Exotic states in long-range spin glasses

We consider Ising spin glasses on Z^d with couplings J_xy = c(y x)Z(x,y) , where the c(y)'s are nonrandom real coefficients and the Z(x,y)'s are independent, identically distributed random variables with E[Z(x,y) ] = 0 and Var[Z(x,y) ] = 1. We prove that if c(y) is a diverging, square summable sequence, then (with probability one) there are uncountably many (infinite volume) ground states σ, each of which has the following property: for any temperature T < oo, there is a Gibbs state supported entirely on (infinite volume) spin configurations which differ from σ only at finitely many sites. This and related results are examples of the bizarre effects that can occur in disordered systems with coupling-dependent boundary conditions.