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We consider random walks on the nonnegative integers in a space-time dependent random environment. We assume that transition probabilities are given by independent 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 aarXiv:2201.07270v3 fatcat:oz4xmr52fbb2temx6svsjdx27i