Abstract
How does removal of sites by a random walk lead to blockage of percolation? To study this problem of correlated site percolation, we consider a random walk (RW) of steps on a -dimensional hypercubic lattice of size (with periodic boundaries). We systematically explore dependence of the probability of percolation (existence of a spanning cluster) of sites not removed by the RW on and . The concentration of unvisited sites decays exponentially with increasing , while the visited sites are highly correlated—their correlations decaying with the distance as (in ). On increasing , the percolation probability approaches a step function, jumping from 1 to 0 when crosses a percolation threshold that is close to 3 for all . Within numerical accuracy, the correlation length associated with percolation diverges with exponents consistent with . There is no percolation threshold at the lower critical dimension of , with the percolation probability approaching a smooth function .
6 More- Received 6 June 2019
DOI:https://doi.org/10.1103/PhysRevE.100.022125
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