Finite size analysis of eigenvalue spectrum for random walks on a critical percolation cluster in four dimensions

Abstract

We study by Markov chain analysis the random walks on a critical percolation cluster embedded in a four-dimensional hypercubic lattice. We calculate the number of dominant eigenvalues of the transition probability matrix and estimate the spectral and fractal dimensions ds and dw of random walks from the eigenvalues and their distribution. The estimates of ds and dw obtained from the data for a given size S of the percolation cluster exhibit some S dependence. Extrapolating the results to S limit, we obtain ds = 1.330±0.010 close to the previous result by other methods and a new result dw = 4.50±0.15. These values are also confirmed by direct Monte Carlo simulations of random walks on a percolation cluster.