Jonathon Peterson

Quenched limits for transient, zero speed one-dimensional random walk in random environment.

We consider a nearest-neighbor, one dimensional random walk {X n } n≥0 in a random i.i.d. environment, in the regime where the walk is transient but with zero speed, so that X n is of order n S for some s 0 and → 0 for x ≤ 0 (a spread out regime).

Weak quenched limiting distributions for transient one-dimensional random walk in a random environment

J. Peterson was partially supported by National Science Foundation grant DMS-0802942. G. Samorodnitsky was partially supported by ARO grant W911NF-10-1-0289 and NSF grant DMS-1005903 at Cornell University

A Probabilistic Proof of a Binomial Identity

Abstract We give an elementary probabilistic proof of a binomial identity. The proof is obtained by computing the probability of a certain event in two different ways, yielding two different expressions for the same quantity.

Current fluctuations of a system of one-dimensional random walks in random environment.

We study the current of particles that move independently in a common static random environment on the one-dimensional integer lattice. A two-level fluctuation picture appears. On the central limit scale the quenched mean of the current process converges to a Brownian motion. On a smaller scale the current process centered at its quenched mean converges to a mixture of Gaussian processes. These Gaussian processes are similar to those arising from classical random walks, but the environment makes itself felt through an additional Brownian random shift in the spatial argument of the limiting current process.

Upper and Lower Bounds on the Speed of a One Dimensional Excited Random Walk

Excited random walks (ERWs) are a self-interacting non-Markovian random walk in which the future behavior of the walk is influenced by the number of times the walk has previously visited its current site. We study the speed of the walk, defined as

Excited Random Walks with Non-nearest Neighbor Steps

V = \lim_{n \rightarrow \infty} \frac{X_n}{n}

Hydrodynamic limit for a system of independent, sub-ballistic random walks in a common random environment

where

Oscillations of quenched slowdown asymptotics for ballistic one-dimensional random walk in a random environment

X_n

The contact process on the complete graph with random vertex-dependent infection rates

is the state of the walk at time

Limiting distributions and large deviations for random walks in random environments

n