Occupation time large deviations for the symmetric simple exclusion process. (English) Zbl 0751.60098

Let \((\eta_ t)\) denote the symmetric simple exclusion process (mean one exponential waiting times) with state space \(X_ d=\{0,1\}^{Z^ d}\), and symmetric transition probabilities \(p(k,j)\), \(k,j\in Z^ d\), the path space being \(D([0,\infty[,X_ d)\). Here, \(\eta_ t(k)\in\{0,1\}\) denotes the number of particles at site \(k\in Z^ d\) at time \(t\). The author is interested in obtaining large deviation results for the occupation time functional \(\int^ t_ 0\eta_ s(0)ds\). For any \(\rho\in[0,1]\) let \(\nu_ \rho\) denote the product measure on \(X_ d\) given by its marginals \(\nu_ \rho\{\eta: \eta(k)=1\}=\rho\), \(k\in Z^ d\). Let \(P_ \rho\) denote the corresponding probability measure on \(D([0,\infty[,X_ d)\). One of the author’s results implies that in the case \(d\geq 3\) the following large deviation result holds: \[ (*)\qquad \limsup_{t\to\infty}{1\over a_ t}\log P_ \rho\left[{1\over t}\int^ t_ 0\eta_ s(0)ds\in F\right]\leq -\inf_{\alpha\in F}\psi_ d(\alpha) \] and \[ (**)\qquad \liminf_{t\to\infty}{1\over a_ t}\log P_ \rho\left[{1\over t}\int^ t_ 0\eta_ s(0)ds\in G\right]\geq -\inf_{\alpha\in G}\psi_ d(\alpha) \] (\(\psi_ d\) being a certain rate function). Here, \(F\) (\(G\)) denote arbitrary closed (open) subsets of \([0,1]\), and \(a_ t=t\), \(t>0\). If \(d=1\), it follows for the nearest neighbour case \((p(k,j)=(2d)^{-1}\) if \(\| k-j\|=1\)) that (*) and (**) hold for \(a_ t=t^{1/2}\), \(t>0\). Finally, if \(d=2\), it follows for the nearest neighbour case that the decay rate \(a_ t\) equals \(t/\log t\), \(t>0\).


60K35 Interacting random processes; statistical mechanics type models; percolation theory
60F10 Large deviations
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