The nature of most probable paths at finite temperatures

TL;DR

This paper investigates the most probable path lengths at finite temperatures on a hypercubic lattice, revealing a phase transition from directed to random walk behavior at a critical temperature, with implications for path probability structures.

Contribution

It introduces a detailed analysis of the transition in path behavior at finite temperatures and derives the critical temperature for this phase change on hypercubic lattices.

Findings

Identifies a critical temperature T_c = 1/( 2 +  D) for the transition.

Shows a crossover in path-length behavior depending on end-to-end distance and temperature.

Finds a maximum end-to-end distance beyond which most probable paths do not exist.

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.