Hawkes processes provide a natural framework to model dependencies between the intensities of point processes. In the context of order-driven financial markets, the relevance of such dependencies has been amply demonstrated from an empirical, as well as theoretical, standpoint. In this work, we build on previous empirical and numerical studies and introduce a mathematical model of limit order books based on Hawkes processes with exponential kernels. After proving a general stationarity result, we focus on the long-time behaviour of the limit order book and the corresponding dynamics of the suitably rescaled price. A formula for the asymptotic (in time) volatility of the price dynamics induced by that of the order book is obtained, involving the average of functions of the various order book events under the stationary distribution.