  We study an annealed model of Uniform Infinite Planar Quadrangulation (UIPQ)
with an infinite two-sided self-avoiding walk (SAW), which can also be
described as the result of glueing together two independent uniform infinite
quadrangulations of the half-plane (UIHPQs). We prove a lower bound on the
displacement of the SAW which, combined with estimates from our previous paper,
shows that the self-avoiding walk is diffusive. As a byproduct this implies
that the volume growth exponent of the lattice in question is $4$ (as is the
case for the standard UIPQ); nevertheless, using our previous work we show its
law to be singular with respect to that of the standard UIPQ, that is -- in the
language of statistical physics -- the fact that disorder holds.
