Multiple Markov chain Monte Carlo study of adsorbing self-avoiding walks in two and in three dimensions

Abstract

A self-avoiding walk adsorbing on a line in the square lattice, and on a plane in the cubic lattice, is studied numerically as a model of an adsorbing polymer in dilute solution. The walk is simulated by a multiple Markov chain Monte Carlo implementation of the pivot algorithm for self-avoiding walks. Vertices in the walk that are visits in the adsorbing line or plane are weighted by e β .T he critical value of β, where the walk adsorbs on the adsorbing line or adsorbing plane, is determined by considering energy ratios and approximations to the free energy. We determine that the critical values of β are βc = 0.565 ± 0.010 in the square lattice 0.288 ± 0.020 in the cubic lattice. In addition, the value of the crossover exponent is determined: φ = 0.501 ± 0.015 in the square lattice 0.5005 ± 0.0036 in the cubic lattice. Metric quantities, including the mean square radius of gyration, are also considered, as well as rescaling of the specific heat and free energy, as the critical point is approached.