Ground States of Disordered Ferromagnets

Abstract

When $\mathop J\nolimits_e^\Lambda = \mathop J\nolimits_e$ (i.e., with no dependence on Λ), an infinite volume ground state for the couplings $\left\{ {\mathop J\nolimits_e :e \in \mathop E\nolimits^d } \right\}$ may be defined as any $s \in S = {\left\{ { - 1, + 1} \right\}^{{Z^d}}}$, such that there exists some sequence ${\bar s^{\left( L \right)}} \in {S_{\partial {\Lambda _L}}}$ (the restriction to cubes Λ _L and to fixed b.c. ’s involves no loss of generality) and some ground state ${s^{\left( L \right)}} \in {S_{{\Lambda _L}}}{\rm{for}}\mathop H\nolimits_{{\Lambda _L}}^{{{\bar s}^{\left( L \right)}}}$ such that for each $x \in {Z^d},{s_x} = \mathop s\nolimits_x^{\left( L \right)}$, for all large L.