About the Long Range Exclusion Process
2004, v.10, Issue 3, 457-476
Introduced by Spitzer [F. Spitzer, Interaction of Markov processes, Adv. Math., 1970, v. 5, 246-290] and studied by Liggett [T.M. Liggett, Long range exclusion processes, Ann. Prob., 1980, v. 8, N 5, 861-889], the Long Range Exclusion Process (LREP) is an interacting particle system with truly long range interaction. Informally speaking: each particle on a lattice hops at independent random times following instantaneously a random dynamic on the lattice until finding a vacant site (if any). These instantaneous, potentially long jumps prevent the process to have the Feller property. In this paper we review the main results about the LREP including recent developments obtained in [X. Zheng, Ergodic theorem for generalized long-range exclusion processes with positive recurrent transition probabilities. Acta Mathematica Sinica (N.S.), 1988, v. 4, N 3, 193-209; H. Guiol, Un resultat pour le processus d'exclusion a longue portee, Ann. Inst. H. Poincare, Probabilites et Statistiques, 1997, v. 33, N 4, 387-405] and [E. Andjel and H. Guiol, Long range exclusion processes, generator and invariant measures, To appear in Ann. Prob., 2004]. New results on Feller approximations and about the regularity set of the LREP are also provided. Finally we briefly discuss some connections of the LREP with the discrete Hammersley process introduced in [P.A. Ferrari, Limit theorems for tagged particles, Markov Processes Relat. Fields, 1996, v. 2, N 1, 17-40] and the sandpile process in infinite volume developed in [C. Maes, F. Redig, E. Saada and A. Van Moffaert, On the thermodynamic limit for a one-dimensional sandpile process, Markov Processes Relat. Fields, 2000, v. 6, N 1, 1-21] and [C. Maes, F. Redig and E. Saada, The abelian sandpile model on an infinite tree, Ann. Prob., 2002, v. 30, N 4, 2081-2107].
Keywords: infinite particle systems,non-Feller process,longrange exclusion,invariant measures,formal generator