Extremal points of high dimensional random walks and mixing times of a Brownian motion on the sphere

We derive asymptotics for the probability of the origin to be an extremal point of a random walk in R^n. We show that in order for the probability to be roughly 1/2, the number of steps of the random walk should be between e^{c n / log n}$ and e^{C n log n}. As a result, we attain a bound for the ?pi/2-covering time of a spherical brownian motion.Comment: 22 Page

Diffusion-limited aggregation on the hyperbolic plane

We consider an analogous version of the diffusion-limited aggregation model defined on the hyperbolic plane. We prove that almost surely the aggregate viewed at time infinity will have a positive density.Comment: Published at http://dx.doi.org/10.1214/14-AOP928 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Pointwise Estimates for Marginals of Convex Bodies

We prove a pointwise version of the multi-dimensional central limit theorem for convex bodies. Namely, let X be an isotropic random vector in R^n with a log-concave density. For a typical subspace E in R^n of dimension n^c, consider the probability density of the projection of X onto E. We show that the ratio between this probability density and the standard gaussian density in E is very close to 1 in large parts of E. Here c > 0 is a universal constant. This complements a recent result by the second named author, where the total-variation metric between the densities was considered.Comment: 17 page