Abstract
The chapter discusses the random cluster representation for spin glasses and how it might be used to rigorously prove the existence of a phase transition in the Edwards and Anderson (EA) spin glass. Highly disordered spin glasses and their relation to invasion percolation and minimal spanning trees are discussed. The chapter discusses on partial proof of the conjecture that there is only a single pair of infinite-volume ground states in the two-dimensional EA model. The highly disordered model discussed in the previous section has an interesting ground state structure however; its relevance to realistic models is unknown. There are, as of yet, no rigorous results on ground state pair that suggests that in 2D there is only a single pair of “incongruent” ground states.
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