Abstract
The shortest paths of self-avoiding walks (SAWs) with bridge lengths b=1, 2, 3 and 2 are studied by exact series expansions and by Monte Carlo simulation methods on the square and simple cubic lattices. In the series work, SAW configurations of up to 24 and 14 steps are generated on the square and simple cubic lattices respectively. Assuming the shortest path SN between two sites, which are separated by N steps along the chain, has the scaling form SN approximately AN+BN1- Delta , it is found that Delta approximately=1 and Delta approximately=1/2 on the square and simple cubic lattices respectively, independent of the bridge length. The problem is also investigated by the Mellin-Pade approximation method using Monte Carlo data on the square lattice. The latter method gives results consistent with those of the series expansion study. The authors' results are not consistent with earlier predictions.
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