Conditioned Two-Dimensional Simple Random Walk: Green’s Function and Harmonic Measure

Multifractal Nature of Two Dimensional Simple Random Walk Paths

The multifractal spectrum of discrete harmonic measure of a two dimensional simple random walk path is considered. It is shown that the spectrum is the same as for Brownian motion, is nontrivial, and can be given in terms of a quantity known as the intersection exponent.

Conditioned One-way Simple Random Walk and Combinatorial Representation Theory

A one-way simple random walk is a random walk in the quadrant Z+ whose increments are elements of the canonical base. In relation with representation theory of Lie algebras and superalgebras, we describe the law of such a random walk conditioned to stay in a closed octant, a semi-open octant, or other types of semi-groups. The combinatorial representation theory of these algebras allows us to d...

Two-dimensional quantum random walk

We analyze several families of two-dimensional quantum random walks. The feasible region (the region where probabilities do not decay exponentially with time) grows linearly with time, as is the case with one-dimensional QRW. The limiting shape of the feasible region is, however, quite different. The limit region turns out to be an algebraic set, which we characterize as the rational image of a...

Random walk on a two-dimensional random environment

Heavy Points of a d-dimensional Simple Random Walk

For a simple symmetric random walk in dimension d ≥ 3, a uniform strong law of large numbers is proved for the number of sites with given local time up to time n. AMS 2000 Subject Classification: Primary 60G50; Secondary 60F15, 60J55.