arXiv subjects

Eugene Strahov

Publications and source records attributed to Eugene Strahov.

At least 19 recordsLinked to original sources

arXiv2026

A refinement of the Ewens sampling formula

We consider an infinitely-many neutral allelic model of population genetics where all alleles are divided into a finite number of classes, and each class is characterized by its own mutation rate. For this model the allelic composition of a sample taken from a very large population of genes is characterized by a random matrix, and the problem is to describe the joint distribution of the matrix entries. The answer is given by a new generalization of the classical Ewens sampling formula called the refined Ewens sampling formula in the present paper. We discuss a Poisson approximation for the refined Ewens sampling formula, and present its derivation by several methods. As an application we obtain limit theorems for the numbers of alleles in different asymptotic regimes.

math.PR↗

arXiv2025

Probability measures on families of partitions related to harmonic analysis on big wreath products

We construct generalized regular representations of the wreath product of a compact group with the infinite symmetric group. The characters of these representations are determined by probability measures on families of partitions called the $z$-measures for the wreath product of a compact group with the symmetric group in the present paper. Our main result is an explicit formula for these $z$-measures which holds true for an arbitrary compact group. The result enables us to describe the spectral measures of the generalized regular representations of big wreath products.

math.RT↗

arXiv2025

Last-passage percolation and product-matrix ensembles

We introduce and study a model of directed last-passage percolation in planar layered environment. This environment is represented by an array of random exponential clocks arranged in blocks, for each block the average waiting times depend only on the local coordinates within the block. The last-passage time, the total time needed to travel from the source to the sink located in a given block, maximized over all the admissible paths, becomes a stochastic process indexed by the number of blocks in the array. We show that this model is integrable, particularly the probability law of the last-passage time process can be determined via a Fredholm determinant of the kernel that also appears in the study of products of random matrices. Further, we identify the scaling limit of the last-passage time process, as the sizes of the blocks become infinitely large and the average waiting times become infinitely small. Finite-dimensional convergence to the continuous-time critical stochastic process of random matrix theory is established.

math.PR↗

arXiv2024

Multiple Partition Structures and Harmonic Functions on Branching Graphs

We introduce and study multiple partition structures which are sequences of probability measures on families of Young diagrams subjected to a consistency condition. The multiple partition structures are generalizations of Kingman's partition structures, and are motivated by a problem of population genetics. They are related to harmonic functions and coherent systems of probability measures on a certain branching graph. The vertices of this graph are multiple Young diagrams (or multiple partitions), and the edges depend on the Jack parameter. Our main result establishes a bijective correspondence between the set of harmonic functions on the graph and probability measures on the generalized Thoma set. The correspondence is determined by a canonical integral representation of harmonic functions. As a consequence we obtain a representation theorem for multiple partition structures. We give an example of a multiple partition structure which is expected to be relevant for a model of population genetics for the genetic variation of a sample of gametes from a large population. Namely, we construct a probability measure on the wreath product of a finite group with the symmetric group. The constructed probability measure defines a multiple partition structure which is a generalization of the Ewens partition structure studied by Kingman. We show that this multiple partition structure can be represented in terms of a multiple analogue of the Poisson-Dirichlet distribution called the multiple Poisson-Dirichlet distribution in the paper.

math.PR↗

arXiv2024

Generalized regular representations of big wreath products

Let $G$ be a finite group with $k$ conjugacy classes, and $S(\infty)$ be the infinite symmetric group, i.e. the group of finite permutations of $\left\{1,2,3,\ldots\right\}$. Then the wreath product $G_{\infty}=G\sim S(\infty)$ of $G$ with $S(\infty)$ (called the big wreath product) can be defined. The group $G_{\infty}$ is a generalization of the infinite symmetric group, and it is an example of a ``big'' group, in Vershik's terminology. For such groups the two-sided regular representations are irreducible, the conventional scheme of harmonic analysis is not applicable, and the problem of harmonic analysis is a nontrivial problem with connections to different areas of mathematics and mathematical physics. Harmonic analysis on the infinite symmetric group was developed in the works by Kerov, Olshanski, and Vershik, and Borodin and Olshanski. The goal of this paper is to extend this theory to the case of $G_{\infty}$. In particular, we construct an analogue $\mathfrak{S}_{G}$ of the space of virtual permutations. We then formulate and prove a theorem characterizing all central probability measures on $\mathfrak{S}_{G}$, and introduce generalized regular representations $T_{z_1,\ldots,z_k}$ of the big wreath product $G_{\infty}$. The paper solves a natural problem of harmonic analysis for the big wreath products: our results describe the decomposition of $T_{z_1,\ldots,z_k}$ into irreducible components.

math.RT↗

arXiv2021

Gap probability for products of random matrices in the critical regime

The singular values of a product of $M$ independent Ginibre matrices of size $N\times N$ form a determinantal point process. Near the soft edge, as both $M$ and $N$ go to infinity in such a way that $M/N\to α$, $α>0$, a scaling limit emerges. We consider a gap probability for the corresponding limiting determinantal process, namely, the probability that there are no particles in the interval $(a,+\infty)$. We derive a Tracy-Widom-like formula in terms of the unique solution of a certain matrix Riemann-Hilbert problem of size $2 \times 2$. The right-tail asymptotics for this solution is obtained by the Deift-Zhou non-linear steepest descent analysis.

math.PR↗

arXiv2021

Product Matrix Processes with Symplectic and Orthogonal Invariance via Symmetric Functions

We apply symmetric function theory to study random processes formed by singular values of products of truncations of Haar distributed symplectic and orthogonal matrices. These product matrix processes are degenerations of Macdonald processes introduced by Borodin and Corwin. Through this connection, we obtain explicit formulae for the distribution of singular values of a deterministic matrix multiplied by a truncated Haar orthogonal or symplectic matrix under conditions where the latter factor acts as a rank $1$ perturbation. Consequently, we generalize the recent Kieburg-Kuijlaars-Stivigny formula for the joint singular value density of a product of truncated unitary matrices to symplectic and orthogonal symmetry classes. Specializing to products of two symplectic matrices with a rank $1$ perturbative factor, we show that the squared singular values form a Pfaffian point process.

math-ph↗

arXiv2020

Averages of products and ratios of characteristic polynomials in polynomial ensembles

Polynomial ensembles are a sub-class of probability measures within determinantal point processes. Examples include products of independent random matrices, with applications to Lyapunov exponents, and random matrices with an external field, that may serve as schematic models of quantum field theories with temperature. We first analyse expectation values of ratios of an equal number of characteristic polynomials in general polynomial ensembles. Using Schur polynomials we show that polynomial ensembles constitute Giambelli compatible point processes, leading to a determinant formula for such ratios as in classical ensembles of random matrices. In the second part we introduce invertible polynomial ensembles given e.g. by random matrices with an external field. Expectation values of arbitrary ratios of characteristic polynomials are expressed in terms of multiple contour integrals. This generalises previous findings by one of the authors for a single ratio in the context of eigenvector statistics in the complex Ginibre ensemble.

math-ph↗

arXiv2018

Product matrix processes as limits of random plane partitions

We consider a random process with discrete time formed by singular values of products of truncations of Haar distributed unitary matrices. We show that this process can be understood as a scaling limit of the Schur process, which gives determinantal formulas for (dynamical) correlation functions and a contour integral representation for the correlation kernel. The relation with the Schur processes implies that the continuous limit of marginals for q-distributed plane partitions coincides with the joint law of singular values for products of truncations of Haar-distributed random unitary matrices. We provide structural reasons for this coincidence that may also extend to other classes of random matrices.

math-ph↗

arXiv2018

Product matrix processes for coupled multi-matrix models and their hard edge scaling limits

Product matrix processes are multi-level point processes formed by the singular values of random matrix products. In this paper we study such processes where the products of up to $m$ complex random matrices are no longer independent, by introducing a coupling term and potentials for each product. We show that such a process still forms a multi-level determinantal point processes, and give formulae for the relevant correlation functions in terms of the corresponding kernels. For a special choice of potential, leading to a Gaussian coupling between the $m$th matrix and the product of all previous $m-1$ matrices, we derive a contour integral representation for the correlation kernels suitable for an asymptotic analysis of large matrix size $n$. Here, the correlations between the first $m-1$ levels equal that of the product of $m-1$ independent matrices, whereas all correlations with the $m$th level are modified. In the hard edge scaling limit at the origin of the spectra of all products we find three different asymptotic regimes. The first regime corresponding to weak coupling agrees with the multi-level process for the product of $m$ independent complex Gaussian matrices for all levels, including the $m$-th. This process was introduced by one of the authors and can be understood as a multi-level extension of the Meijer $G$-kernel introduced by Kuijlaars and Zhang. In the second asymptotic regime at strong coupling the point process on level $m$ collapses onto level $m-1$, thus leading to the process of $m-1$ independent matrices. Finally, in an intermediate regime where the coupling is proportional to $n^{\frac12}$, we obtain a family of parameter dependent kernels, interpolating between the limiting processes in the weak and strong coupling regime. These findings generalise previous results of the authors and their coworkers for $m=2$.

math-ph↗

arXiv2018

On characteristic polynomials for a generalized chiral random matrix ensemble with a source

We evaluate averages involving characteristic polynomials, inverse characteristic polynomials and ratios of characteristic polynomials for a $N\times N$ random matrix taken from a $L$-deformed Chiral Gaussian Unitary Ensemble with an external source $Ω$. Relation to a recently studied statistics of bi-orthogonal eigenvectors in the complex Ginibre ensemble, see Y.V. Fyodorov arXiv:1710.04699, is briefly discussed as a motivation to study asymptotics of these objects in the case of external source proportional to the identity matrix. In particular, for an associated 'complex bulk/chiral edge' scaling regime we retrieve the kernel related to Bessel/Macdonald functions.

math-ph↗

arXiv2018

Finite rank perturbations in products of coupled random matrices: From one correlated to two Wishart ensembles

We compare finite rank perturbations of the following three ensembles of complex rectangular random matrices: First, a generalised Wishart ensemble with one random and two fixed correlation matrices introduced by Borodin and Péché, second, the product of two independent random matrices where one has correlated entries, and third, the case when the two random matrices become also coupled through a fixed matrix. The singular value statistics of all three ensembles is shown to be determinantal and we derive double contour integral representations for their respective kernels. Three different kernels are found in the limit of infinite matrix dimension at the origin of the spectrum. They depend on finite rank perturbations of the correlation and coupling matrices and are shown to be integrable. The first kernel (I) is found for two independent matrices from the second, and two weakly coupled matrices from the third ensemble. It generalises the Meijer $G$-kernel for two independent and uncorrelated matrices. The third kernel (III) is obtained for the generalised Wishart ensemble and for two strongly coupled matrices. It further generalises the perturbed Bessel kernel of Desrosiers and Forrester. Finally, kernel (II), found for the ensemble of two coupled matrices, provides an interpolation between the kernels (I) and (III), generalising previous findings of part of the authors.

math-ph↗

arXiv2016

Hard edge limit of the product of two strongly coupled random matrices

We investigate the hard edge scaling limit of the ensemble defined by the squared singular values of the product of two coupled complex random matrices. When taking the coupling parameter to be dependent on the size of the product matrix, in a certain double scaling regime at the origin the two matrices become strongly coupled and we obtain a new hard edge limiting kernel. It interpolates between the classical Bessel-kernel describing the hard edge scaling limit of the Laguerre ensemble of a single matrix, and the Meijer G-kernel of Kuijlaars and Zhang describing the hard edge scaling limit for the product of two independent Gaussian complex matrices. It differs from the interpolating kernel of Borodin to which we compare as well.

math-ph↗

arXiv2015

Dynamical correlation functions for products of random matrices

We introduce and study a family of random processes with a discrete time related to products of random matrices. Such processes are formed by singular values of random matrix products, and the number of factors in a random matrix product plays a role of a discrete time. We consider in detail the case when the (squared) singular values of the initial random matrix form a polynomial ensemble, and the initial random matrix is multiplied by standard complex Gaussian matrices. In this case we show that the random process is a discrete-time determinantal point process. For three special cases (the case when the initial random matrix is a standard complex Gaussian matrix, the case when it is a truncated unitary matrix, or the case when it is a standard complex Gaussian matrix with a source) we compute the dynamical correlations functions explicitly, and find the hard edge scaling limits of the correlation kernels. The proofs rely on the Eynard-Mehta theorem, and on contour integral representations for the correlation kernels suitable for an asymptotic analysis.

math-ph↗

arXiv2015

Dropping the independence: singular values for products of two coupled random matrices

We study the singular values of the product of two coupled rectangular random matrices as a determinantal point process. Each of the two factors is given by a parameter dependent linear combination of two independent, complex Gaussian random matrices, which is equivalent to a coupling of the two factors via an Itzykson-Zuber term. We prove that the squared singular values of such a product form a biorthogonal ensemble and establish its exact solvability. The parameter dependence allows us to interpolate between the singular value statistics of the Laguerre ensemble and that of the product of two independent complex Ginibre ensembles which are both known. We give exact formulae for the correlation kernel in terms of a complex double contour integral, suitable for the subsequent asymptotic analysis. In particular, we derive a Christoffel-Darboux type formula for the correlation kernel, based on a five term recurrence relation for our biorthogonal functions. It enables us to find its scaling limit at the origin representing a hard edge. The resulting limiting kernel coincides with the universal Meijer G-kernel found by several authors in different ensembles. We show that the central limit theorem holds for the linear statistics of the singular values and give the limiting variance explicitly.

math-ph↗

arXiv2014

Differential equations for singular values of products of Ginibre random matrices

It was proved by Akemann, Ipsen and Kieburg that squared singular values of products of $M$ complex Ginibre random matrices form a determinantal point process whose correlation kernel is expressible in terms of Meijer's $G$-functions. Kuijlaars and Zhang recently showed that at the edge of the spectrum, this correlation kernel has a remarkable scaling limit $K_M(x,y)$ which can be understood as a generalization of the classical Bessel kernel of Random Matrix Theory. In this paper we investigate the Fredholm determinant of the operator with the kernel $K_M(x,y)χ_J(y)$, where $J$ is a disjoint union of intervals, $J=\cup_j(a_{2j-1},a_{2j})$, and $χ_J$ is the characteristic function of the set $J$. This Fredholm determinant is equal to the probability that $J$ contains no particles of the limiting determinantal point process defined by $K_M(x,y)$ (the gap probability). We derive Hamiltonian differential equations associated with the corresponding Fredholm determinant, and relate them with the monodromy preserving deformation equations of the Jimbo, Miwa, Mori, Ueno and Sato theory. In the special case $J=(0,s)$ we give a formula for the gap probability in terms of a solution of a system of non-linear ordinary differential equations.

math-ph↗

arXiv2012

Hole probabilities and overcrowding estimates for products of complex Gaussian matrices

We consider eigenvalues of a product of n non-Hermitian, independent random matrices. Each matrix in this product is of size N\times N with independent standard complex Gaussian variables. The eigenvalues of such a product form a determinantal point process on the complex plane (Akemann and Burda J. Phys A: Math. Theor. 45 (2012) 465201), which can be understood as a generalization of the finite Ginibre ensemble. As N\rightarrow\infty, a generalized infinite Ginibre ensemble arises. We show that the set of absolute values of the points of this determinantal process has the same distribution as {R_1^{(n)},R_2^{(n)},...}, where R_k^{(n)} are independent, and (R_k^{(n)})^2 is distributed as the product of n independent Gamma variables Gamma(k,1). This enables us to find the asymptotics for the hole probabilities, i.e. for the probabilities of the events that there are no points of the process in a disc of radius r with its center at 0, as r\rightarrow\infty. In addition, we solve the relevant overcrowding problem: we derive an asymptotic formula for the probability that there are more than m points of the process in a fixed disk of radius r with its center at 0, as m\rightarrow\infty.

math-ph↗

arXiv2012

Representation theory of the infinite symmetric group and Pfaffian point processes

We construct a family of Pfaffian point processes relevant for the harmonic analysis on the infinite symmetric group. The correlation functions of these processes are representable as Pfaffians with matrix valued kernels. We give explicit formulae for the matrix valued kernels in terms of the classical Whittaker functions. The obtained formulae have the same structure as that arising in the study of symplectic ensembles of Random Matrix Theory. The paper is an extended version of the author's talk at Fall 2010 MSRI Random Matrix Theory program.

math.RT↗