Surface roughening with quenched disorder in high dimensions: Exact results for the Cayley tree

Discrete models describing pinning of a growing self-affine interface due to geometrical hindrances can be mapped to the diode-resistor percolation problem in all dimensions. We present the solution of this percolation problem on the Cayley tree. We find that the order parameter P varies near the critical point pc as exp(-A/ pc-p), where p is the fraction of bonds occupied by diodes. This result suggests that the critical exponent βp of P diverges for d→, and that there is no finite upper critical dimension. The exponent ν characterizing the parallel correlation length changes its value from ν=3/4 below pc to ν=1/4 above pc. Other critical exponents of the diode-resistor problem on the Cayley tree are γ=0 and ν=0, suggesting that ν/ν→0 when d→. Simulation results in finite dimensions 2≤d≤5 are also presented.