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O. Khorunzhiy

Publications and source records attributed to O. Khorunzhiy.

15 recordsLinked to original sources
arXiv2026

Limit theorems for walks and triangles on Erdös-Rényi random graphs with large interaction radius

We study cumulants of numbers of $q$-step walks on Erd\H os-Rényi random graphs with distance-dependent edge probability in the limit when the number of vertices $N$, concentration $c$, and interaction radius $R$ tend to infinity. These cumulants can be associated with a formal cumulant expansion of the free energy of matrix models of exponential random graphs widely known in mathematical and theoretical physics. We show that in three different asymptotic regimes, the limiting values of $k$-th cumulants ${\cal F}_k^{(q)}$ exist and can be associated with one or another family of tree-type diagrams, in dependence on the asymptotic behavior of parameters $cR/N$ for $q$-step non-closed walks and $c^2R/N^2$ for 3-step closed walks, respectively. In some cases, we obtain explicit expressions for ${\cal F}_k^{(q)}$ with the help of a modified Prüfer codification. These results allow us to prove Limit Theorems for the number of non-closed walks and for the number of triangles in corresponding ensembles of large random graphs. We indicate an asymptotic threshold that separates the normal probability distribution and the Poisson one for the number of triangles in random graphs. We show that in the random graph ensemble that we consider the average vertex degree can be bounded from above while the total number of triangles infinitely increases, thus rigorously solving a graph collapse problem known in applications.

math.PR↗
arXiv2024

Enumeration of tree-type diagrams assembled from oriented chains of edges

We study a family of tree-type diagrams that arise in studies of the cumulant expansion in discrete Erd\H os-Rényi random matrix models. Using a version of the Pr\" ufer code, we obtain an explicit expression for the number of tree-type diagrams assembled from $k$ oriented chains of $q$ edges. Using this modified Prüfer codification, we get an explicit expression for sum overs weighted tree-type diagrams with a weight depending on multiplicity of edges. We describe similar results for tree-type diagrams assembled from chains that are not necessarily regular.

math.CO↗
arXiv2024

On asymptotic properties of high moments of compound Poisson distribution

We study asymptotic behavior of the moments $M_k(λ)$ of the sum $X_1+\dots+X_{N_λ}$, where $N_λ$ follows the Poisson probability distribution with mean value $λ$ and $\{X_j\}$ is a family of i.i.d. random variables also independent from $N_λ$. We obtain an explicit expression for the leading term of $M_k(λ)$ as $k\to\infty$ and study it in dependence of the asymptotic behavior of $λ= λ_k$. In application, we establish a concentration property of maximal vertex degree of large weighted random graphs. Another application is related with a variable that arises in the studies of high moments of large random matrices. Finally, regarding three particular cases of probability distribution of $X_j$, we comment on the asymptotic behavior of certain combinatorial polynomials, including the Bell polynomials of even partitions.

math.PR↗
arXiv2020

On asymptotic behavior of Bell polynomials and concentration of vertex degree of large random graphs

We study concentration properties of vertex degrees of $n$-dimensional Erdos-Rényi random graphs with the edge probability $ρ/n$ by means of high moments of these random variables in the limit when $n$ and $ρ$ tend to infinity. These moments are asymptotically close to one-variable Bell polynomials ${\cal B}_k(ρ), k\in {\bf N}$ that represent moments of the Poisson probability distribution ${\cal P}(ρ)$. We study asymptotic behavior of the Bell polynomials and modified Bell polynomials for large values of $k$ and $ρ$ with the help of the local limit theorem for auxiliary random variables. Using the results obtained, we get the upper bounds for the deviation probabilities of the normalized maximal vertex degree of the Erdos-Rényi random graphs in the limit $n,ρ\to\infty$ such that the ratio $ρ/\log n $ remains finite or infinitely increases.

math.PR↗
arXiv2016

On eigenvalue distribution of random matrices of Ihara zeta function of large random graphs

We consider the ensemble of real symmetric random matrices $H^{(n,ρ)}$ obtained from the determinant form of the Ihara zeta function of random graphs that have $n$ vertices with the edge probability $ρ/n$. We prove that the normalized eigenvalue counting function of $H^{(n,ρ)}$ weakly converges in average as $n,ρ\to\infty$ and $ρ=o(n^α)$ for any $α>0$ to a shift of the Wigner semi-circle distribution. Our results support a conjecture that the large Erdos-Rényi random graphs satisfy in average the weak graph theory Riemann Hypothesis.

math-ph↗
arXiv2015

On high moments of strongly diluted large Wigner random matrices

We consider a dilute version of the Wigner ensemble of nxn random matrices $H$ and study the asymptotic behavior of their moments $M_{2s}$ in the limit of infinite $n$, $s$ and $ρ$, where $ρ$ is the dilution parameter. We show that in the asymptotic regime of the strong dilution, the moments $M_{2s}$ with $s=χρ$ depend on the second and the fourth moments of the random entries $H_{ij}$ and do not depend on other even moments of $H_{ij}$. This fact can be regarded as an evidence of a new type of the universal behavior of the local eigenvalue distribution of strongly dilute random matrices at the border of the limiting spectrum. As a by-product of the proof, we describe a new kind of Catalan-type numbers related with the tree-type walks.

math-ph↗
arXiv2011

High Moments of Large Wigner Random Matrices and Asymptotic Properties of the Spectral Norm

We consider an ensemble of nxn real symmetric random matrices A whose entries are determined by independent identically distributed random variables that have symmetric probability distribution. Assuming that the moment 12+2delta of these random variables exists, we prove that the probability distribution of the spectral norm of A rescaled to n^{-2/3} is bounded by a universal expression. The proof is based on the completed and modified version of the approach proposed and developed by Ya. Sinai and A. Soshnikov to study high moments of Wigner random matrices.

math.PR↗
arXiv2010

A class of even walks and divergence of high moments of large Wigner random matrices

We study high moments of truncated Wigner nxn random matrices by using their representation as the sums over the set W of weighted even closed walks. We construct the subset W' of W such that the corresponding sum diverges in the limit of large n and the number of moments proportional to n^{2/3} for any truncation of the order n^{1/6+epsilon}, epsilon>0 provided the probability distribution of the matrix elements is such that its twelfth moment does not exist. This allows us to put forward a hypothesis that the finiteness of the twelfth moment represents the necessary condition for the universal upper bound of the high moments of large Wigner random matrices.

math.PR↗
arXiv2009

Uniform bounds for exponential moment of maximum of a Dyck paths

Let D be a Dyck path chosen uniformly from the set of Dyck paths with 2n steps. We prove that the sequence of the exponential moments of the maximum of D normalized by the square root of n converges in the limit of infinite n, and therefore is bounded uniformly in n. This result justifies corresponding assumption used to prove certain estimates of high moments of large random matrices.

math.PR↗
arXiv2009

Asymptotic properties of resolvents of large dilute Wigner matrices

We study the spectral properties of the dilute Wigner random real symmetric n-dimensional matrices H such that the entries H(i,j) take zero value with probability 1-p/n. We prove that under rather general conditions on the probability distribution of H(i,j) the semicircle law is valid for the dilute Wigner ensemble in the limit of infinite n and p. In the second part of the paper we study the leading term of the correlation function of the resolvent G(z) of H with large enough Im z in the limit of infinite n and p such that 3/5 log n <log p < log n. We show that this leading term, when considered on the local spectral scale, converges to the same limit as that of the resolvent correlation function of the Wigner ensemble of random matrices. This shows that the moderate dilution of the Wigner ensemble does not alter its universality class.

math-ph↗
arXiv2008

Even Walks and Estimates of High Moments of Large Wigner Random Matrices

We revisit the problem of estimates of moments of random n-dimensional matrices of Wigner ensemble by using the approach elaborated by Ya. Sinai and A. Soshnikov and further developed by A. Ruzmaikina. Our main subject is given by the structure of closed even walks and their graphs that arise in these studies. We show that the total degree of a vertex of such a graph depends not only on the self-intersections degree of but also on the total number of all non-closed instants of self-intersections of the walk. This result is used to fill the gaps of earlier considerations.

math-ph↗
arXiv2007

On Connected Diagrams and Cumulants of Erdos-Renyi Matrix Models

Regarding the adjacency matrices of n-vertex graphs and related graph Laplacian, we introduce two families of discrete matrix models constructed both with the help of the Erdos-Renyi ensemble of random graphs. Corresponding matrix sums represent the characteristic functions of the average number of walks and closed walks over the random graph. These sums can be considered as discrete analogs of the matrix integrals of random matrix theory. We study the diagram structure of the cumulant expansions of logarithms of these matrix sums and analyze the limiting expressions in the cases of constant and vanishing edge probabilities as n tends to infinity.

math-ph↗
arXiv2006

Estimates for moments of random matrices with Gaussian elements

We describe an elementary method to get non-asymptotic estimates for the moments of Hermitian random matrices whose elements are Gaussian independent random variables. As the basic example, we consider the GUE matrices. Immediate applications include GOE and the ensemble of Gaussian anti-symmetric Hermitian matrices. The estimates we derive give asymptotically exact expressions for the first terms of 1/N-expansions of the moments and covariance terms. We apply our method to the ensemble of Gaussian Hermitian random band matrices whose elements are zero outside of the band of width b. The estimates we obtain show that the spectral norm of these matrices remains bounded in the limit of infinite N when b is much greater than log N to the power 3/2.

math-ph↗
arXiv2002

Limit Theorem for Sums of Random Products

We study a generalization of the Random Energy Model to the case when the number of exponential factors varies at random. Also a relation between REM and the Erd"os-R'enyi limit theorem for maximums of partial sums is considered.

math.PR↗