A. Gandolfi

Uniqueness of the infinite component in a random graph with applications to percolation and spin glasses

SummaryWe extend the theorem of Burton and Keane on uniqueness of the infinite component in dependent percolation to cover random graphs on ℤd or ℤd × ℕ with long-range edges. We also study a short-range percolation model related to nearest-neighbor spin glasses on ℤd or on a slab ℤd × {0,...K} and prove both that percolation occurs and that the infinite component is unique forV=ℤ2×{0,1} or larger.

On the uniqueness of the infinite cluster in the percolation model

We simplify the recent proof by Aizenman, Kesten and Newman of the uniqueness of the infinite open cluster in the percolation model. Our new proof is more suitable for generalization in the direction of percolation-type processes with dependent site variables.

Zero-Temperature Dynamics of ± J Spin Glasses and Related Models

Abstract: We study zero-temperature, stochastic Ising models σt on Zd with (disordered) nearest-neighbor couplings independently chosen from a distribution μ on R and an initial spin configuration chosen uniformly at random. Given d, call μ type ℐ (resp., type ℱ) if, for everyx in Zd, σxt flips infinitely (resp., only finitely) many times as t→∞ (with probability one) – or else mixed type ℳ. Models of type ℒ and ℳ exhibit a zero-temperature version of “local non-equilibration”. For d=1, all types occur and the type of any μ is easy to determine. The main result of this paper is a proof that for d=2, ±J models (where μ=αδJ+(1-α)δ-J) are type ℳ, unlike homogeneous models (type ℐ) or continuous (finite mean) μs (type ℳ). We also prove that all other noncontinuous disordered systems are type ℳ for any d≥ 2. The ±J proof is noteworthy in that it is much less “local” than the other (simpler) proof. Homogeneous and ±J models for d≥ 3 remain an open problem.

Extremal two-correlations of two-valued stationary one-dependent processes

SummaryThe maximal value of the two-correlation for two-valued stationary one-dependent processes with fixed probability α of a single symbol is determined. We show that the process attaining this bound is unique except when α=1/2, when there are exactly two different processes. The analogous problem for minimal two-correlation is discussed, and partial results are obtained.

Exotic states in long-range spin glasses

AbstractWe consider Ising spin glasses onZd with couplingsJxy=cy−xZxy, where thecys are nonrandom real coefficients and theZxys are independent, identically distributed random variables withE[Zxy]=0 andE[Zxy2]=1. We prove that if ∑y|cy|=∞ while ∑y|cy|2=∞, then (with probability one) there are uncountably many (infinite volume) ground statesn

On the Uniqueness of the Infinite Occupied Cluster in Dependent Two- Dimensional Site Percolation

tilde sigma

Greedy lattice animals: negative values and unconstrained maxima

n, each of which has the following property: forany temperatureT<∞, there is a Gibbs state supported entirely on (infinite volume) spin configurations which differ fromn