Motion by curvature and large deviations for an interface dynamics on $\mathbb{Z}^2$

Abstract

We study large deviations for a Markov process on curves in $\mathbb{Z}^2$ mimicking the motion of an interface. Our dynamics can be tuned with a parameter $β$, which plays the role of an inverse temperature, and coincides at $β$ = $\infty$ with the zero-temperature Ising model with Glauber dynamics, where curves correspond to the boundaries of droplets of one phase immersed in a sea of the other one. We prove that contours typically follow a motion by curvature with an influence of the parameter $β$, and establish large deviations bounds at all large enough $β$ < $\infty$. The diffusion coefficient and mobility of the model are identified and correspond to those predicted in the literature.

Figures (6)

• Figure 1: Some possible updates for the zero temperature Glauber dynamics. Red squares represent $-$ spins and white squares $+$ spins, assimilating a square with its centre, a point of $(\mathbb{Z}^*)^2$. If either of the lowest two red squares disappear (at rate 1/2, since both have two neighbours of each colour), then the remaining square has three neighbours of opposite colour and is flipped instantaneously. Both these squares thus become white at rate $1$. After that move, the Glauber rules preclude any square of the line from becoming red: the droplet shrinks.
• Figure 2: The convex interface $\gamma$ and the droplet $\Gamma^N$ associated with the discretisation $\gamma^N := \partial\Gamma^N$ of $\gamma$. The four regions of both $\gamma$ and $\gamma^N$ are materialised by opening and closing brackets (for $\gamma$) or parentheses (for $\gamma^N$), with each region starting with an opening symbol and ending with a closing one. Note that regions overlap at the poles. This is the case for the north pole of $\gamma$ and all poles of $\gamma^N$ as the latter must contain at least two edges. To avoid confusion the delimiters of regions 1,3 are in grey and the other two in black. As an example, region 2 of $\gamma^N$ corresponds to the thick dashed lines. In each region, the quarterplane to which the tangent vector ${\bf T}$ belongs is indicated by a cyan square. The spins (i.e. the blocks) which have an edge belonging to a pole are coloured in green.
• Figure 3: Some moves and associated jump rates for the contour dynamics acting on an element of $\Omega^N_{\text{mic}}$. Positions of the extremities $L_k,R_k$, $1\leq k\leq4$ of the poles are represented by dark dots. Possible positions of $L_k,R_k$ after a jump are represented by light dots. Dynamical moves amount to adding or deleting squares of side-length $1/N$ ("blocks"). The pole $P_3$ contains two blocks, i.e. $p_3=2$, thus both of its blocks can be simultaneously deleted by an update.
• Figure 4: Left figure: two microscopic curves equal everywhere except at the north pole: the configuration represented by the black line has a pole containing $2$ blocks, the one with the red line a pole containing $6$ blocks. Initially, the droplet delimited by the black line contains the one delimited in red. A possible update after which the inclusion does not hold is represented in dashed red lines: the contour dynamics at $\beta<\infty$ is not monotonous. Right-figure: only looking at points of the interface in a neighbourhood of $x$, the update indicated by an arrow should be allowed, as the corresponding block has two neighbours in and two neighbours out of the droplet. However, this update is forbidden as the resulting curve would not belong to $\Omega^N_{\text{mic}}$ (it would not be simple). The contour dynamics is therefore non local. The vectors ${\bf e}^{\pm}_y,{\bf e}^{\pm}_z$ are indicated for two points $y,z$ of the interface. The edges $[y+{\bf e}^-_y,y]$ and $[y,y+{\bf e}^+_y]$ are perpendicular: a block can be added or removed at $y$ (in this example, added). The same situation occurs at site $w$: the edge $[w,w+{\bf e}^+_w]$ is vertical, corresponding to $\xi_w = 1$, while the edge $[w',w] = [w+{\bf e}^-_{w},w]$ is horizontal, i.e. $\xi_{w'} = 0$.
• Figure 5: North pole of a curve with the proportion $\xi^{+,\varepsilon N}_{L_1}$ of vertical edges to the right of the pole. For drawing convenience, $\xi^{+,\varepsilon N}_{L_1}$ is assumed to be close to $e^{-\beta}$ (this is only shown to be true for its time average in Proposition ). The corresponding angles $\theta(L_1)_\pm$ are also drawn.

Theorems & Definitions (17)

• Definition 2.1: The set $\Omega$
• Definition 2.3: State space $\Omega^N_{\text{mic}}$
• Remark 2.4
• proof
• Definition 2.5
• Remark 2.6
• Lemma 2.7
• Definition 2.8: Initial condition of the dynamics
• Definition 2.10: Effective state space $\mathcal{E}$
• Proposition 2.11