Uniform Bessel bounds for endpoint crossings in the persistent random walk

A ball bounces down a Galton board, where its first bounce’s direction is fair while every later bounce repeats the previous bounce’s direction with probability $p$, all the way down to some bin. We ask for which $p$ the middle bin is exactly as likely as the end bins, a question originally posed by Kagey. We find that for $N$ bounces, the tie occurs when the ball changes direction (turns) approximately $\log N$ times on average. We compare the central bin with a Bessel function, with an error bound that is explicit for all $N$. This yields an explicit interval for the tie at every $N\ge 2$, rather than just for large $N$. This is done analytically for $N\ge 175$, and for smaller $N$ exactly by integer arithmetic, cross-verified in Lean. We also find the tie of every other bin with the end bins and the shift caused by periodic boundaries.