Universal crossing probabilities and incipient spanning clusters in directed percolation

Shape-dependent universal crossing probabilities are studied, via Monte Carlo simulations, for bond and site directed percolation on the square lattice in the diagonal direction, at the percolation threshold. Since the system is strongly anisotropic, the shape-dependence enters through the effective aspect ratio r_eff=cL^z/t, where c is a non-universal constant and z the anisotropy exponent. A particular attention is paid to the influence of the initial state on the universal behaviour of the crossing probability. Using anisotropic finite-size scaling and generalizing a simple argument given by Aizenman for isotropic percolation, we obtain the behaviour of the probability to find n incipient spanning clusters on a finite system at time t. The numerical results are in good agreement with the conjecture.Comment: 8 pages, 12 figures; minor correction

Aperiodic Extended Surface Perturbations in the Ising Model

We study the influence of an aperiodic extended surface perturbation on the surface critical behaviour of the two-dimensional Ising model in the extreme anisotropic limit. The perturbation decays as a power of the distance from the free surface with an oscillating amplitude following some aperiodic sequence. The asymptotic density is 1/2 so that the mean ampltitude vanishes. The relevance of the perturbation is discussed by combining scaling arguments of Cordery and Burkhardt for the Hilhorst-van Leeuwen model and Luck for aperiodic perturbations. The relevance-irrelevance criterion involves the decay exponent of the perturbation, the wandering exponent which governs the fluctuation of the sequence and the bulk correlation length exponent. Analytical results are obtained for the surface magnetization which displays a rich variety of critical behaviours. The results are checked through a numerical finite-size-scaling study. They show that second-order effects must be taken into account in the discussion of the relevance-irrelevance criterion. The scaling behaviours of the first gap and the surface energy are also discussed.Comment: 13 pages, 13 figures, LaTeX2e, EPJ macro

Multiscaling in Ising quantum chains with random Hilhorst-van Leeuwen perturbations

We consider the influence on the surface critical behaviour of a quantum Ising chain of quenched random surface perturbations decaying as a power of the distance from the surface (random Hilhorst-van Leeuwen models). We study, analytically and numerically, the multiscaling behaviour of the surface magnetization and the surface energy density in the case of marginal perturbations.Comment: 14 pages, 5 figures, LaTeX2e, epsf, elsar