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.