Random walk on nonnegative integers in beta distributed random environment

Abstract

We consider random walks on the nonnegative integers in a space-time dependent random environment. We assume that transition probabilities are given by independent $\mathrm{Beta}(\mu ,\mu )$ distributed random variables, with a specific behaviour at the boundary, controlled by an extra parameter $\eta$. We show that this model is exactly solvable and prove a formula for the mixed moments of the random heat kernel. We then provide two formulas that allow us to study the large-scale behaviour. The first involves a Fredholm Pfaffian, which we use to prove a local limit theorem describing how the boundary parameter $\eta$ affects the return probabilities. The second is an explicit series of integrals, and we show that non-rigorous critical point asymptotics suggest that the large deviation behaviour of this half-space random walk in random environment is the same as for the analogous random walk on $\mathbb{Z}$.

arXiv comment44 pages. v3: minor edits

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• The crossover from the Macroscopic Fluctuation Theory to the Kardar-Parisi-Zhang equation controls the large deviations beyond Einstein's diffusion
• First Passage Time for Many Particle Diffusion in Space-Time Random Environments
• Boundary current fluctuations for the half space ASEP and six vertex model
• Markov duality and Bethe ansatz formula for half-line open ASEP