Mathematics Colloquia and Seminars

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Description

The Sherrington-Kirkpatrick model is a "simple" model of a random spin system; i.e. an Ising model where the spin-spin couplings are random. The distribution of the random couplings is permutation-invariant, and the couplings are Gaussian-distributed. According to physicists, the model is supposedly solved by the creative, but hard-to-confirm ansatz of Giorgio Parisi. Recent developments by Francesco Guerra and others resulted in new activity, culminating with Michel Talagrand's announced proof of Parisi's ansatz. In this talk I will describe the rather simple results of Guerra and Toninelli and an equally simple generalization by Aizenman, Sims and myself, which we call an "extended variational principle". In the second part of the talk, I will explain the relation to Kingman's exchangeable partition structures, particularly for the random energy model. I will tell you a conjectured characterization I have arrived at for the Choquet simplex of random partition structures spanned by (a branch of) the Poisson-Dirichlet processes of Pitman and Yor: \{\textrm{PD}(\alpha,0) : 0 \leq \alpha \leq 1\}. This conjecture is a "REM-level" specialization of a recent conjecture of Guerra. Guerra's conjecture, if proved correct, would yield a rather intuitive proof of Parisi's ansatz a different way than Talagrand's, quite similar to Parisi's way of thinking.