We develop efficient algorithms to approximate the partition function and sample from the hard-core and Potts models on lattices at sufficiently low temperatures in the phase coexistence regime. In contrast, the Glauber dynamics are known to take exponential time to mix in this regime.  Our algorithms are based on the cluster expansion and Pirogov-Sinai theory, classical tools from statistical physics for understanding phase transitions, as well as Barvinok's approach to polynomial approximation.  Joint work with Tyler Helmuth and Guus Regts.