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 ${\mathit{P}}_{\mathrm{\infty }}$ varies near the critical point ${\mathit{p}}_{\mathit{c}}$ as exp(-A/ √${\mathit{p}}_{\mathit{c}}$-p ), where p is the fraction of bonds occupied by diodes. This result suggests that the critical exponent ${\mathrm{\beta }}_{\mathit{p}}$ of ${\mathit{P}}_{\mathrm{\infty }}$ diverges for d→∞, and that there is no finite upper critical dimension. The exponent ${\nu }_{\mathrm{\parallel }}$ characterizing the parallel correlation length changes its value from ${\nu }_{\mathrm{\parallel }}$=3/4 below ${\mathit{p}}_{\mathit{c}}$ to ${\nu }_{\mathrm{\parallel }}$=1/4 above ${\mathit{p}}_{\mathit{c}}$. Other critical exponents of the diode-resistor problem on the Cayley tree are γ=0 and ${\nu }_{\mathrm{\perp }}$=0, suggesting that ${\nu }_{\mathrm{\perp }}$/${\nu }_{\mathrm{\parallel }}$→0 when d→∞. Simulation results in finite dimensions 2≤d≤5 are also presented.