Most probable paths in homogeneous and disordered lattices at finite temperature

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 ∼ r²) 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.