The nature of most probable paths at finite temperatures

Abstract

We determine the most probable length of paths at finite temperatures, with a preassigned end-to-end distance and a unit of energy assigned to every step on a $D$-dimensional hypercubic lattice. The asymptotic form of the most probable path-length shows a transition from the directed walk nature at low temperatures to the random walk nature as the temperature is raised to a critical value $T_c$. We find $T_c=1/(\ln 2+\ln D)$. Below $T_c$ the most probable path-length shows a crossover from the random walk nature for small end-to-end distance to the directed walk nature for large end-to-end distance; the crossover length diverges as the temperature approaches $T_c$. For every temperature above $T_c$ we find that there is a maximum end-to-end distance beyond which a most probable path-length does not exist.

arXiv comment4 pages (REVTeX); Eq.7 simplified; typing error in Eq.12 corrected; to appear in Physica Scripta