  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
$\mathbb{Z}$.
