Dimensional Interpolation for Random Walk

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...

Tail estimates for one-dimensional random walk in random environment

Suppose that the integers are assigned i.i.d. random variables f! x g (taking values in the unit interval), which serve as an environment. This environment deenes a random walk fX k g (called a RWRE) which, when at x, moves one step to the right with probability ! x , and one step to the left with probability 1 ? ! x. Solomon (1975) determined the almost-sure asymptotic speed (=rate of escape) ...

Erosion by a one-dimensional random walk.

We consider a model introduced by Baker et al. [Phys. Rev. E 88, 042113 (2013)] of a single lattice random walker moving on a domain of allowed sites, surrounded by blocked sites. The walker enlarges the allowed domain by eroding the boundary at its random encounters with blocked boundary sites: attempts to step onto blocked sites succeed with a given probability and convert these sites to allo...

Random walk on a two-dimensional random environment

Universal Persistence for Local Time of One-dimensional Random Walk

We prove the power law decay p(t, x) ∼ t−φ(x,b)/2 in which p(t, x) is the probability that the fraction of time up to t in which a random walk S of i.i.d. zero-mean increments taking finitely many values, is non-negative, exceeds x throughout s ∈ [1, t]. Here φ(x, b) = P(Lévy(1/2, κ(x, b)) < 0) for κ(x, b) = √ 1−xb− √ 1+x √ 1−xb+ √ 1+x and b = bS > 0 measuring the asymptotic asymmetry between p...