Motion by curvature and large deviations for an interface dynamics on \(\mathbb{Z}^2\) (Q6589462)

The paper presents a family of interface dynamics (contour dynamics). In the scaling limit, this dynamics typically evolves according to motion by curvature, and the goal is to characterize the large deviations. The contour dynamics is closely related to the zero-temperature Glauber dynamics for the Ising model: it has the same updates, except that additional moves depending on a parameter \(\beta\)>0 are allowed. This parameter \(\beta\) plays the role of an inverse temperature acting on local portions of the contours. The model at each \(\beta>0\) has reversible dynamics and, contrary to the Glauber dynamics for the zero-temperature Ising model, the dynamics is not monotonous. When \(\beta=\infty\), the update rules of the contour dynamics are exactly the same as the Ising ones. Section 2 gives the microscopic model and some notation. The dynamics is introduced in details using the zero-temperature Glauber dynamics as comparison, while useful topological facts are collected in Appendix B. \N\NThe main results of the paper are listed in Section 2, with Section 2D presenting the structure of the proof as well as a connection of the contour dynamics with the exclusion process. Section 3 specifies Radon-Nikodym derivatives for a large class of tilted dynamics. Under the assumption that trajectories live in a nice enough space, it shows how motion by curvature emerges from the microscopic computations as well as the influence of the parameter \(\beta\). The computations of the Radon-Nikodym derivatives are then used to prove large deviations, with the upper bound in Section 4 and the lower bound in Section 5. A number of technical results and subexponential estimates are postponed to Section 6 and Appendices A--B. In particular, Section 6 is a collection of estimates that are genuinely particular to our model, concerning the dynamical behavior of the poles, that is, the sections of the contours on which the parameter \(\beta\) affects the dynamics.