Criticality of self-avoiding walks with an excluded infinite needle

Abstract

The authors study the effect of repulsion for self-avoiding walks and random walks from excluded sets. They show, in particular, that the mean displacement away from an excluded infinite needle of self-avoiding random walks in three dimensions has to diverge along the privileged axis as Nsigma , where N is the number of steps and sigma is a sub-leading critical exponent for the two-point function. This exponent has been determined by using a high-precision Monte Carlo simulation ( sigma =0.370+or-0.011). Its knowledge is used to improve the measure of universal quantities, like the exponent nu ( nu =0.5867+or-0.0025, in agreement with the in -expansion estimate and with experimental data) and amplitude ratios. They verify also that for simple random walks the excluded needle introduces instead logarithmic violations to scaling.