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We compare several growth models on the two dimensional lattice. In some models, like internal DLA and rotor-router aggregation, the scaling limits are universal; in particular, starting from a point source yields a disk. In the abelian sandpile, particles are added at the origin and whenever a site has four particles or more, the top four particles topple, with one going to each neighbor. Despite similarities to other models, for the sandpile, the intriguing pattern that arises is not circular and depends on the particular lattice. A scaling limit exists for the sandpile, as was recently shown by Pegden and Smart, but it is not universal and is still mysterious. This research has been greatly influenced by pictures of the relevant sets, which I will show in the talk. They suggest a connection to conformal mapping which has not been established yet. (Talk based on joint works with Lionel Levine.)

A planar set that contains a unit segment in every direction is called a Kakeya set. These sets have been studied intensively in geometric measure theory and harmonic analysis since the work of Besicovich (1928); we find a new connection to game theory and probability. A hunter and a rabbit move on an n-vertex cycle without seeing each other until they meet. At each step, the hunter moves to a neighboring vertex or stays in place, while the rabbit is free to jump to any node. Thus they are engaged in a zero sum game, where the payoff is the capture time. We show that every rabbit strategy yields a Kakeya set; the optimal rabbit strategy is based on a discretized Cauchy random walk, and it yields a Kakeya set K consisting of 4n triangles, that has minimal area among such Kakeya sets (the area of K is of order 1/log(n)). Passing to the scaling limit yields a simple construction of a random Kakeya set with zero area from two Brownian motions. (Joint work with Y. Babichenko, R. Peretz, P. Sousi and P. Winkler.)

Yuval Peres

Yuval Peres is a Principal Researcher in the Theory group at Microsoft Research. Before joining MSR in 2006, he was a Professor at UC Berkeley. He has also taught at Yale and at the Hebrew University.

Yuval has published more than 200 papers with 100 co-authors and has mentored 19 PhD theses. His research encompasses many areas of probability theory, including random walks, Brownian motion, percolation, point processes and random graphs as well as connections with Ergodic Theory, PDEs, Combinatorics, Fractals, and Theoretical Computer Science. He has recently co-authored books on Markov chains and mixing times, on zeros of Gaussian analytic functions, and on Brownian motion. Yuval is a fellow of the AMS and a recipient of the Rollo Davidson Prize. In 2001 he received the Loeve Prize, awarded once every two years to a leading probabilist. Yuval was an invited speaker at the International Congress of Mathematics (2002) and in the European Congress of Mathematics (2008). His favorite quote is from his son Alon, who was overheard at age 6 asking a friend: "Leo, do you have a religion? You know, a religion, like Christian, or Jewish, or Mathematics....?"

DeLong Lecture Series