Equilibrium fluctuations for zero range processes in random environment
A central limit theorem for the density field for stationary zero range processes in a random environment is proved. The evolution of a zero range process in a random environment on a lattice \(Z^d\) is described as follows. If there are \(n\) particles at site \(x\), at rate \( \alpha_x p(y) g(n) \) one of them jumps to site \(x+y\). Here \(\alpha =\{\alpha_x,x\in Z^d\}\) is a family of i.i.d. random variables (random environment), \(p\) is a finite range, symmetric transition probability on \(Z^d\), and \(g:N\to R_+\) with \(0=g(0)<g(k)\) for \(k\geq 1\) is a jump rate. The density field \(Y_t\) is defined as follows: for each compact supported smooth function \(H:R^d\to R\) and each \(t\geq 0\), \[ Y_t^N(H) = N^{-{d/2}} \sum_{x\in Z^d} H(x/N) \left \{\eta_{tN^2}(x) - E_{\overline {\nu}_{\alpha ,\varphi}} [\eta (x)]\right\}, \] where \(\eta_t(x)\) is the number of particles at time \(t\) at site \(x\), \(N^{-1}\) is the scale parameter, and \(\overline {\nu}_{\alpha ,\varphi}\) is an invariant measure (one from a parametric class \(\{\overline {\nu}_{\alpha ,\varphi}\}_{\varphi \geq 0}\)) of the process for the fixed environment \(\alpha \). The main result of the paper states that, averaging over \(\alpha \), the density field converges in distribution to Ornstein-Uhlenbeck process.
scientific article; zbMATH DE number 1364508