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 εi associated with bond i. The most probable path-length tmp in a homogeneous medium (εi=ε, for all i) is found to undergo a phase transition, from an optimal-like form (tmp∼r) at low temperatures to a random walk form (tmp∼r2) near the critical temperature Tc=ε/ln2. At T>Tc the most probable path-length diverges, discontinuously, for all finite endpoint separations greater than a particular value r∗(T). In disordered lattices, with εi homogeneously distributed between ε−δ/2 and ε+δ/2, the random walk phase is absent, but a phase transition to diverging tmp 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.