  We consider random walkers that deform the medium as they move, enabling a
faster motion in regions which have been recently visited. This induces an
effective attraction between walkers mediated by the medium, which can be
regarded as a space metric, giving rise to a statistical mechanics toy model
either for gravity, motion through deformable matter or adaptable geometry. In
the strong-deformability regime, we find that diffusion is initially described
by the porous medium equation, thus yielding subdiffusive behavior of an
initially localized cloud of particles. Indeed, while the average width of a
single cloud will sustain a $\sigma\sim t^{1/2}$ growth, the combined width of
the whole ensemble will grow like $\sigma\sim t^{1/3}$ in a certain time
regime. This difference can be accounted for by the strong correlations between
the particles, which we explore indirectly through the fluctuations of the
center of mass of the cloud and the expected value of the experienced density,
defined as the average density measured by the particles themselves.

Statistical Mechanics