arXiv subjects

Zdzisław Burda

Publications and source records attributed to Zdzisław Burda.

7 recordsLinked to original sources

arXiv2026

Heider balance in square versus triangular lattices under an external field

We discuss the Heider model in the presence of an external social field. This field breaks the possible symmetry between the probabilities of hostile and friendly relationships. We consider the system in the presence of fluctuations generated by thermal noise and present the results of a comparative study of two-dimensional triangular and square networks with periodic boundary conditions. The results were obtained using three different methods: exact calculations for small systems ($3 \times 3$ square lattice and $2 \times 2$ triangular lattice), Monte Carlo simulations of medium-sized systems ($16 \times 16$ lattices), and exact calculations in the thermodynamic limit (corresponding to infinite size) of certain limiting cases for which analytical solutions are possible. In particular, we exploit the recently discovered equivalence between structurally balanced systems and the Ising model to derive an exact form of the edge magnetization susceptibility for systems in Heider equilibrium.

cond-mat.stat-mech↗

arXiv2026

Relationship between Heider links and Ising spins

We show that the Heider model with an external field is equivalent, in the limit of structural balance, to the Ising model with nearest-neighbor interactions without an external field. More precisely, we claim that the signs of the Heider relations that maintain structural equilibrium in the system can be represented as nearest neighbor Ising spin products. We demonstrate this explicitly for a complete graph and provide a general argument for an arbitrary graph. A consequence of the equivalence is that the system of balanced Heider states undergoes a phase transition, inherited from the Ising model, at a critical value of the social field at which the fluctuations of edge magnetization are maximal.

cond-mat.stat-mech↗

arXiv2017

Eigenvector statistics of the product of Ginibre matrices

We develop a method to calculate left-right eigenvector correlations of the product of $m$ independent $N\times N$ complex Ginibre matrices. For illustration, we present explicit analytical results for the vector overlap for a couple of examples for small $m$ and $N$. We conjecture that the integrated overlap between left and right eigenvectors is given by the formula $O = 1 + (m/2)(N-1)$ and support this conjecture by analytical and numerical calculations. We derive an analytical expression for the limiting correlation density as $N\rightarrow \infty$ for the product of Ginibre matrices as well as for the product of elliptic matrices. In the latter case, we find that the correlation function is independent of the eccentricities of the elliptic laws.

cond-mat.stat-mech↗

arXiv2015

Invariant sums of random matrices and the onset of level repulsion

We compute analytically the joint probability density of eigenvalues and the level spacing statistics for an ensemble of random matrices with interesting features. It is invariant under the standard symmetry groups (orthogonal and unitary) and yet the interaction between eigenvalues is not Vandermondian. The ensemble contains real symmetric or complex hermitian matrices $\mathbf{S}$ of the form $\mathbf{S}=\sum_{i=1}^M \langle \mathbf{O}_i \mathbf{D}_i\mathbf{O}_i^{\mathrm{T}}\rangle$ or $\mathbf{S}=\sum_{i=1}^M \langle \mathbf{U}_i \mathbf{D}_i\mathbf{U}_i^\dagger\rangle$ respectively. The diagonal matrices $\mathbf{D}_i=\mathrm{diag}\{λ_1^{(i)},\ldots,λ_N^{(i)}\}$ are constructed from real eigenvalues drawn \emph{independently} from distributions $p^{(i)}(x)$, while the matrices $\mathbf{O}_i$ and $\mathbf{U}_i$ are all orthogonal or unitary. The average $\langle\cdot\rangle$ is simultaneously performed over the symmetry group and the joint distribution of $\{λ_j^{(i)}\}$. We focus on the limits i.) $N\to\infty$ and ii.) $M\to\infty$, with $N=2$. In the limit i.), the resulting sum $\mathbf{S}$ develops level repulsion even though the original matrices do not feature it, and classical RMT universality is restored asymptotically. In the limit ii.) the spacing distribution attains scaling forms that are computed exactly: for the orthogonal case, we recover the $β=1$ Wigner's surmise, while for the unitary case an entirely new universal distribution is obtained. Our results allow to probe analytically the microscopic statistics of the sum of random matrices that become asymptotically free. We also give an interpretation of this model in terms of radial random walks in a matrix space. The analytical results are corroborated by numerical simulations.

cond-mat.stat-mech↗

arXiv2014

Dysonian dynamics of the Ginibre ensemble

We study the time evolution of Ginibre matrices whose elements undergo Brownian motion. The non-Hermitian character of the Ginibre ensemble binds the dynamics of eigenvalues to the evolution of eigenvectors in a non-trivial way, leading to a system of coupled nonlinear equations resembling those for turbulent systems. We formulate a mathematical framework allowing simultaneous description of the flow of eigenvalues and eigenvectors, and we unravel a hidden dynamics as a function of new complex variable, which in the standard description is treated as a regulator only. We solve the evolution equations for large matrices and demonstrate that the non-analytic behavior of the Green's functions is associated with a shock wave stemming from a Burgers-like equation describing correlations of eigenvectors. We conjecture that the hidden dynamics, that we observe for the Ginibre ensemble, is a general feature of non-Hermitian random matrix models and is relevant to related physical applications.

math-ph↗

arXiv2012

Maximal entropy random walk in community finding

The aim of this paper is to check feasibility of using the maximal-entropy random walk in algorithms finding communities in complex networks. A number of such algorithms exploit an ordinary or a biased random walk for this purpose. Their key part is a (dis)similarity matrix, according to which nodes are grouped. This study encompasses the use of the stochastic matrix of a random walk, its mean first-passage time matrix, and a matrix of weighted paths count. We briefly indicate the connection between those quantities and propose substituting the maximal-entropy random walk for the previously chosen models. This unique random walk maximises the entropy of ensembles of paths of given length and endpoints, which results in equiprobability of those paths. We compare performance of the selected algorithms on LFR benchmark graphs. The results show that the change in performance depends very strongly on the particular algorithm, and can lead to slight improvements as well as significant deterioration.

physics.soc-ph↗

arXiv2010

A Random Matrix Approach to VARMA Processes

We apply random matrix theory to derive spectral density of large sample covariance matrices generated by multivariate VMA(q), VAR(q) and VARMA(q1,q2) processes. In particular, we consider a limit where the number of random variables N and the number of consecutive time measurements T are large but the ratio N/T is fixed. In this regime the underlying random matrices are asymptotically equivalent to Free Random Variables (FRV). We apply the FRV calculus to calculate the eigenvalue density of the sample covariance for several VARMA-type processes. We explicitly solve the VARMA(1,1) case and demonstrate a perfect agreement between the analytical result and the spectra obtained by Monte Carlo simulations. The proposed method is purely algebraic and can be easily generalized to q1>1 and q2>1.

q-fin.ST↗