Critical slowdown for Ising model on the two-dimensional lattice
Allan Sly (opens in new tab) (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 (opens in new tab) (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.
The continuous limit of large random planar maps
Jean-François Le Gall (opens in new tab)(Université Paris-Sud, Orsay and Institut Universitaire de France).
Abstract: Planar maps are graphs embedded in the plane, considered up to continuous deformation. They have been studied extensively in combinatorics, and they also have significant geometrical applications. Random planar maps have been used in theoretical physics, where they serve as models of random geometry. Our goal is to discuss the convergence in distribution of rescaled random planar maps viewed as random metric spaces. More precisely, we consider a random planar map M(n) which is uniformly distributed over the set of all planar maps with n vertices in a certain class. We equip the set of vertices of M(n) with the graph distance rescaled by the factor n^{-1/4}. We then discuss the convergence in distribution of the resulting random metric spaces as n tends to infinity, in the sense of the Gromov-Hausdorff distance between compact metric spaces. This problem was stated by Oded Schramm in his 2006 ICM paper, in the special case of triangulations. In the case of bipartite planar maps, we first establish a compactness result showing that a limit exists along a suitable subsequence. We then prove that this limit, which is called the Brownian map, can be written as a quotient space of Aldous’ Continuum Random Tree (the CRT) for an equivalence relation which has a simple definition in terms of Brownian labels assigned to the vertices of the CRT. We discuss various properties of the Brownian map.
A renormalisation group analysis of the 4-dimensional continuous-time weakly self-avoiding walk
Gordon Slade (opens in new tab) (U British Columbia)
Abstract: We discuss recent joint work with David Brydges which proves |x|^{-2} decay of the critical two-point function for the continuous-time weakly self-avoiding walk on Z^4. The walk two-point function is identified as the two-point function of a supersymmetric field theory with quartic self-interaction, and the field theory is then analysed using renormalisation group methods.
Martingales from pairs of randomized Poisson, Gamma, negative binomial and hyperbolic secant processes
Włodzimierz Bryc (opens in new tab) (U Cincinnati)
Abstract: Consider a pair of independent Poisson processes, or a pair of Negative Binomial processes, or Gamma, or hyperbolic secant processes with a shared randomly selected parameter. Under appropriate randomization, one can deterministically re-parametrize the time and scale for both processes so that the first process runs on time interval $(0,1)$, the second process runs on time interval $(1,infty)$, and the two processes seamlessly join into one Markov martingale on $(0,infty)$. In fact, a property stronger than martingale holds: we stitch together two processes into a single quadratic harness on $(0,infty)$.
This talk is based on joint work in progress with J. Wesolowski.