  The laws of Physics are time-reversible, making no qualitative distinction
between the past and the future -- yet we can only go towards the future. This
apparent contradiction is known as the `arrow of time problem'. Its resolution
states that the future is the direction of increasing entropy. But entropy can
only increase towards the future if it was low in the past, and past low
entropy is a very strong assumption to make, because low entropy states are
rather improbable, non-generic. Recent works, however, suggest we can do away
with this so-called `past hypothesis', in the presence of reversible dynamical
laws featuring expansion. We prove that this is the case for a toy model, set
in a 1+1 discrete spacetime. It consists in graphs upon which particles
circulate and interact according to local reversible rules. Some rules locally
shrink or expand the graph. Generic states always expand; entropy always
increases -- thereby providing a local explanation for the arrow of time.
