Infinite-range mean-field percolation: Transfer matrix study of longitudinal correlation length

Two infinite-range directed percolation models, equivalent also to epidemic models, are considered for a finite population (finite number of sites)N at each time (directed axis) step. The general features of the transfer matrix spectrum (evolution operator spectrum for the epidemic) are studied numerically, and compared with analytical predictions in the limitN = . One of the models is devised to allow numerical results to be obtained forN as high as nearly 800 for the largest longitudinal percolation correlation length (relaxation time for epidemic). The finite-N behavior of this correlation length is studied in detail, including scaling near the percolation transition, exponential divergence (withN) above the percolation transition, as well as other noncritical and critical-point properties.