Exact enumeration of self-avoiding walks on lattices with random site energies

Abstract

The self-avoiding random walk on lattices with quenched random site energies is studied using exact enumeration in d=2 and 3. For each configuration we compute the size R and energy E of the minimum-energy self-avoiding walk (SAW). Configuration averages yield the exponents \ensuremath{\nu} and \ensuremath{\chi}, defined by ${\mathit{R}}^{2}$\ifmmode\bar\else\textasciimacron\fi{}\ensuremath{\sim}${\mathit{N}}^{2\ensuremath{\nu}}$ and \ensuremath{\delta}${\mathit{E}}^{2}$\ifmmode\bar\else\textasciimacron\fi{}\ensuremath{\sim}${\mathit{N}}^{2\mathrm{\ensuremath{\chi}}}$. These calculations indicate that \ensuremath{\nu} is significantly larger than its value in the pure system. Finite-temperature studies support the notion that the system is controlled by a zero-temperature fixed point. Consequently, exponents obtained from minimum-energy SAW's characterize the properties of finite temperature SAW's on disordered lattices.