Linear fluctuation of interfaces in Glauber-Kawasaki dynamics

Abstract

In this article, we find a scaling limit of the space-time mass fluctuation field of Glauber + Kawasaki particle dynamics around its hydrodynamic mean curvature interface limit. Here, the Glauber rates are scaled by $K=K_N$, the Kawasaki rates by $N^2$ and space by $1/N$. We start the process so that the interface ${\Gamma }_t$ formed is stationary that is, ${\Gamma }_t$ is `flat'. When the Glauber rates are balanced on $T^d$, ${\Gamma }_t=\Gamma =\{x:x_1=0\}$ is immobile and the hydrodynamic limit is given by $\rho (t,v)={\rho }_{+}$ for $v_1\in (0,1/2)$ and $\rho (t,v)={\rho }_{-}$ for $v_1\in (-1/2,0)$ for all $t\ge 0$, where $v=(v_1,\dots ,v_d)\in T^d$ identified with $[-1/2,1/2{)}^d$. Since in the formation the boundary region about the interface has width $O(1/\sqrt{K_N})$, we will scale the $v_1$ coordinate in the fluctuation field by $\sqrt{K_N}$ so that the scaling limit will capture information `near' the interface. We identify the fluctuation limit as a Gaussian field when $K_N\uparrow \infty$ and $K_N=O(\sqrt{\log (N)})$ in $d\le 2$. In the one dimensional case, the field limit is given by $\mathbf{e}(v_1)B_t$ where $B_t$ is a Brownian motion and $\mathbf{e}$ is the normalized derivative of a decreasing `standing wave' solution $\varphi$ of ${\partial }_{v_1}^2\varphi -V^{\prime }(\varphi )=0$ on $R$, where $V^{\prime }$ is the homogenization of the Glauber rates. In two dimensions, the limit is $\mathbf{e}(v_1)Z_t(v_2)$ where $Z_t$ is the solution of a one dimensional stochastic heat equation. The appearance of the function $\mathbf{e}(\cdot )$ in the limit field indicates that the interface fluctuation retains the shape of the transition layer $\varphi$.