Saturday October 16, 2010
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Supported by Microsoft Research and the Pacific Institute for the Mathematical Sciences (PIMS).
The Birnbaum Lecture in Probability will be 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).
This is 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. This year the conference is being hosted at Microsoft. There is no registration fee. Participants are requested to email David.Wilson@microsoft.com in advance so that adequate food may be arranged for and so that wireless access can be provided. Breakfast, lunch, and coffee will be free.
The Scientific Committee for the 12th NW Probability Seminar (2010) consists 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).
The talks will take place in Building 99 at Microsoft. Parking at Microsoft is free.
Schedule & Recordings & Slides
9:45 - 11:00 Coffee and muffins 11:00 - 11:40
Allan Sly (Microsoft Research)
Critical slowdown for Ising model on the two-dimensional lattice11:55 - 12:35 
Edward Waymire (Oregon State)
Interfacial Phenomena and Skew Diffusion12:35 - 2:00 Lunch (catered) 1:35 - 2:00 Open problems (overlaps with lunch) 2:05 - 2:55 
Jean-François Le Gall (Orsay)
The continuous limit of large random planar maps3:05 - 3:45
Gordon Slade (U British Columbia)
A renormalisation group analysis of the
4-dimensional continuous-time weakly self-avoiding walk3:45 - 4:20 Tea and snacks 4:20 - 5:00
Włodzimierz Bryc (U Cincinnati)
Martingales from pairs of randomized Poisson, Gamma,
negative binomial and hyperbolic secant processes5:45 - Dinner (catered)
Talk abstracts
Critical slowdown for Ising model on the two-dimensional lattice
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.
The continuous limit of large random planar maps
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.
A renormalisation group analysis of the 4-dimensional continuous-time weakly self-avoiding walk
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.
Martingales from pairs of randomized Poisson, Gamma, negative binomial and hyperbolic secant processes
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.
Directions
From the north: Take I-5 south, then I-405 south, then WA-520 east.
From the south: Take I-5 north, then I-405 north, then WA-520 east.
From Seattle: Take WA-520 east.
By airplane: Fly to Seattle's airport, take I-405 north, then WA-520 east.
From WA-520 east, take the 148th Ave NE North exit (this is the second 148th Ave NE exit). Turn right (north) onto 148th Ave NE, proceed a few blocks, and turn right onto NE 36th St. Building 99 will be on the left. The address is 14820 NE 36th St, Redmond, WA 98052-5319. Click here for a map.
Hotels
Many people make the NW Probability Seminar a day trip, but for those wishing to stay longer, some nearby hotels include
- Homestead Studio Suites (probably the best choice)
15805 NE 28th Street
Bellevue, WA 98008
(425) 885-6675 - Silver Cloud Inn
10621 NE 12th Street
Bellevue, WA 98004
(800) 205-6937 - Courtyard Marriott (extremely close)
14615 NE 29th Place
Bellevue, WA 98007
(425) 869-5300 - Fairfield Inn Marriott (extremely close)
14595 NE 29th Place
Bellevue, WA 98007
(425) 869-6548 - Residence Inn Marriott-Redmond
7575 164th Ave NE
Redmond, WA 98052
(425) 497-9226 - Silver Cloud Inn (near the University of Washington in Seattle)
5036 25th Avenue NE
Seattle, WA 98105
(800) 205-6940 - University Inn (near the University of Washington)
4140 Roosevelt Way NE
Seattle, WA 98105
(800) 733-3855 - Watertown (near the University of Washington)
4242 Roosevelt Way NE
Seattle, WA 98105
(866) 944-4242
