Embedded Markov chain at level exits

Suppose motion within each level ends on hitting one of two boundaries, and each boundary transition resets the process to a fixed entry state of an adjacent level. Observe only the successive level exits. The resulting level process is a Markov chain, with transition probabilities equal to the within-level boundary-hitting probabilities. The fixed-entry condition matters: without it, retaining only the level may discard information needed for the next transition law.

For a simple symmetric random walk on local rooms $0,\dots ,r$, starting each excursion at room $1$, the gambler's ruin probability of reaching $r$ first is $1/r$. Thus the level chain moves up with probability $1/r$ and down with probability $(r-1)/r$. Independent future decisions, or more generally the Strong Markov property, justify restarting this calculation after each exit.

For two active levels between absorbing levels $0$ and $3$, starting on level $1$, put $p=1/r$ and $q=1-p$. The success probabilities satisfy $u=pv$ and $v=p+qu$, henceFor $r=4$, the result is $1/13$. This compression illustrates how unbiased microscopic choices can produce a biased coarse random walk when entry points are close to one boundary.