论文标题
在beta分布式随机环境中的非负整数上随机步行
Random walk on nonnegative integers in beta distributed random environment
论文作者
论文摘要
我们考虑在时空依赖的随机环境中随机步行。我们假设过渡概率由独立$ \ mathrm {beta}（μ，μ）$分布式随机变量给出，边界处具有特定行为，由额外的参数$η$控制。我们表明，该模型是可以解决的，并且证明了随机热核的混合力矩的公式。然后，我们提供两个公式，使我们能够研究大规模行为。第一个涉及Fredholm Pfaffian，我们用来证明局部限制定理描述边界参数$η$如何影响回报概率。第二个是一系列的一系列积分，我们表明，非矛盾的临界点渐近学表明，在随机环境中，此半空间随机行走的较大偏差行为与$ \ mathbb {Z} $上的类似随机步行相同。
We consider random walks on the nonnegative integers in a space-time dependent random environment. We assume that transition probabilities are given by independent $\mathrm{Beta}(μ,μ)$ distributed random variables, with a specific behaviour at the boundary, controlled by an extra parameter $η$. 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 $η$ 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 $\mathbb{Z}$.
