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Brownian survival among Gibbsian traps. (English) Zbl 0769.60104
The author studies the long time survival probability of a Brownian motion \(Z\) on \(R^ d\) \((d\geq 1)\) which moves among random obstacles constructed by translating a model nonpolar compact set at the points of a Gibbs point process independent of \(Z\). It is assumed that the law \(\tilde P\) of this point process satisfies the Dobrushin-Lanford-Ruelle equations relative to an activity number \(\nu>0\) and a suitable transformation invariant pair potential \(V(x-y)\). Let \(T\) denote the entrance time of \(Z\) into the obstacles and let \(P_ 0\) denote standard Wiener measure. The author shows that under additional assumptions on \(\tilde P\), the limit \[ \lim_{t\to\infty}t^{-d/(d+2)}\log(\tilde P\otimes P_ 0(T>t))=-c(d,p) \] exists where \(c(d,p)\) depends on a certain pressure \(p\in(0,\infty)\) whose existence follows from the hypotheses on \(V(\cdot)\).

MSC:
60K40 Other physical applications of random processes
82D30 Statistical mechanical studies of random media, disordered materials (including liquid crystals and spin glasses)
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