Shortest path of SAWs with bridges: series results

Abstract

Recent Monte Carlo simulation results of Yang and Chakrabarti (1990) suggest that the shortest path S_N of an N-stepped self-avoiding walk (SAW), with finite range interactions of bridges, has a finite size scaling behaviour S_N/N approximately=A+N^{- δ} (B+C/N), where the exponent δ is superuniversal; δ approximately=0.19 for all dimensions δ studied (2<or=δ<or=5). The authors report the small-N series enumeration results for S_N for SAWs with nearest-neighbour bridges on square (up to N=18), triangular (up to N=11) and simple cubic (up to N=12) lattices. The estimated values of Delta for different lattices (approximately=0.22+or−0.01 for d=2 and approximately=0.26+or−0.01 for d=3) have been compared with the above Monte Carlo estimate and the indication of a superuniversal behaviour for δ has been discussed.