Probability Seminar --- 2005 -- 2006 Sept. 5 --- Greg Lawler Conformal Invariance and Two-dimensional Polymers This will be a "colloquium level" talk reviewing work in the last few years on the scaling limit of self-avoiding walks. I will be giving this talk at a meeting in Pittsburgh later in the week, so I thought I could start the semester by giving it here. Sept. 12 --- Joan Lind, The geometry of Loewner evolution Sept. 19 --- David White, Processes with inert drift Sept. 26 --- Josh Rushton, Cornell, A Chung-type functional LIL for $\alpha$-stable processes, and related invariance results. Oct. 3 --- Paul Jung, Cornell Oct. 10 --- break Oct. 17 --- Lea Popovic, Continuum trees from catalytic population models Oct 24 --- Raazesh Sainudiin, A randomized enclosure algorithm: Moore rejection sampling Oct. 31 Kazumasa Kuwada, Large deviations for random currents induced from stochastic line integrals Nov, 7 Gady Kozma, The scaling limit of loop-erased walk in three dimensions Nov, 11 (Special time), 11:00 am, Julien Dubedat Nov. 21 --- Rick Durrett, Waiting for ATCAAAG One possible explanation for the substantial organismal differences that have developed in the 6 million years sincee the divergence of humans and chimpanzees is that there have been changes in gene regulation. The word in the title is a sample transcription factor binding sites and motivates the following probability question: given a 1000 nucleotide region in our genome, how long does it take for a specified six to nine letter word to appear in that region in some individual? Stone and Wray (2001) computed 5949 years as the answer for six letter words in the human population. We will show that for words of length 6, the average waiting time is 100,000 years while for words of length 8, the waiting time is roughly a 1/3 - 2/3 mixture of exponentials with means 375,000 years and 625 million years. In biological reality, the match to the target word does not have to be perfect for binding to occur. If we model this by saying that we allow mismatch, then almost all of the mass in the probability distribution shifts to the smaller mean. This is joint work with Deena Schmidt. SPRING 2006 TENTATIVE SCHEDULE Jan 23 --- N. Lanchier Jan 30 --- Two talks: S. Cohen and A. Sturm (details later) Feb 6 -- no talk, but see http://www.cs.cornell.edu/courses/cs789/2006sp/Ludek.Kucera.htm for an interesting, related talk. Feb 13 -- Antar Bandyopadhyay, Chalmers University, Random Walk in Dynamic Markovian Random Environment Thursday, Feb 16, 3:00, 206 Malott (note unusual time and place) Davar Khoshnevisan, Utah A Coupling and the Darling - Erdos Conjectures We present a coupling of the 1-dimensional Ornstein-Uhlenbeck process with an i.i.d. sequence. We then apply this coupling to resolve two conjectures of Darling and Erd￿s (1956). Interestingly enough, we prove one and disprove the other conjecture. [This is joint work with David Levin.] Time-permitting, we may use the ideas of this talk to describe precisely the rate of convergence in the classical law of the iterated logarithm of Khintchine for Brownian motion (1933). [This portion is joint work with David Levin and Zhan Shi, and has recently appeared in the Electr. Comm. of Probab. (2005)] Feb. 27 -- Mihai Sirbu, Columbia Mar 6 -- Ted Cox, Syracuse Mar. 13 -- X. Chen, Tennessee Large deviations for Brownian intersection local times and related problems It has been known that the study of intersection local times is partially motivated by the needs in mathematical physics. In this talk I will report some recent progress on the large deviations for the intersection local times run by Brownian motions. In particular, I will give answer to a conjecture raised by the physicist Dulpantier. In addition, I will talk about some impact of our results in some related areas such as ranges of random walks and local times of multi-parameter processes. Part of the talk is based on the joint work with Richard Bass, Wenbo Li and Jay Rosen. Mar. 27 -- J. Schweinsburg, UCSD The loop-erased random walk and the uniform spanning tree on the four-dimensional discrete torus. Let x and y be points chosen uniformly at random from the four-dimensional discrete torus with side length n. We show that the length of the loop-erased random walk from x to y is of order n^2 (log n)^{1/6}, resolving a conjecture of Benjamini and Kozma. We also show that the scaling limit of the uniform spanning tree on the four-dimensional discrete torus is the Brownian continuum random tree of Aldous. Our proofs use the techniques developed by Peres and Revelle, who studied the scaling limits of the uniform spanning tree on a large class of finite graphs that includes the d-dimensional discrete torus for d >= 5, in combination with results of Lawler concerning intersections of four-dimensional random walks Apr. 3, Benedek Valko, U of Toronto, Limits of random trees from real-world networks The following random tree model is often used to describe real-world networks. We start with a single vertex and in every step we connect a new vertex randomly to one of the old ones with probability proportional to a function of its degree. We find asymptotic degree distribution, and prove a limit theorem for the tree itself as viewed from a random point. (Joint work with A. Rudas and B. Toth) Apr. 10 -- M. Reed, Duke Apr. 17 -- J. Yukich, Lehigh Apr, 24 -- M. Roeckner, Purdue "Stochastic porous media and fast diffusion equations" We first present a new existence and uniqueness result for stochastic evolution equations on Hilbert spaces. This is a generalization of a classical result by Krylov and Rozovskii based on the so-called variational approach to stochastic partial differential equations (SPDE). The main motivation are applications to nonlinear SPDE of porous media type which also include cases where the nonlinear functions grow slowly at infinity ("fast diffusion equations"). Generally, the main problem is to find the appropriate Gelfand triple to work on. In our case Orlics spaces turn out to be convenient. We show how one must choose the defining Young function for a given nonlinearity. After presenting these applications, we shall summarize results about the qualitative behaviour of solutions and about their invariant measures.