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Random walks on random systems: Eigenspectrum of large Markov matrices.

We study the eigenvalue spectrum of the hopping transition probability matrix W for the random walk in randomly diluted clusters, both in critical and weak dilution. We obtain the relations between the spectrum in the limit where the eigenvalue |λ|→1 and the long-time behavior of the velocity autocorrelation function and mean probability of the random walk to return to the starting point after t steps by scaling and by numerical diagonalization of W