Regular Spacings of Complex Eigenvalues in the One-dimensional non-Hermitian Anderson Model

We prove that in dimension one the non-real eigenvalues of the non-Hermitian Anderson (NHA) model with a selfaveraging potential are regularly spaced. The class of selfaveraging potentials which we introduce in this paper is very wide and in particular includes stationary potentials (with probability one) as well as all quasi-periodic potentials. It should be emphasized that our approach here is much simpler than the one we used before. It allows us a) to investigate the above mentioned spacings, b) to establish certain properties of the integrated density of states of the Hermitian Anderson models with selfaveraging potentials, and c) to obtain (as a by-product) much simpler proofs of our previous results concerned with non-real eigenvalues of the NHA model.Comment: 21 pages, 1 figur

Induced Ginibre ensemble of random matrices and quantum operations

A generalisation of the Ginibre ensemble of non-Hermitian random square matrices is introduced. The corresponding probability measure is induced by the ensemble of rectangular Gaussian matrices via a quadratisation procedure. We derive the joint probability density of eigenvalues for such induced Ginibre ensemble and study various spectral correlation functions for complex and real matrices, and analyse universal behaviour in the limit of large dimensions. In this limit the eigenvalues of the induced Ginibre ensemble cover uniformly a ring in the complex plane. The real induced Ginibre ensemble is shown to be useful to describe statistical properties of evolution operators associated with random quantum operations, for which the dimensions of the input state and the output state do differ.Comment: 2nd version, 34 pages, 5 figure

Density of State in a Complex Random Matrix Theory with External Source

The density of state for a complex $N\times N$ random matrix coupled to an external deterministic source is considered for a finite N, and a compact expression in an integral representation is obtained.Comment: 7 pages, late

Fractional Brownian motion with Hurst index H=0 and the Gaussian Unitary Ensemble

The goal of this paper is to establish a relation between characteristic polynomials of N×N GUE random matrices H as N→∞, and Gaussian processes with logarithmic correlations. We introduce a regularized version of fractional Brownian motion with zero Hurst index, which is a Gaussian process with stationary increments and logarithmic increment structure. Then we prove that this process appears as a limit of DN(z)=−log|det(H−zI)| on mesoscopic scales as N→∞. By employing a Fourier integral representation, we use this to prove a continuous analogue of a result by Diaconis and Shahshahani [J. Appl. Probab. 31A (1994) 49–62]. On the macroscopic scale, DN(x) gives rise to yet another type of Gaussian process with logarithmic correlations. We give an explicit construction of the latter in terms of a Chebyshev–Fourier random series

Interpolation between Airy and Poisson statistics for unitary chiral non-Hermitian random matrix ensembles

We consider a family of chiral non-Hermitian Gaussian random matrices in the unitarily invariant symmetry class. The eigenvalue distribution in this model is expressed in terms of Laguerre polynomials in the complex plane. These are orthogonal with respect to a non-Gaussian weight including a modified Bessel function of the second kind, and we give an elementary proof for this. In the large n limit, the eigenvalue statistics at the spectral edge close to the real axis are described by the same family of kernels interpolating between Airy and Poisson that was recently found by one of the authors for the elliptic Ginibre ensemble. We conclude that this scaling limit is universal, appearing for two different non-Hermitian random matrix ensembles with unitary symmetry. As a second result we give an equivalent form for the interpolating Airy kernel in terms of a single real integral, similar to representations for the asymptotic kernel in the bulk and at the hard edge of the spectrum. This makes its structure as a one-parameter deformation of the Airy kernel more transparent

Summing free unitary random matrices

I use quaternion free probability calculus - an extension of free probability to non-Hermitian matrices (which is introduced in a succinct but self-contained way) - to derive in the large-size limit the mean densities of the eigenvalues and singular values of sums of independent unitary random matrices, weighted by complex numbers. In the case of CUE summands, I write them in terms of two "master equations," which I then solve and numerically test in four specific cases. I conjecture a finite-size extension of these results, exploiting the complementary error function. I prove a central limit theorem, and its first sub-leading correction, for independent identically-distributed zero-drift unitary random matrices.Comment: 17 pages, 15 figure

A Folklore Archive Offline and Online: Digitisation Problems

The article was submitted on 12.07.2022.Цель статьи – выявить сложности, связанные с представлением фольклорного архива в цифровом виде, и наметить перспективы цифровизации материалов. Исследователи фольклора признают значительные затруднения при переходе к новым стандартам фиксации контекста, что показал анализ представленных на сегодняшний день в сети интернет-архивов, фольклорных электронных ресурсов, баз данных и материалов фольклорных лабораторий России. Рассмотрена связь технических процедур по разметке фольклорных текстов с возможностью использовать интерпретационные моменты этой процедуры для восстановления логики программ полевых исследований и контекстных связей собранных данных. Понятие контекста, понимаемого как разнообразные «горизонтальные связи» фольклорного/фольклоризованного текста при его функционировании в пространстве локальной культуры, приобретает все большее значение. Перевод архивных данных в цифровой формат может иметь практическое применение не только для структурного упорядочивания (создания) поисковой системы, размеченной определенными маркерами/тэгами, но и как возможность реконструкции частично утраченного научно-исследовательского контекста. В материалах 1960–2000 гг. восполнить из живого общения с собирателями, участниками экспедиций такой контекст часто не удается в связи с межпоколенческими разрывами традиций в полевой фольклористике. Предложенный вариант представляет собой разработку одного из практических способов применения оцифровки массива полевых материалов для уточнения истории местных научных школ и объединений. Статья содержит обзор и оценку наполнения доступного онлайн-контента нескольких десятков фольклорных архивов различных институций.This article aims to identify the main problems associated with the digitisation of a folklore archive and formulate some prospects of digitising materials. The authors analyse Internet archives, folklore and similar electronic resources, databases, and materials from Russian folklore laboratories. The second part of the article is devoted to the connection of technical procedures for labeling folklore texts with the possibility of using the interpretive techniques of this procedure to restore the logic of field research programmes and the contextual connections of the collected data. The concept of context understood as a variety of “horizontal connections” of a folklore text during its functioning in the space of local culture is becoming increasingly important. The conversion of archival data into a digital format can be of practical use not only as a structural ordering – creating a search engine marked with certain markers / tags, but also as an opportunity to reconstruct a partially lost research context. In the materials from the 1960s–2000s, it is often impossible to make up for context from live communication with specialists and members of expeditions due to intergenerational breaks in tradition that have occurred in Russian folklore studies. This article offers a solution which aims to develop one of the practical ways to use the digitisation of an array of field materials to clarify the history of local research schools and associations. The article contains an overview and assessment of the content of the available online folklore archives.Исследование выполнено при финансовой поддержке Министерства науки и высшего образования Российской Федерации в рамках Программы развития Уральского федерального университета имени первого Президента России Б. Н. Ельцина в соответствии с программой стратегического академического лидерства «Приоритет‑2030»

Almost-Hermitian Random Matrices: Crossover from Wigner-Dyson to Ginibre eigenvalue statistics

By using the method of orthogonal polynomials we analyze the statistical properties of complex eigenvalues of random matrices describing a crossover from Hermitian matrices characterized by the Wigner- Dyson statistics of real eigenvalues to strongly non-Hermitian ones whose complex eigenvalues were studied by Ginibre. Two-point statistical measures (as e.g. spectral form factor, number variance and small distance behavior of the nearest neighbor distance distribution $p(s)$) are studied in more detail. In particular, we found that the latter function may exhibit unusual behavior $p(s)\propto s^{5/2}$ for some parameter values.Comment: 4 pages, RevTE