Seymour Sherman Lecture and Conference
Probability and Statistical Physics
April 21-23, 2006
Indiana University, Bloomington

Sherman Lecturer: Tom Liggett (UCLA)

Conference Speakers:

Omer Angel (UBC)
Noam Berger (UCLA)
Sourav Chatterjee (UCB)
Julien Dubédat (NYU)
Sharad Goel (Stanford)
Dan Romik (UCB)
Scott Sheffield (NYU)
Dapeng Zhan (UCB)

Full support will be provided for a limited number of graduate students. To apply for support, please send email to Russell Lyons at rdlyons@indiana.edu and also have a brief letter of recommendation sent there as well. Materials are due by March 10, 2006. We will notify applicants of financial support by March 15, 2006. Late applications will be considered as long as money is still available.

There is no registration fee nor need to register except for the banquet at Samira Afghan Restaurant Friday, April 21, 2006 at 6:30pm. If you are interested in the banquet, which will cost about $20, send email to Russell Lyons by April 10, 2006. There will be many choices of food, including vegetarian. Although these are not official conference events, you may be interested in the following musical events on Saturday and Sunday evenings: jazz and classical. Both should be superb.

We will provide hotel rooms for all supported participants. There is a block of rooms reserved at the Courtyard Marriott Hotel, where the speakers and supported grad students will stay. There are not many rooms there, but there will be some available to others. When you call, tell them you are with the IU Math Dept./Seymour Sherman Conference and ask for group code SSW. If you don't get a room there, see here for a list of hotels. Note that this is the same weekend as the Little 500 Bicycle Race (immortalized by the 1979 Academy Award-winning movie, Breaking Away), so that rooms will be very scarce. However, we have also reserved a block of rooms at a new hotel, TownePlace Suites by Marriott. If you call them, tell them you are with the "math conference". They are not in walking distance of the campus, however.

The Sherman Lecture will be at 4pm, Friday. The Conference will be Sat. 9:30-6 and Sun. 9:30-11:30. All talks will be held in Swain East 140. Registration and breaks will be in the Rawles Hall lounge. These buildings are next to each other. "Registration" means picking up a name badge, talk schedule, restaurant information, and computer information.

For restaurant recommendations, go here.

Schedule

Friday, April 21
 3:30 registration and tea
 4:00 Liggett (Sherman Lecture)
 6:30 banquet

All later talks are 45 minutes, which leaves time for interaction between them.

Saturday, April 22
 9:00 coffee, OJ, muffins, bagels
 9:30 Sheffield
10:45 Dubédat
 1:30 Angel
 2:30 Goel
 3:20 coffee, fruit, nuts
 4:00 Chatterjee
 5:15 Romik

Sunday, April 23
 9:00 coffee, OJ, muffins, bagels
 9:30 Zhan
10:45 Berger

Talk Titles and Abstracts:

Omer Angel: Log^* and divergence of spatial coalescents

We consider a process where each particle performs a simple random walk in continuous time on a given graph, with some mechanism for coalescing particles that are at the same site. This mechanism could be Kingman's coalescent where each pair coalesce at rate 1 or more general Lambda-coalescents. We show that if started with infinitely many particles at a single site, the total number of particles remains infinite. Starting with N particles, at any given time the number of particles is typically at least log^*(N), and this bound is sharp.

Noam Berger: Detecting the trail of a random walker

Let G be a transient graph, and flip a fair coin at each vertex. This gives a distribution P. Now start a random walk from a vertex v, and retoss the coin at each visited vertex, this time with probability 0.75 for heads and probability 0.25 for tails. The eventual configuration of the coins gives a distribution Q. Are P and Q absolutely continuous w.r.t. each other? are they singular? (i.e. can you tell whether a random walker had tampered with the coins or not?) In the talk I'll answer to this question for various graphs and various types of random walk. Based on joint work with Y. Peres.

Sourav Chatterjee: Normal approximation: A new insight and a simple method

Although it is difficult to believe that there may be something new to say about normal approximation in the twenty-first century, the fact still remains that there exists no simple and certain method of proving central limit theorems. All techniques known to statisticians and probabilists require extensive information about the nature of the system, and some good luck. On the other hand, variance bounds (and hence weak laws) are routine issues which daunt no one. The purpose of this talk is to introduce a new method by which it is possible to reduce every normal approximation problem to a variance bounding exercise. Applications, including one involving a nontrivial statistical problem (due to Peter Bickel), will be described.

Julien Dubédat: Commutation of SLEs

Schramm-Loewner Evolutions (SLEs) have proved a powerful tool to describe the scaling limit of a conformally invariant simple curve. In several instances (percolation, uniform spanning tree, ...), one can define in a discrete setting several simple curves. We will discuss questions pertaining to the joint law of these curves in the scaling limit.

Sharad Goel: Estimating Mixing Times via the Spectral Profile

Given 52 playing cards, how many shuffles does it take to approximately randomize the deck? More generally, how long does it take a finite Markov chain to get close to its stationary distribution? In this talk, I'll introduce the spectral profile as a tool for proving upper and lower bounds on convergence rates. This approach extends the commonly used spectral gap method, and allows us to recover and refine previous conductance-based estimates of mixing time. I will illustrate how the spectral profile technique is applied in several models, including groups with moderate growth, the fractal-like Viscek graphs, and the torus. This work is joint with Ravi Montenegro and Prasad Tetali.

Tom Liggett: Symmetry, Moment Problems, and Statistical Mechanical Systems on Complete Graphs

The Ising model on a complete graph with n vertices is known as the Curie-Weiss model. The corresponding Gibbs state is a simple probability measure on {0,1}ⁿ that is invariant under permutations of the vertices. Such a measure is said to be exchangeable. De Finetti's Theorem states that if n were infinite, an exchangeable measure would be a mixture of homogeneous product measures. For finite n, this is the case if and only if a certain finite sequence can be expressed as the first n moments of a probability distribution on [0,1]. This provides a connection to the classical Hausdorff moment problem. Using this connection, we will show that the Curie-Weiss model is a mixture of homogeneous product measures if and only if the model is ferromagnetic. In the antiferromagnetic case, we will ask whether the measure can be extended to {0,1}^l for l>n. This question leads to a new moment problem, which we will solve and apply to get a rather precise asymptotic result for the Curie-Weiss model. This talk is based on joint work with J. Steif and B. Toth.

Dan Romik: Random square Young tableaux with applications

A square Young tableau is a sequence of instructions for building an NxN square wall in [0,N]² by laying N² unit square bricks, with the requirement that each new brick that is laid must be supported from below and from the left by an existing brick or the x- or y-axes. If one chooses such a sequence of instructions uniformly at random among all possible ones, what is the typical growth profile of the wall of bricks? I will describe the answer to this question, obtained in joint work with Boris Pittel. The solution reveals a striking similarity between the random square Young tableau model and the well-known Plancherel measure, which has been extensively studied because of its connections to increasing subsequences in random permutations and random matrix theory. I will then describe some applications of the square Young tableau limit shape result to other combinatorial probability models.

Scott Sheffield: Gaussian free fields and the height gap

We give an overview of the proof of the fact that, in a certain fine-mesh-limiting sense, there is a constant expected height gap between either side of each contour line of the (interpolated) discrete Gaussian free field. Joint work with Oded Schramm.

Dapeng Zhan: SLE and LERW in multiply connected domains

We define SLE(&kappa) in multiply connected domains using radial or chordal Loewner equation and some suitable coordinate system. The definition is invariant under the change of coordinates. In the case that &kappa=2, SLE(2) preserves some continuous local martingales which resemble the discrete martingales preserved by loop-erased random walk (LERW). So SLE(2) is very likely to be the scaling limits of the corresponding LERW.

Seymour Sherman Lecture and Conference Probability and Statistical Physics April 21-23, 2006 Indiana University, Bloomington Sherman Lecturer: Tom Liggett (UCLA)