Geometric properties of optimal and most probable paths on randomly disordered lattices

Abstract

This chapter describes the geometry of optimal paths and most probable paths, expressed as the relation between the length and the end-to-end distance, on lattices with randomly disordered bond energies and the methods of determining them. The shortest geometrical path between two points in a given space is the path along which the distance between them is minimum; it is the optimal path between the two points when the weight of the path is defined as its length. Optimal paths are defined at zero temperature as the paths of minimum energy and most probable paths are defined at finite temperatures as the paths of minimum free energy. Optimal paths are always self avoiding. While directed optimal paths are always self-affine, irrespective of the strength of the disorder, non-directed optimal paths are self-affine only when the bond energies are weakly disordered and these have self-similar forms when the bond energies are strongly disordered. The most probable paths are the paths of minimum free energy-the free energy of a path is considered as its weight. Most probable paths undergo a phase transition at a finite temperature: the paths conform to the condition of minimum energy in the low temperature phase and to the condition of maximum entropy in the high temperature phase. For the most probable directed paths the phase transition occurs in 3 + 1 and higher dimensions whereas for the most probable non-directed paths the phase transition occurs in all dimensions. . The chapter ends with a discussion on the classic optimization problem of the shortest path of a travelling salesman through a set of randomly chosen sites on a lattice.