We present an exact algorithm for finding all the ground states of the two-dimensional Edwards-Anderson +/-J spin glass and characterize its performance. We investigate how the ground states change with increasing system size and antiferromagnetic bond ratio x. We find that the ground-state distribution is log normal and that its dependence on x is nontrivial. We also study the breakdown of ferromagnetic order and show how percolating paths are created by the overlap of different ground-state solutions.
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