i consider a markov process inspired by a toy model of a limit order book. "bid" and "ask" orders arrive in time; the prices are iid uniform on [0,1]. (i'll discuss some extensions.) when a match is possible (bid > ask), the highest bid and lowest ask leave the system. this process turns out to have surprising dynamics, with three limiting behaviours occurring with probability one. at low prices (< 0.21...), bids eventually never leave; at high prices (>0.78...), asks eventually never leave; and in between, the system "ought to" be positive recurrent. i will show how we can derive explicitly the limiting distribution of certain marginals for the middle prices; this makes it possible to extract the numerical values above from a 0-1 law result.