Conduction and connection properties of self-avoiding walks with bridges.

Abstract

The voltage drop distribution in the random-resistor networks constituted by all nearest-neighbor bonds connecting points visited by a lattice self-avoiding walk (SAW) is studied by accurate numerical techniques in d=2. An analysis of the moments of the distribution allows one to establish the value \ensuremath{\zeta}=1.333\ifmmode\pm\else\textpm\fi{}0.007 for the resistance exponent and to exclude the possibility of multifractal behavior. These results are also consistent with a topological investigation of the connection properties of SAW, which independently yields \ensuremath{\zeta}=1.33\ifmmode\pm\else\textpm\fi{}0.01. This clearly supports \ensuremath{\zeta}=(4/3 and a spectral dimension d\ifmmode \tilde{}\else \~{}\fi{}=1 for these walks, solving an old controversy. A possible extension of such an analysis to SAW at the FTHETA point is also discussed.