Occupation time large deviations of two-dimensional symmetric simple exclusion process.
The article considers the nearest-neighbour symmetric simple exclusion process on the square lattice \( \mathbb{Z}^2 \). This process can be seen as follows. Distribute particle on \( \mathbb{Z}^2 \), such that each site is occupied by at most one particle. Each particle, independently from the others, waits a mean-\(1\) exponential time before it chooses one of its neighbour sites with uniform probability. If the site is unoccupied, the particle jumps to this site, otherwise, it stays where it is. Thus each lattice point is occupied by at most one particle. For this stochastic dynamics one has a one-parameter family of invariant states, the Bernoulli product measures with given probability density \( \alpha\in[0,1] \). This measure is obtained if one places at each site particle with probability \( \alpha \) independently from the others. A Bernoulli product measure is an ergodic, invariant, reversible state for the exclusion process. The occupation time at the origin is defined as the random proportion of time that the origin is occupied in a given time interval. This random variable converges almost surely for diverging time with respect to a Bernoulli product measure to the parameter \( \alpha \) of that measure. The authors prove a large deviations result for this occupation time at the origin for diverging time, where the decay rate is \( t/\log t \) in the time variable. This is the first example of an interacting particle system where the large deviations rate function associated to the occupation time can be explicitly computed. Second, the method of the proof is remarkable. The authors show that the large deviations in dimension two are related to the large deviations for the polar empirical measure (recall that in dimension one the large deviations of the occupation measure are related to the large deviations for the empirical measure in the hydrodynamical limit). The polar empirical measure is a random measure on \( \mathbb{R}_+ \), and for this article all fluctuations occur in the interval \( [0,1/2) \). For the hydrodynamic scale and above, i.e., on \( [1/2,\infty) \), the random measure is fixed and equals the Lebesgue measure with constant weight given by the parameter of the Bernoulli starting measure. The proof of the large deviations goes via the corresponding large deviations results for the polar empirical measure, and a proof of a superexponential estimate.
scientific article; zbMATH DE number 2100730