Revisiting Trade-sign Long-memory and Square-root Law price impact

Starting with a coupled discrete reaction--diffusion formulation for the lit and latent order books with non-uniformly sampled event times and meta-order source terms we show how two familiar market-microstructure regularities can emerge from this framework: the long-memory of trade signs associated with the Lillo--Mike--Farmer (LMF) theory and the square-root law (SQRL) of meta-order impact. This uses the well known locally linear order book and constant participation-rate execution in the front dynamics to reduce the dynamics to a Volterra equation whose leading-order solution then yields the well know result of concave impact trajectory, and a completion impact proportional to the square root of the meta-order size. We then re-derive the heavy-tailed Pareto-distributed meta-order lengths induce a power-law trade-sign autocorrelation through the interface source term to discuss the interface representation. These are familiar derivations, what is slightly different here is that we reinterpret these known derivations to make it clear that LMF law is an event-time sign-memory statement, whereas the square-root law is a physical-time viability statement where non-uniform event waiting times can alter the calendar-time impact trajectories depending on the mappings and interpolation used to set continuum operational time.