### Critical slowdown for Ising model on the two-dimensional lattice ; Interfacial Phenomena and Skew Diffusion

#### Date

October 20, 2010

#### Speakers

Allan Sly and Edward Waymire

#### Affiliation

U.C. Berkeley, Oregon State University

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This is a recap of the 12th Northwest Probability Seminar, a one-day mini-conference organized by the University of Washington, the Oregon State University, the University of British Columbia, the University of Oregon, and the Theory Group at Microsoft Research. The conference was hosted at Microsoft.

Supported by Microsoft Research and the **Pacific Institute for the Mathematical Sciences** (PIMS).

The **Birnbaum**** Lecture in Probability** was given by **Jean-François Le Gall** (Université Paris-Sud, Orsay and Institut Universitaire de France). [Past Birnbaum speakers] The other speakers will be Włodzimierz Bryc (U Cincinnati), Gordon Slade (U British Columbia), and Allan Sly (Microsoft Research), and Edward Waymire (Oregon State U).

The Scientific Committee for the 12th NW Probability Seminar (2010) consisted of Martin Barlow (U British Columbia), Chris Burdzy (U Washington), Zhen-Qing Chen (U Washington), Yevgeniy Kovchegov (Oregon State U), David Levin (U Oregon), and Yuval Peres (Microsoft).

9:45 – 11:00 | Coffee and muffins |
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11:00 – 11:40 | Allan Sly (Microsoft Research)Critical slowdown for Ising model on the two-dimensional lattice |
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11:55 – 12:35 | Edward Waymire (Oregon State)Interfacial Phenomena and Skew Diffusion |
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12:35 – 2:00 | Lunch (catered) |
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1:35 – 2:00 | Open problems (overlaps with lunch) |
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2:05 – 2:55 | Jean-François Le Gall (Orsay)The continuous limit of large random planar maps |
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3:05 – 3:45 | Gordon Slade (U British Columbia)A renormalisation group analysis of the 4-dimensional continuous-time weakly self-avoiding walk |
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3:45 – 4:20 | Tea and snacks |
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4:20 – 5:00 | Włodzimierz Bryc (U Cincinnati)Martingales from pairs of randomized Poisson, Gamma, negative binomial and hyperbolic secant processes |
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5:45 – | Dinner (catered) |

**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.

**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.

**Jean-François Le Gall **(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.

**Gordon Slade** (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.

**Włodzimierz Bryc** (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.

October 20, 2010

Allan Sly and Edward Waymire

U.C. Berkeley, Oregon State University

October 20, 2010

Jean-Francois Le Gall and Gordon Slade

University of British Columbia

October 20, 2010

Wlodzimierz Bruc

U Cincinnati