## Probability Seminar

**Date:** October 9, 2017

**Time:** 10:00AM - 4:00PM

**Location:** BLOC 220

**Speaker:** Eviatar Procaccia and Yuan Zhang, TAMU

**Title:** *On covering monotonic paths with simple random walk*

**Abstract:** We study the probability that a $d$ dimensional simple random walk (or the first $L$ steps of it) covers each point in a nearest neighbor path connecting 0 and the boundary of an $L_1$ ball. We show that among all such paths, the one that maximizes the covering probability is the monotonic increasing one that stays within distance 1 from the diagonal. As a result, we can obtain an exponential upper bound on the decaying rate of covering probability of any such path when dâ‰¥4 and a $\log$ correction for $d=3$. Interesting conjectures and open questions will be presented. The talk will be split between Eviatar and Yuan. Eviatar will present the background and Yuan the combinatorics part.