The microstructure of ultrametricity

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Abstract
The hierarchical organization of the pure states of a S.K. spin glass (ultrametricity) is analysed in terms of self-averaging distributions of local magnetizations. We show that every pure state α defines an ultrametric distance Dα(i, j) among the N sites. Given two states α, β with overlap q there is a minimum distance dm such that for two sites i, j with Dα(i, j) ≽ d m the two distances Dα and D β coincide. It follows that the sites can be partitioned in disjoint cells inside which the total magnetization is the same for all the states with mutual overlap q. For this same family of states we then define an « ancestor » that has, inside each cell, constant local magnetization equal to the average magnetization of the descendants. The ancestors satisfy mean field like equations. The functional dependence of the local magnetization in terms of the local field is given by the solution of the diffusion equation in x space which is given a purely static interpretation

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