Statistics of self-avoiding walks on random lattices

Abstract

Self-avoiding-walk (SAW) statistics on randomly dilute lattices is reviewed. The phase diagrams, giving the variations of the SAW connectivity constant and of the tricritical or θ-point, with lattice dilution, are obtained by employing various numerical methods. The numerical estimates are compared with those obtained using various analytical and mean field-like estimates of the phase diagram. The critical behaviour, given especially by the SAW size exponents in the high temperature limit and at the θ-point, are studied on dilute lattices. Specifically, the size exponents on the percolating fractals are investigated and discussed. Fractal effects on the statistics of SAWs are identified, and the numerical values obtained for the size exponents are compared with various analytic (renormalisation group etc.) and Flory-like (mean field) estimates.