Random walks in disordered systems with long-range transitions. Asymptotically exactly solvable models

F. S. Dzheparov ; V. E. Shestopal

Teoretičeskaâ i matematičeskaâ fizika, Tome 94 (1993) no. 3, pp. 496-514

Résumé

Random-jump models of transport in disordered system are studied. They are described by the master equation $\dot P=-{\mathcal A}\xi P$, where $-{\mathcal A}$ is the generator of the spatially and temporary uniform random walks process on a regular lattice, $\xi$ is the diagonal operator, $\xi _{xy}=\xi _x \delta _{xy}$, where $\{\xi _x\}$ are independent positive random variables with the same distribution. The case is elaborated when ${\mathcal A}_{xy}={\mathcal A}_{yx}={\mathcal A}_{x-y,0}$, the transition rates are determined by multipole-type interactions and $\{\xi _x\}$ have several first negative moments (the random-jump-rate model – with unbounded jumps). Methods of asymptotical expansion of the propagator for small Laplace parameter values and for long times are developed. It is made also by means of functional integral representation. The influence of disorder and of interaction power on the long-time asymptotics is considered. A method of investigation for system with a forced drift along a certain direction is suggested. Some methods of reciprocal transformation of asymptotically–exactly solvable problems are discussed and linkages with other known models are demonstrated. The $l_1$-norm of resolvent is obtained for any Markov process with a countable set of states and $l_1$-bounded generator.