  We analyze several high dimensional generalizations of the toric code at
nonzero temperature. We find that in large enough dimension, there can be a
distinct separation between the critical temperature $T_c$, given by
thermodynamic singularities, and the percolation temperature $T_p$, given by
the percolation of defects. We argue that the regime $T_p<T<T_c$ is a range of
temperatures where a self-correcting quantum memory can operate despite having
percolating defects. We present analytic arguments and numerical evidence in
support of this scenario, including a mean-field treatment and Monte Carlo
simulations. Near $T_c$, simulations observe a large hysteretic behavior, which
may have applications by allowing the self-correcting phase to survive in a
"superheated" regime.
