Most probable paths in homogeneous and disordered lattices at finite temperature

Abstract

We determine the geometrical properties of the most probable paths at finite temperatures T, between two points separated by a distance r, in one-dimensional lattices with positive energies of interaction ε i associated with bond i. The most probable path-length t mp in a homogeneous medium ( ε i = ε, for all i) is found to undergo a phase transition, from an optimal-like form ( t mp ∼ r) at low temperatures to a random walk form ( t mp ∼ r 2) near the critical temperature T c=ε/ ln 2 . At T> T c the most probable path-length diverges, discontinuously, for all finite endpoint separations greater than a particular value r ∗(T) . In disordered lattices, with ε i homogeneously distributed between ε− δ/2 and ε+ δ/2, the random walk phase is absent, but a phase transition to diverging t mp still takes place. Different disorder configurations have different transition points. A way to characterize the whole ensemble of disorder, for a given distribution, is suggested.