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Probability that a modified $c$-bounded lattice walk is also $c$-bounded

Let $\mathcal{B}_n^{(c)}$ be the set of $n$ step lattice bridges (starts at $(0,0)$ ends at $(n,0)$), taking steps $\{(1,1), (-1,1)\}$ bounded between $y = c$ and $y = -c$ for a constant $c \geq 0$. F …