The Sherrington-Kirkpatrick Model: An Overview
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The Sherrington-Kirkpatrick Model: An Overview Dmitry Panchenko
Received: 29 August 2012 / Accepted: 31 August 2012 / Published online: 18 September 2012 © Springer Science+Business Media, LLC 2012
Abstract The goal of this paper is to review some of the main ideas that emerged from the attempts to confirm mathematically the predictions of the celebrated Parisi ansatz in the Sherrington-Kirkpatrick model. We try to focus on the big picture while sketching the proofs of only a few selected results, but an interested reader can find most of the missing details in Panchenko (The Sherrington-Kirkpatrick Model, Manuscript, 2012) and Talagrand (Mean-Field Models for Spin Glasses, Springer, Berlin, 2011). Keywords Sherrington-Kirkpatrick model · Parisi ansatz
1 Introduction The Sherrington-Kirkpatrick Model In 1975, Sherrington and Kirkpatrick [37] introduced a mean field model for a spin glass—a disordered magnetic alloy that exhibits unusual magnetic behavior. Given a configuration of N Ising spins, σ = (σ1 , . . . , σN ) ∈ ΣN = {−1, +1}N , the Hamiltonian of the model is given by N 1 HN (σ ) = √ gij σi σj , N i,j =1
(1)
where (gij ) are i.i.d. standard Gaussian random variables, collectively called the disorder of the model. The fact that the distribution of HN (σ ) is invariant under the permutations of the coordinates of σ is called the symmetry between sites, which is what one usually
The author is partially supported by NSF grant. D. Panchenko () Texas A&M University, Mailstop 3386, College Station, TX, 77843, USA e-mail: [email protected]
The Sherrington-Kirkpatrick Model: An Overview
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understands by a mean field model. The Hamiltonian (1) is a Gaussian process with the covariance 2 N N 1 2 1 1 1 2 1 1 2 2 2 EHN σ HN σ = σ σ σ σ =N σ σ = N R1,2 (2) N i,j =1 i j i j N i=1 i i that depends on the spin configurations σ 1 , σ 2 only through their normalized scalar product R1,2 =
N 1 1 2 1 1 2 σ ·σ = σ σ , N N i=1 i i
(3)
called the overlap of σ 1 and σ 2 . Since the distribution of a Gaussian process is determined by its covariance, it is not surprising that the overlaps play a central role in the analysis of the model. One can also consider a generalization of the Sherrington-Kirkpatrick model, the so-called mixed p-spin model, which corresponds to the Hamiltonian βp HN,p (σ ) (4) HN (σ ) = p≥1
given by a linear combination of pure p-spin Hamiltonians HN,p (σ ) =
1 N (p−1)/2
N
gi1 ...ip σi1 · · · σip ,
(5)
i1 ,...,ip =1
where the random variables (gi1 ...ip ) are standard Gaussian, independent for all p ≥ 1 and all (i1 , . . . , ip ). Similarly to (2), it is easy to check that the covariance is, again, a function of the overlap, βp2 x p . (6) EHN σ 1 HN σ 2 = N ξ(R1,2 ), where ξ(x) = p≥1
One usually assumes that the coefficients (βp ) decrease fast enough to ensure that the process is well defined when the sum in (4) includes infinitely many terms. The model may also include the external field term h(σ1 + · · · + σN ) with the external field parameter h ∈ R.
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