Activities This Week

The random-field Ising model (RFIM) is a standard model for a disordered magnetic system, obtained by placing the standard ferromagnetic Ising model in a random external magnetic field. Imry-Ma (1975) predicted, and Aizenman-Wehr (1989) proved, that the two-dimensional RFIM has a unique Gibbs state at any positive intensity of the random field and at all temperatures. Thus, the addition of an arbitrarily weak random field suffices to destroy the famed phase transition of the two-dimensional Ising model. We study quantitative features of this phenomenon, bounding the decay rate of the effect of boundary conditions on the magnetization in finite systems. This is known to decay exponentially fast for a strong random field. The main new result is a power-law upper bound which is valid at all field strengths and at all temperatures, including zero. Our analysis proceeds through a streamlined and quantified version of the Aizenman-Wehr proof. Several open problems will be mentioned. Joint work with Michael Aizenman.

In 2005, Bergelson, Host and Kra showed that if $(X,\mu,T)$ is an ergodic measure preserving system and $A\subset X$, then for every $\epsilon>0$ there exists a syndetic set of $n\in\mathbb{N}$ such that $\mu(A\cap T^{-n}A\cap\dots\cap T^{-kn}A)>\mu^{k+1}(A)-\epsilon$ for $k\leq3$, extending Khintchine’s theorem. This phenomenon is called multiple recurrence with good lower bounds. Good lower bounds for certain polynomial expressions was studied by Frantzikinakis but several questions remain open. In this talk I will survey this topic, and present some progress regarding polynomial expressions, commuting transformations, and configurations involving the prime numbers. This is work in progress with Joel Moreira, Ahn Le and Wenbo Sun.

Experimentally, the convex-layer decomposition of subsets of the integer grid (“grid peeling”) seems to behave at the limit like the affine curve-shortening flow. We offer some theoretical arguments to explain this phenomenon. In particular, we derive some rigorous results for the special case of peeling the quarter-infinite grid: We prove that, in this case, the number of grid points removed up to iteration $n$ is $\Theta(n^{3/2}\log{ n})$ and moreover, the boundary at iteration $n$ is sandwiched between two hyperbolas that are separated from each other by a constant factor. Joint work with David Eppstein and Sariel Har-Peled