  Motivated by empirical data, we develop a statistical description of the
queue dynamics for large tick assets based on a two-dimensional Fokker-Planck
(diffusion) equation, that explicitly includes state dependence, i.e. the fact
that the drift and diffusion depends on the volume present on both sides of the
spread. "Jump" events, corresponding to sudden changes of the best limit price,
must also be included as birth-death terms in the Fokker-Planck equation. All
quantities involved in the equation can be calibrated using high-frequency data
on best quotes. One of our central finding is the the dynamical process is
approximately scale invariant, i.e., the only relevant variable is the ratio of
the current volume in the queue to its average value. While the latter shows
intraday seasonalities and strong variability across stocks and time periods,
the dynamics of the rescaled volumes is universal. In terms of rescaled
volumes, we found that the drift has a complex two-dimensional structure, which
is a sum of a gradient contribution and a rotational contribution, both stable
across stocks and time. This drift term is entirely responsible for the
dynamical correlations between the ask queue and the bid queue.
