisation, i.e. exponentially decaying bounds on the transition amplitude in the mean. They are derived through the study of fractional moments of the resolvent, which are finite due to resonance-diffusing effects of the disorder. One of the byproducts of the analysis is a nearly optimal Wegner estimate. A particular example of the class of systems covered by our results is the discrete alloy-type Anderson model.

math-ph↗
arXiv2010

On the Measure of the Absolutely Continuous Spectrum for Jacobi Matrices

We apply the methods of classical approximation theory (extreme properties of polynomials) to study the essential support $Σ_{ac}$ of the absolutely continuous spectrum of Jacobi matrices. First, we prove an upper bound on the measure of $Σ_{ac}$ which takes into account the value distribution of the diagonal elements, and implies the bound due to Deift-Simon and Poltoratski-Remling. Second, we generalise the differential inequality of Deift-Simon for the integrated density of states associated with the absolutely continuous spectrum to general Jacobi matrices.

math.SP↗
arXiv2010

The spectral edge of some random band matrices

We study the asymptotic distribution of the eigenvalues of random Hermitian periodic band matrices, focusing on the spectral edges. The eigenvalues close to the edges converge in distribution to the Airy point process if (and only if) the band is sufficiently wide (W >> N^{5/6}.) Otherwise, a different limiting distribution appears.

math-ph↗
arXiv2010

Variations on the Berry-Esseen theorem

We analyze the quality of the gaussian approximation to linear combinations of n independent, identically-distributed random variables with finite fourth moments. It turns out that there exist universal, simple linear combinations that perform better than the sum of the variables. We also investigate the case in which the random variables are independent, yet they are not necessarily identically distributed.

math.PR↗