Resistance and spectral dimension of self-avoiding walks with local bridges

Abstract

Self-avoiding-walk (SAW) configurations are considered by incorporating local bridges, i.e., connecting any two nearest-neighbor sites visited by the SAW by massless cross links. Introducing a new elegant method of estimating the resistance of an arbitrary network, we find the resistance exponent δ=0.920±0.005 in d=2 by enumerating the random samples of SAW’s using a Monte Carlo method. We also find the shortest-connecting-path-length exponent t to be equal to 0.975±0.005 using a simulation technique in two dimension. Random walks on SAW networks with local bridges are studied using a scaling formalism in the ‘‘grand canonical ensemble’’ picture of SAW’s. We fit the mean end-to-end distance 〈$R_t$〉 of random walks of t steps with a scaling form 〈$R_t$〉∼$t^1$/$d_w$F((f-$f_c$)$t^x$) (f being the fugacity of SAW’s) and find $d_w$=$d_F$(1+δ). In two dimensions this predicts the spectral dimension $d_S$ (=2$d_F$/$d_w$) of SAW networks with bridges to be equal to 1.042.