### Berkeley Probability Seminar, Spring 2010

 Date Speaker Title Abstract 20 Jan. Dimitri Shlyakhtenko (UCLA) Random Matrices and Free Stochastic Calculus We discuss applications of free stochastic calculus (a free probability analog of stochastic calculus). These include applications to random matrix theory as well as questions around L^2 cohomology of discrete groups. Based in part on joint work with A. Guionnet. 27 Jan. Lauren Williams (UCB Math) A Combinatorial Approach to the Asymmetric Exclusion Process (ASEP) The ASEP is a model from statistical mechanics which describes a system of interacting particles hopping left and right on a one-dimensional lattice of n sites with open boundaries. In the bulk, the rate of hopping left is q times the rate of hopping right, and particles may enter and exit at both sides of the lattice at rates alpha, beta, gamma, and delta. We introduce some new tableaux (staircase tableaux) and use them to describe the stationary distribution of the ASEP with all parameters general. In special cases (gamma=delta=0) we can also "lift" the dynamics of the ASEP to a process on the tableaux themselves. Finally, using work of Uchiyama-Sasamoto-Wadati, we use our tableaux to give a combinatorial formula for moments of the Askey-Wilson weight function. This is joint work w/ Sylvie Corteel. 3 Feb. Gerónimo Uribe Bravo (UCB Stat) A Lamperti Type Representation of Continuous State Branching Processes with Immigration Continuous-state branching process with immigration (CBI) form the class of limits of sequences of Galton-Watson process with immigration which are accelerated and normalized. In 1968, Lamperti stated that if immigration is not allowed, these process are in one to one correspondence with Lévy processes with no-negative jumps through a time change but provided no proof. This puzzling bijection has some heuristic explanations which will help us survey the strategies for proofs of the Lamperti representation of CB processes. We will then give a simple relationship between the breadth-first walk associated to a tree and the sequence of its generation sizes to motivate a relationship between CBI processes and Lévy processes which generalizes the Lamperti representation of CB processes and gives another proof of it. Finally, we will discuss some consequences of this Lamperti type representation. This talk will refer to the article: Proof(s) of the Lamperti representation of Continuous-State Branching processes, Probability Surveys 6, 2009 (with Ma. Emilia Caballero and Amaury Lambert) as well as work in progress with Ma. Emilia Caballero and José Luis Pérez. 10 Feb. No seminar. 17 Feb. Ben Morris (UC Davis) Improved Mixing Time Bounds for the Thorp Shuffle The Thorp shuffle is defined as follows. Cut the deck into two equal piles. Drop the first card from the left pile or the right pile according to a fair coin flip; then drop from the other pile. Continue this way until both piles are empty. We show that the mixing time for the Thorp shuffle with 2^d cards is O(d^3). This improves on the best known bound of O(d^4). 24 Feb. Atilla Yilmaz (UCB Math) Large Deviations for Random Walk in a Random Environment I will talk about large deviations for nearest-neighbor random walk in an i.i.d. environment on $Z^d$. There exist variational formulae for the quenched and the averaged rate functions $I_q$ and $I_a$, obtained by Rosenbluth and Varadhan, respectively. $I_q$ and $I_a$ are not identically equal. However, when $d\geq4$ and the walk satisfies the so-called (T) condition of Sznitman, they are equal on an open set $A_{eq}$. For every $\xi$ in $A_{eq}$, there exists a positive solution to a Laplace-like equation involving $\xi$ and the original transition kernel of the walk. This solution lets us define a new transition kernel via the h-transform technique of Doob. This new kernel corresponds to the unique minimizer of Varadhan's variational formula at $\xi$. It also corresponds to the unique minimizer of Rosenbluth's variational formula provided that the latter is slightly modified. In other words, when the limiting average velocity of the walk is conditioned to be equal to $\xi$, the walk chooses to tilt its original transition kernel by an h-transform. 3 Mar. Partha Dey (UCB Stat) Central Limit Theorem for First-Passage Percolation Across Thin Cylinders We consider first-passage percolation on the graph $\mathbb {Z} \times \{ - h_n, -h_n+1, \ldots, h_n \}^{d-1}$ for fixed $d\ge 2$ where each edge has an i.i.d.~nonnegative weight. The passage time for a path is defined as the sum of weights of all the edges in that path and the first-passage time between two vertices is defined as the minimum passage time over all paths joining the two vertices. We will show that the first-passage time $T_n$ between the origin and the vertex $(n,0,\ldots,0)$ satisfies a Gaussian CLT as long as $h_n=o (n^{\alpha})$ with $\alpha < 1/(d+1)$. The proof will be based on a decomposition of $T_n$ as a sum of independent random variables, certain moment estimates and a renormalization type argument. Joint work with Sourav Chatterjee. 10 Mar. Subhankar Ghosh (USC) Concentration of Measures via Size-biased Couplings Abstract here (PDF). 17 Mar. James Propp (UMass Lowell) Rotor walks and Markov chains For any discrete Markov chain one can construct deterministic “rotor-router” analogues. Such an analogue typically has many of the same properties as the random process it mimics but is more sharply concentrated around its average-case behavior. I will discuss the general theory of derandomization of Markov chains via rotor-routers, as well as the particular example of walk in Z2. Let p be the probability that a random walk in Z2 that walks from source vertex (0,0) until it hits the finite target set B stops at a particular vertex b in B. If one performs N successive runs of a suitable rotor-router walk in Z2 from (0,0) to B, the number of runs that stop at b is Np plus or minus O(log N). This is joint work with Alexander Holroyd. 24 Mar. Spring break - no seminar. 31 Mar. Dan Romik (UC Davis) Arctic Circles, Random Domino Tilings and Square Young Tableaux It is well-known that domino tilings and certain other random combinatorial or statistical-physics models on a two-dimensional lattice exhibit a spatial phase transition between an "arctic" or "frozen" region where the behavior of the random object is asymptotically deterministic and a "temperate" region where truly random behavior is observed. One famous example of such a phenomenon is the Arctic Circle Theorem due to Jockusch, Propp and Shor, which shows that for uniformly random domino tilings of a particular region known as the Aztec Diamond, the curve that forms the interface between the frozen and temperate regions converges to a circle. Cohn, Elkies and Propp later derived a more detailed result about the limiting height function of the typical domino tiling of the Aztec diamond. In this talk, I will present a new proof of the Cohn-Elkies-Propp limit shape result for the height function based on a connection to alternating-sign matrices and a variational analysis. The proof highlights a surprising connection of this result to another arctic-circle type phenomenon observed in a different problem involving uniformly random square Young tableaux. 7 Apr. Mykhaylo Shkolnikov (Stanford) On Competing Particle Systems The talk will deal with processes of gaps between competing particles on the real line and half-line. Such evolutions were studied first by Ruzmaikina and Aizenman (2005) in the context of the Sherrington-Kirkpatrick model of spin glasses and later by Chatterjee and Pal (2007) and Pal and Pitman (2008) in the context of models for the capital distribution in a financial market. I will give several extensions of their results and discuss applications to financial mathematics and queueing theory. 14 Apr. Arnab Sen (UCB Stat) Coalescing Systems of Non-Brownian Particles A well-known result of Arratia shows that one can make rigorous the notion of starting an independent Brownian motion at every point of an arbitrary closed subset of the real line and then building a set-valued process by requiring particles to coalesce when they collide. Arratia noted that the value of this process will be almost surely a locally finite set at all positive times, and a finite set almost surely if the initial value is compact. We investigate whether such instantaneous coalescence still occurs for coalescing systems of particles where either the state space of the individual particles is not locally homeomorphic to an interval or the sample paths of the individual particles are discontinuous. We show that Arratia's conclusion is valid for Brownian motions on the Sierpinski gasket and for stable processes on the real line with stable index greater than one. Joint work with Steve Evans and Ben Morris. 21 Apr. Nathan Ross (UCB Stat) Geometric Approximation via the Equilibrium Distribution Transformation In this talk we will show how to use the equilibrium distribution of an integer valued random variable in order to establish bounds on the total variation distance to a geometric distribution. The key tool is a new formulation of Stein's method in the spirit of the recent work of Peköz and Röllin for exponential approximation. We will first discuss the theorem and then apply it to obtain an error in the geometric approximation of compound geometric sums and also the generation size of a critical Galton-Watson tree conditioned on non-extinction. This is joint work with Erol Peköz and Adrian Röllin. 28 Apr. Jason Miller (Stanford) Universality for SLE(4) We resolve a conjecture of Sheffield that SLE(4), a conformally invariant random curve, is the universal limit of the chordal zero-height contours of random surfaces with isotropic, uniformly convex potentials. Time permitting, we will also explain how the estimates developed for this work can be used to prove a new central limit theorem for linear functionals of such models.