Critical slowdown for Ising model on the two-dimensional lattice

Опубликовано 15 июля 2016, 18:35

Critical slowdown for Ising model on the two-dimensional lattice ; Interfacial Phenomena and Skew Diffusion
11:00 - 11:40 Allan Sly (Microsoft Research) Critical slowdown for Ising model on the two-dimensional lattice
11:55 - 12:35 Edward Waymire (Oregon State) Interfacial Phenomena and Skew Diffusion Allan Sly (Microsoft Research) Abstract: Intensive study throughout the last three decades has yielded a rigorous understanding of the spectral-gap of the Glauber dynamics for the Ising model on $\Z_2$ everywhere except at criticality. At the static phase-transition for Ising, the dynamics is conjectured to undergo a critical slowdown: At high temperature the inverse-gap is $O(1)$, at the critical $\beta_c$ it is polynomial in the side-length and at low temperature it is exponential in it. A long series of works verified this picture on $\Z_2$ except at $\beta=\beta_c$ where the behavior remained unknown. In this work we establish the first rigorous polynomial upper bound for the critical mixing, thus confirming the critical slowdown for the Ising model in $\Z_2$. Namely, we show that on a finite box with arbitrary boundary conditions, the inverse-gap at $\beta=\beta_c$ is polynomial in the side-length. The proof harnesses recent understanding of the scaling limit of critical Fortuin-Kasteleyn representation of the Ising model together with classical tools from the analysis of Markov chains. Joint work with Eyal Lubetzky. Interfacial Phenomena and Skew Diffusion Edward Waymire (Oregon State) Abstract: Skew diffusion refers to stochastic processes whose infinitesimal generators are second order advection-dispersion elliptic operators having piecewise constant coefficients. Such processes arise naturally in connection with macroscopic mass balance and flux laws in highly heterogeneous environments. We shall discuss some recent results pertaining to interfacial effects in terms of martingale properties, local time and first passage time properties. This is based on joint work with Thilanka Appuhamillage, Vrushali Bokil, Enrique Thomann, and Brian Wood. Critical slowdown for Ising model on the two-dimensional lattice