INTERDISCIPLINARY STOCHASTIC PROCESSES COLLOQUIUM Tuesday March 18, room 60 Evans, 4.10 - 5.00pm Speaker: Amir Dembo (Mathematics and Statistics Departments, Stanford University) Title: Fractal geometry of simple random covering: late and favorite points Abstract: A simple random walk on large d-dimensional lattice torus visits most points within a negligible fraction of the cover time of the torus. When d>2 the exceptional sets of (p-late) points not visited till a fraction p of the cover time behave like samples of uniformly independent drawing. However, for d=2 they exhibit a surprisingly subtle fractal structure, with clustering of p-late points in mesoscopic discs centered at a p-late point, and way more p-late pairs within prescribed mesoscopic distances than attributed to discs around typical p-late points. A site x on the d-dimensional lattice is p-favorite if the simple random walk visits x during its first N>>1 steps at least a fraction p of the number of visits it makes to the most frequently visited site by that time. The fractal geometry of the set of p-favorite points has a lot to do with that of p-late points. As a demonstration we find the growth rate of the largest p-favorite ball, of the largest completely k-covered ball, and of other related objects. Again the picture is more interesting for d=2 than for d>2. This talk is based on joint works with Yuval Peres, Jay Rosen and Ofer Zeitouni.