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

2011/06/02 by Eldan, Ronen
#FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1106.0470

Abstract

We derive asymptotics for the probability of the origin to be an extremal point of a random walk in Rⁿ. 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.

Related

• Limit theorems for projections of random walk on a hypersphere
• Cover Times for Brownian Motion and Random Walks in Two Dimensions
• On the separation cut-off phenomenon for Brownian motions on high dimensional spheres
• On the first hitting time of a high-dimensional orthant
• Kac's Walk on n-sphere mixes in nlog n steps