We consider a spin glass at temperature T = 0 where the underlying graph is a locally finite tree. We prove for a wide range of coupling distributions that uniqueness of ground states is equivalent to the maximal flow from any vertex to ∞ (where each edge e has capacity |J e |) being equal to zero which is equivalent to recurrence of the simple random walk on the tree.Let G = (V, E) be a locally finite graph, for a given finite set B ⊂ V define E(B) as the set of edges with at least one end in B. For any finite setIn the Edwards-Anderson spin glass model [8] one considers nearest neighbor interactions and the case where the J xy ′ s, also called the couplings, are i.i.d. random variables. The distribution of a single coupling will be denoted by ν, throughout we assume that ν({0}) = 0. With ν E we denote the distribution of (J xy , (x, y) ∈ E). We call an edge e = (x, y) ∈ E satisfied for a configuration σ if J xy σ x σ y > 0, if J xy σ x σ y < 0 we call it unsatisfied. Note that ν E (∃e ∈ E such that J xy = 0) = 0. Ground states are local minima of the Hamiltonian defined in (1), i.e. the Hamiltonian (1) can not be lowered by flipping the spins for some finite B ⊂ V . This means
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