The Edwards-Anderson model on Z^d corresponds to the Ising model where the couplings are random, e.g. independent standard Gaussians. Unlike the Ising model, the behavior of the model at low temperature is widely misunderstood from a rigorous (and non-rigorous) point of view for d greater or equal to 2. A related open problem is to determine the number of ground states of the model, i.e. the number of configurations minimizing the energy (in a suitable sense). This optimization problem is intricate due to the presence of both ferromagnetic and antiferromagnetic couplings. In this talk, I will review the possible scenarios at low temperature and explain recent rigorous results on the uniqueness of the ground state (zero-temperature case) on the half-plane for d=2. This is joint work with M. Damron, C. Newman, and D. Stein.
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