The Sherrington-Kirkpatrick model
نویسندگان
چکیده
The Sherrington-Kirkpatrick (SK) model was introduced by David Sherrington and Scott Kirkpatrick in 1975 as a simple ‘solvable’ (in their words) model for spin-glasses. Spin-glasses are some type of magnetic alloys, and ‘solvable’ meant that the asymptotic free entropy density could be computed exactly. It turns out that the original SK solution was incorrect and in fact inconsistent (the authors knew this). A consistent conjecture for the asymptotic free energy per spin was put forward by Giorgio Parisi in 1982, and derived through the non-rigorous replica method. It took 24 years to prove this conjecture. The final proof is due to Michel Talagrand (2006) and is a real tour de force. In these two lectures we will prove that the asymptotic free entropy density exists and that it is upper bounded by the Parisi formula. The first result is due to Francesco Guerra and Fabio Toninelli [GT02], and the second to Guerra [Gue03]. They are based on an interpolation trick that was a authentic breakthrough eventually leading to Talagrand’s proof.
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In [11] Michel Talagrand gave a rigorous proof of the Parisi formula in the classical Sherrington-Kirkpatrick (SK) model. In this paper we build upon the methodology developed in [11] and extend Talagrand’s result to the class of SK type models in which the spins have arbitrary prior distribution on a bounded subset of the real line.
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