Uniqueness and non-uniqueness for spin-glass ground states on trees

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 ∞ (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

Universal exploration dynamics of random walks

“…In the context of symmetric Markovian RWs characterized by a fractal dimension 𝑑 f and a walk dimension 𝑑 w , it was shown in Refs. [5,26,27,28] that there exist two main different classes of exploration: either 𝜇 ≤ 1, and the RW visits every site infinitely often, or 𝜇 > 1, and the RW will never visit some sites. The first type of RWs is said to be recurrent, while the second type is said to be transient.…”

On Absence of disorder chaos for spin glasses on $\mathbb{Z} ^{d}$

“…The theorem was first proved by Chatterjee [8] in the case of Gaussian couplings with an explicit decay in t. Namely, if J(t) and J (t) are two Ornstein-Uhlenbeck processes both starting at J(0) then evolving independently, he proved for some c > 0 that (5) E[Q Λ (σ(J(t)), σ(J (t)))] ≥ c 4d 2 e −t/(4d 2 c) . The result is to be compared to the Sherrington-Kirkpatrick model on the complete graph with Gaussian coupling for which the average overlap goes to 0 as Λ → Z d for any fixed t [8] (the proof there is given at positive temperature, but the result is expected to hold at zero temperature as well).…”

“…This is still an important open question to be resolved related to the existence and the nature of the spin glass phase transition in finite dimension. The situation is more tractable for the model on trees, see [5], and on the half-plane, see [3,2]. We expect that, at least for d = 2, the critical droplets of all edges have finite size (uniformly in Λ), in which case a stronger version of Theorem 1.1 should hold:…”