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Iranian Journal of Physics Research

A random hermitian matrix representation for two-dimensional percolation model

Saber, Sina, Saberi, Abbas Ali

Published May 1, 2022• 0 citations
Publisher

Abstract

In this letter, the random matrix theory representation of a bond-percolation model on square lattice is presented. The behavior of random matrix model can be determined only by its two largest eigenvalues. The second largest eigenvalue sits exactly on the edge of semicircle part of eigenvalue’s distribution and its position is a function of p, on the other hand the first largest eigenvalue is disjointed from other eigenvalues and its distribution is Gaussian. Also the first largest eigenvalue is responsible for scaling properties near criticality. Numerical simulations show power-law divergences emerged from coalescence of two largest eigenvalues near critical point at the thermodynamic limit. In this letter a scaling formalism is presented which describes complete scaling behavior of largest eigenvalue’s fluctuations with a set of scaling exponents in finite-size systems.

DOI
10.47176/ijpr.22.1.71279

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In this letter, the random matrix theory representation of a bond-percolation model on square lattice is presented. The behavior of random matrix model can be determined only by its two largest eigenvalues. The second largest eigenvalue sits exactly on the edge of semicircle part of eigenvalue’s distribution and its position is a function of p, on the other hand the first largest eigenvalue is disjointed from other eigenvalues and its distribution is Gaussian. Also the first largest eigenvalue is responsible for scaling properties near criticality. Numerical simulations show power-law divergences emerged from coalescence of two largest eigenvalues near critical point at the thermodynamic limit. In this letter a scaling formalism is presented which describes complete scaling behavior of largest eigenvalue’s fluctuations with a set of scaling exponents in finite-size systems.
We report on the exact treatment of a random-matrix representation of a bond-percolation model on a square lattice in two dimensions with occupation probability p. The percolation problem is mapped onto a random complex matrix composed of two random real-valued matrices of elements +1 and -1 with probability p and 1-p, respectively. We find that the onset of percolation transition can be detected by the emergence of power-law divergences due to the coalescence of the first two extreme eigenvalues in the thermodynamic limit. We develop a universal finite-size scaling law that fully characterizes the scaling behavior of the extreme eigenvalue's fluctuation in terms of a set of universal scaling exponents and amplitudes. We make use of the relative entropy as an index of the disparity between two distributions of the first and second-largest extreme eigenvalues to show that its minimum underlies the scaling framework. Our study may provide an inroad for developing new methods and algorithms with diverse applications in machine learning, complex systems, and statistical physics.
Abstract Complex systems consisting of N agents can be investigated from the aspect of principal fluctuation modes of agents. From the correlations between agents, an N × N correlation matrix C can be obtained. The principal fluctuation modes are defined by the eigenvectors of C. Near the critical point of a complex system, we anticipate that the principal fluctuation modes have the critical behaviors similar to that of the susceptibity. With the Ising model on a two-dimensional square lattice as an example, the critical behaviors of principal fluctuation modes have been studied. The eigenvalues of the first 9 principal fluctuation modes have been invesitigated. Our Monte Carlo data demonstrate that these eigenvalues of the system with size L and the reduced temperature t follow a finite-size scaling form λ n (L, t) = L γ/ν f n (tL 1/ν ), where γ is critical exponent of susceptibility and ν is the critical exponent of the correlation length. Using eigenvalues λ 1 , λ 2 and λ 6 , we get the finite-size scaling form of the second moment correlation length ξ ( L , t ) = L ξ ˜ ( t L 1 / ν ) . It is shown that the second moment correlation length in the two-dimensional square lattice is anisotropic.
We examine the average cluster distribution as a function of lattice probability for a very small (L=6)(L=6)(L=6) lattice and determine the scaling function of three-dimensional percolation. The behavior of the second moment, calculated from the average cluster distribution of L=6L=6L=6 and L=63L=63L=63 lattices, is compared to power-law behavior predicted by the scaling function. We also examine the finite-size scaling of the critical point and the size of the largest cluster at the critical point. This analysis leads to estimates of the critical exponent $\ensuremath{\nu}$ and the ratio of critical exponents $\ensuremath{\beta}/\ensuremath{\nu}$.
A transfer-matrix scaling technique is developed for randomly diluted systems and applied to the site-diluted Ising model on a square lattice. For each connected configuration between adjacent columns, the contribution of the respective transfer matrix to the decay of correlations is considered only as far as the ratio of the two largest eigenvalues, allowing an economical incorporation of configurational averages. Predictions for the phase boundary at and near the percolation and pure Ising limits, and for the correlation exponent \ensuremath{\eta} at those limits, agree with exactly known results to within 1% error with largest strip widths of only L=5. The exponent \ensuremath{\eta} remains near the pure value (1/4) for all intermediate concentrations until it turns over to the percolation value at the threshold.
We theoretically investigate the quantum percolation problem on Lieb lattices in two and three dimensions. We study the statistics of the energy levels through random matrix theory and determine the level spacing distributions, which, with the aid of finite-size scaling theory, allows us to obtain accurate estimates for site- and bond-percolation thresholds and critical exponents. Our numerical investigation supports a localized-delocalized transition at finite threshold, which decreases as the average coordination number increases. The precise determination of the localization length exponent enables us to claim that quantum site- and bond-percolation problems on Lieb lattices belong to the same universality class, with ν decreasing with lattice dimensionality d, similarly to the classical percolation problem. In addition, we verify that, in three dimensions, quantum percolation on Lieb lattices belongs to the same universality class as the Anderson impurity model.
A renormalization-group approach using a scaling transformation in real space is applied to the critical behavior of two-dimensional percolation systems. In various approximations for the triangular-site lattice and the square-bond lattice, the location of the fixed point and the correlation-length exponent are calculated by determining the behavior of the probabilities under a scale transformation. The fixed points for all approximations for the two lattices are found to be in complete agreement with known exact results for the critical percolation probability.