Abstract
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 $\infty$ (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.
Highlights
Introduction and definitionsGround states are local minima of the Hamiltonian defined in (1.1), i.e. the Hamiltonian (1.1) can not be lowered by flipping the spins for some finite B ⊂ V
Introduction and definitionsLet 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
We can apply the construction of the proof of Lemma 2.2 to the tree T1 instead of T and get subtrees T1,1 and T1,2 of T1 and an edge h1 ∈ E1,2, such that T1,1 is the tree connected to the root 0 and (2.3) and (2.4) hold true in T1
Summary
Ground states are local minima of the Hamiltonian defined in (1.1), i.e. the Hamiltonian (1.1) can not be lowered by flipping the spins for some finite B ⊂ V. For some edge e = (x, y), we denote the shortest path connecting the root 0 to y by Pe. A subset Π ⊂ E is called a cutset separating x and infinity if every infinite non self-intersecting path starting at x contains at least one edge in Π. For a function g : {F ⊂ E : |F | < ∞} → R we set lim inf g(Π)
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