Abstract:  Once-reinforced random walk (ORRW), rotor walk, and Kozma's centrally exited walk share the same growth exponent 2/3 for their range size in the plane (for ORRW only the lower bound is proved).In the Eulerian walker model, also called rotor walk, every site of the square lattice starts with an arrow pointing at one of its four neighbors. A walker repeatedly turns the arrow at its current site 90 degrees clockwise and steps in the new direction. Priezzhev, Dhar, Dhar, and Krishnamurthy introduced this walk in 1996 and conjectured that for independent uniform initial arrows, the radius of the range (i.e., the visited region) after the first t steps is of order t^(2/3). The lower bound is easy and general (Florescu-Levine-P., 2016); the upper bound is not.

I will explain the proof of this conjecture. We also show that the range normalized by t^{1/3} converges to a convex limit shape. Our proof extends to the Hexagonal lattice. Remaining open questions:

Is the limit shape a disk? Can the proof be extended to the Triangular lattice? 

Is the rotor walk transient in higher dimensions? 

 (Joint work with  Ahmed Bou-Rabee, UPenn).

Bio: Yuval Peres obtained his PhD in 1990 from the Hebrew University, Jerusalem. He was a postdoctoral fellow at Stanford and Yale, and was then a Professor of Mathematics and Statistics in Jerusalem and in Berkeley. Later, he was a Principal researcher at Microsoft. In 2023, he joined Beijing Institute of Mathematical Sciences and Applications. He has published more than 350 papers in most areas of probability theory, including random walks, Brownian motion, percolation, and random graphs. He has co-authored books on Markov chains, probability on graphs, game theory and Brownian motion, which can be found at https://www.yuval-peres-books.com. His presentations are available at https://yuval-peres-presentations.com. He is a recipient of the Rollo Davidson prize and the Loeve prize. He has mentored 21 PhD students including Elchanan Mossel (MIT, AMS fellow), Jian Ding (PKU, ICCM gold medal and Rollo Davidson prize), Balint Virag and Gabor Pete (Rollo Davidson prize). He was an invited speaker at the 2002 International Congress of Mathematicians in Beijing, at the 2008 European congress of Math, and at the 2017 Math Congress of the Americas. In 2016, he was elected to the US National Academy of Science.