  Motivated by the growing interest in self-correcting quantum memories, we
study the feasibility of self-correction in classical lattice systems composed
of bounded degrees of freedom with local interactions. We argue that
self-correction, including a requirement of stability against external
perturbation, cannot be realized in system with broken global symmetries such
as the 2d Ising model, but that systems with local, i.e. gauge, symmetries have
the required properties. Previous work gave a three-dimensional quantum system
which realized a self-correcting classical memory. Here we show that a purely
classical three dimensional system, Wegner's 3D Ising lattice gauge model, can
also realize this self-correction despite having an extensive ground state
degeneracy. We give a detailed numerical study to support the existence of a
self-correcting phase in this system, even when the gauge symmetry is
explicitly broken. More generally, our results obtained by studying the memory
lifetime of the system are in quantitative agreement with the phase diagram
obtained from conventional analysis of the system's specific heat, except that
self-correction extends beyond the topological phase, past the lower critical
temperature.
