Probability Seminar: Lionel Levine (MIT)

Start with n particles at the origin in the square grid Z2, and let each particle in turn perform simple random walk until reaching an unoccupied site. Lawler, Bramson and Griffeath proved that with high probability the resulting random set of n occupied sites is close to a disk. We show that its fluctuations from circularity are, with high probability, at most logarithmic in the radius of the disk, answering a question posed by Lawler in 1995. These logarithmic fluctuations were predicted numerically by chemical physicists in the 1980's. Joint work with David Jerison and Scott Sheffield.