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 TMF, 1993, Volume 94, Number 3, Pages 496–514 (Mi tmf1439)

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

F. S. Dzheparov, V. E. Shestopal

Institute for Theoretical and Experimental Physics (Russian Federation State Scientific Center)

Abstract: 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.

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English version:
Theoretical and Mathematical Physics, 1993, 94:3, 345–357

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Citation: F. S. Dzheparov, V. E. Shestopal, “Random walks in disordered systems with long-range transitions. Asymptotically exactly solvable models”, TMF, 94:3 (1993), 496–514; Theoret. and Math. Phys., 94:3 (1993), 345–357

Citation in format AMSBIB
\Bibitem{DzhShe93} \by F.~S.~Dzheparov, V.~E.~Shestopal \paper Random walks in disordered systems with long-range transitions. Asymptotically exactly solvable models \jour TMF \yr 1993 \vol 94 \issue 3 \pages 496--514 \mathnet{http://mi.mathnet.ru/tmf1439} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1226226} \transl \jour Theoret. and Math. Phys. \yr 1993 \vol 94 \issue 3 \pages 345--357 \crossref{https://doi.org/10.1007/BF01017267} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=A1993MA71000013} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. V. E. Shestopal, “Multiparameter models for random walks in disordered lattice systems”, Theoret. and Math. Phys., 119:2 (1999), 660–669
2. Dzheparov F., Gul'ko A., Heitjans P., L'vov D., Schirmer A., Shestopal V., Stepanov S., Trostin S., “Spin dynamics and beta-NMR after polarized neutrons capture”, Physica B, 297:1–4 (2001), 288–292
3. F. S. Dzheparov, V. E. Shestopal, “Asymptotically Exactly Solvable Models of Processes in Stochastically Homogeneous Disordered Lattice Media”, Theoret. and Math. Phys., 135:1 (2003), 549–565
4. F. S. Dzheparov, “Delocalization of excitations in disordered media with dipole transfer”, JETP Letters, 82:8 (2005), 521–525
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