# Northwest Probability Seminar

### and Pacific Institute for Mathematical Sciences

The Birnbaum Lecture in Probability will be delivered by Rich Bass (University of Connecticut) this year.

Northwest Probability Seminars are one-day mini-conferences held at the University of Washington and organized in collaboration with the Oregon State University, the University of British Columbia, University of Victoria, the University of Oregon, and the Theory Group at the Microsoft Research. There is no registration fee. Participants are requested to contact Zhen-Qing Chen (zchen@math.washington.edu ) in advance so that adequate facilities may be arranged for.

The Scientific Committee for the NW Probability Seminar 2007 consists of Chris Burdzy (U Washington), Zhenqing Chen (U Washington), Edwin Perkins (U British Columbia), David Levin (U Oregon) and Yevgeniy Kovchegov (Oregon State U).

The talks will take place in Thomson Hall 125. See the map of north-central campus for the location of Thomson Hall and Padelford Hall (the Department of Mathematics is in the Padelford Hall). More campus maps are available at the UW Web site.

Parking on UW campus is free on Saturdays after 12:00 (noon). More information is available at a parking Web site provided by UW. This year, October 20, 2007 is also the Husky game day so the traffic may be slow before and after the game.

### Schedule

• 10:00 Coffee and Registration - Thomson Hall 119
• 11:00 David Levin, University of Oregon
• On Mixing of mean-field Glauber dynamics
Abstract: I will discuss the Glauber dynamics for the Ising model on the complete graph on n vertices, reporting on joint work with M. Luczak and Y. Peres. At high temperature, we show that the dynamics exhibits a cut-off (in a window of size order n) at the (well-known) mixing time of order n \log n. At the critical temperature, we prove the mixing time is order n^{3/2}. Finally, if the dynamics are restricted to states with non-negative magnetization, at low temperature the mixing time is order n\log n, contrasting with exponential mixing time for the unrestricted dynamics.

• 12:00 Anthony Quas, University of Victoria
• Distances in Positive Density Sets
Abstract: Given a set of distances D, one can consider the graph G_{d,D} on R^d where two points are adjacent if they are separated by a distance belonging to D and ask for its chromatic number. The case where D={1} is the Hadwiger-Nelson problem and it is known that 4<=chi(G_{2,{1}})<=7. If the colour classes are required to be measurable, we obtain the measurable chromatic number \chi_m(G_{d,D}). It is known that 5<=\chi_m(G_{2,{1}})<=7.
In the case where D is unbounded, it turns out that \chi_m(G_{d,D})=\infty. We give a conceptual new proof of this and discuss possible extensions to the general (non-measurable) case.

• 1:00 - 2:30 Lunch

• 2:30 David Wilson, Microsoft Research
• Boundary Partitions in Trees and Dimers
Abstract: Given a finite planar graph, a grove is a spanning forest in which every component tree contains one or more of a specified set of vertices (called nodes) on the outer face. For the uniform measure on groves, we compute the probabilities of the different possible node connections in a grove. These probabilities only depend on boundary measurements of the graph and not on the actual graph structure, i.e., the probabilities can be expressed as functions of the pairwise electrical resistances between the nodes, or equivalently, as functions of the Dirichlet-to-Neumann operator (or response matrix) on the nodes. These formulae can be likened to generalizations (for spanning forests) of Cardy's percolation crossing probabilities, and generalize Kirchhoff's formula for the electrical resistance. Remarkably, when appropriately normalized, the connection probabilities are in fact integer-coefficient polynomials in the matrix entries, where the coefficients have a natural combinatorial interpretation. A similar phenomenon holds in the so-called double-dimer model: connection probabilities of boundary nodes are polynomial functions of certain boundary measurements, and as formal polynomials, they are specializations of the grove polynomials. Upon taking scaling limits, we show that the double-dimer connection probabilities coincide with those of the contour lines in the Gaussian free field with certain natural boundary conditions. These results have direct application to connection probabilities for multiple-strand SLE_2, SLE_8, and SLE_4.
Joint work with Richard Kenyon.

• 3:30-3:45 Coffee - Thomson Hall 119

• 3:45 Rich Bass, University of Connecticut
• "Birnbaum Lecture": Random sampling and probability
Abstract: The random sampling'' in the title has nothing whatsoever to do with statistics. Instead, it refers to the perfect reconstruction of a band-limited function from  samples, a classical problem of Fourier analysis and signal processing. With deterministic sampling, almost everything is known in one dimension and almost nothing is known in higher dimensions. It turns out that one loses very little in  efficiency by using random sampling, and in return one can use  probabilistic techniques to get some interesting theoretical results.
This is joint work with Karlheinz Groechenig.

• 5:30 No host dinner at Cedars Restaurant on Brooklyn. Click on the restaurant name to go to its Web page.