Symmetric exclusion on random sets and a related problem for random walks in random environment
We study symmetric exclusion on a random set, where the underlying kernel p(x,y) is strictly positive. The random set is generated by Bernoulli experiments with success probability q. We prove that for certain values of the involved parameters the transport of particles through the system is drastically different from the classical situation on \({\mathbb{Z}}\). In dimension one and \[ r:=\lim_{| x| \to \infty}| (| x|^{-1}\log p(0,x))| >| \log (1-q)| \] the transport of particles occurs on a nonclassical scale and is (on a macroscopic scale) not governed by the heat equation as in the case: \(r<| \log (1- q)|\) on a random set, or in the classical situation on \({\mathbb{Z}}.\) The reason for this behaviour is, that a random walk with jump rates p(x,y) restricted to a random set, converges to Brownian motion in the usual scaling if \(r<| \log (1-q)|\) but yields nontrivial limit behaviour only in the scaling \[ x\to u^{-1}x,\quad t\to u^{1+\alpha}t\quad (\alpha >1)\quad if\quad +\infty >r>| \log (1- q)|. \] We calculate \(\alpha\) and study the limiting processes for the various scalings for fixed random sets. We shortly discuss the case \(r=+\infty\), here in general a great variety of scales yields nontrivial limits. Finally we discuss the case of a ``stationary'' random set.
scientific article; zbMATH DE number 4102262