Limiting operations for sequences of quantum random variables and a convergence theorem for quantum martingales

We study quantum random variables and generalize several classical limit results to the quantum setting. We prove a quantum analogue of Lebesgue's dominated convergence theorem and use it to prove a quantum martingale convergence theorem. This quantum martingale convergence theorem is of particular interest since it exhibits non-classical behaviour; even though the limit of the martingale exists and is unique, it is not explicitly identifiable. However, we provide a partial classification of the limit through a study of the space of all quantum random variables having quantum expectation zero.Comment: 11 pages, 0 figure

Estimates of random walk exit probabilities and application to loop-erased random walk

We prove an estimate for the probability that a simple random walk in a simply connected subset A of Z^2 starting on the boundary exits A at another specified boundary point. The estimates are uniform over all domains of a given inradius. We apply these estimates to prove a conjecture of S. Fomin in 2001 concerning a relationship between crossing probabilities of loop-erased random walk and Brownian motion.Comment: 26 pages, 0 figure