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A simple model of a limit order book

We formulate a simplified model of a limit order book, in which the arrival process is independent of the current state. We prove a phase transition result: there exist prices $κ_b$ and $κ_a$ such that, for any $ε> 0$, only finitely many bid (ask) departures occur at prices below $κ_b-ε$ (above $κ_a+ε$), while the interval $(κ_b+ε, κ_a - ε)$ infinitely often contains no bids, and infinitely often contains no asks. We derive expressions for $κ_b$ and $κ_a$, which we solve in the case of uniform arrivals. We conjecture the positive recurrence of a modified model, and find the steady-state distribution of the highest bid and of the lowest ask assuming the positive recurrence.