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Article Dans Une Revue Communications in Mathematical Physics Année : 2023

Random walk on nonnegative integers in beta distributed random environment

Résumé

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 $\mathbb{Z}$.

Dates et versions

hal-03766758 , version 1 (01-09-2022)

Identifiants

Citer

Guillaume Barraquand, Mark Rychnovsky. Random walk on nonnegative integers in beta distributed random environment. Communications in Mathematical Physics, 2023, 398 (2), pp.823-875. ⟨10.1007/s00220-022-04536-1⟩. ⟨hal-03766758⟩
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