  We revisit the "epsilon-intelligence" model of Toth et al.(2011), that was
proposed as a minimal framework to understand the square-root dependence of the
impact of meta-orders on volume in financial markets. The basic idea is that
most of the daily liquidity is "latent" and furthermore vanishes linearly
around the current price, as a consequence of the diffusion of the price
itself. However, the numerical implementation of Toth et al. was criticised as
being unrealistic, in particular because all the "intelligence" was conferred
to market orders, while limit orders were passive and random. In this work, we
study various alternative specifications of the model, for example allowing
limit orders to react to the order flow, or changing the execution protocols.
By and large, our study lends strong support to the idea that the square-root
impact law is a very generic and robust property that requires very few
ingredients to be valid. We also show that the transition from super-diffusion
to sub-diffusion reported in Toth et al. is in fact a cross-over, but that the
original model can be slightly altered in order to give rise to a genuine phase
transition, which is of interest on its own. We finally propose a general
theoretical framework to understand how a non-linear impact may appear even in
the limit where the bias in the order flow is vanishingly small.
