The talk will present some interesting combinatorial and geometrical symmetries of the distribution of random walks on domains with boundary. Naively, a random walk of length n consists of n consecutive "moves" (up, down, left, or right) on the lattice of integer points in the plane. Let Δ be an infinite domain in the Euclidean plane with boundary ∂Δ. I will give explicit formulas for the probability of a random walk to exit Δ at a precise point on ∂Δ. The justification of such formulas presents meaningful connections with linear algebra, probability, geometry, and real analysis. The talk is geared toward an undergraduate audience and exemplifies a possible "capstone" topic for our curriculum.

This is joint work with Jamie Brandon, currently at Brandeis University.