\((\log t)^{2/3}\) law of the two-dimensional asymmetric simple exclusion process. (English) Zbl 1060.60099

Yau, Horng-Tzer

Ann. Math. (2) 159, No. 1, 377-405 (2004).

The author studies the diffusion coefficient for the two-dimensional asymmetric simple exclusion process, specifically, the Markov process on \(\{ 0, 1 \}^{{\mathbb{Z}}^2}\) which has the generator \[ ({\mathcal L}f)(\eta) = \sum_{x\in {\mathbb{Z}}^2} \bigl[ \eta_x(1-\eta_{x+e_1})(f(\eta^{x,x+e_1}) - f(\eta)) + \tfrac 12(f(\eta^{x,x+e_2}) - f(\eta))\bigr] \] and the Bernoulli measure with density \(1/2\) as the equilibrium distribution. The result of the paper is that \( \int_0^{\infty}e^{-\lambda t}tD_{11}(t)\,dt \sim \lambda^{-2}| \log\lambda| ^{2/3} \) holds for sufficiently small \(\lambda > 0\), where \(D_{11}\) is the only nonzero entry of the diffusion coefficient matrix.

Reviewer: Hiroshi Tamura (Kanazawa)

MSC:

60K35	Interacting random processes; statistical mechanics type models; percolation theory
82C22	Interacting particle systems in time-dependent statistical mechanics

Keywords:

diffusion coefficient