The Sherrington-Kirkpatrick model: A challenge for mathematicians
This is the first in a series of papers by the author in which he attacks a variety of models for disordered systems from statistical mechanics and/or combinatorial optimisation by rigourous probabilistic methods. Such models are `solved' in the physics literature by powerful methods, before all the so-called ``replica trick'', but no mathematical justification of these techniques and their results has so far been found. Rigorous results on these systems are sparse [see the collection of review papers in \textit{A. Bovier, P. Picco} (eds.), ``Mathematical aspects of spin glasses and neural networks'' (Progress in Probability 41, Boston, 1998)[, and the present paper contains some remarkable new contributions in this directions. The model considered here is the most classical model of a mean field spin glass, the Sherrington-Kirkpatrick model. This model is defined through the Hamiltonian function \(H_N\) on the set \(\{-1\}^N\) given by \[ H_N(\sigma)=-\sum_{1\leq i<j\leq N} J_{ij}\sigma_i\sigma_j +h\sum_{1\leq i\leq N} \sigma_i, \] where \(J_{ij}\) are a family of independent standard normal variables. An object of particular concern is the partition function \[ Z_N(\overline b,h)\equiv \sum_{\sigma\in\{-1\}^N}\exp(-\beta H_N(\sigma)). \] The parameters \(\overline b\) and \(h\) are called the inverse temperature and the magnetic field, respectively. While completely understood on the heuristic level [see the book by \textit{M. Mézar, G. Parisi} and \textit{M. A. Virasoro}, ``Spin glass theory and beyond'' (Singapore, 1987)] prior rigorous results were only available for the case \(h=0\) and \(\overline b\leq 1\), where \textit{M. Aizenman, J. L. Lebowitz} and \textit{D. Ruelle} [Commun. Math. Phys. 112, 3-20 (1987)] proved that \[ \lim_{N\uparrow\infty} -\tfrac 1 N\ln Z_N(\overline b,0) = \overline b^2/2-\ln 2 \] and that \(Z_N(\overline b,0)/E Z_N(\overline b,0)\) converges to a log-Gaussian random variable. In the present paper a number of additional results is proven: Theorem 1.2 replaces the central limit theorem of Aizenman, Lebowitz and Ruelle (loc. cit.) by exponential upper and lower bounds on the deviations of \(\ln Z_N(\overline b,0)\) from \(-\overline b^2/4\). More importantly, Theorem 1.4 gives control over exponential moments of the overlap parameter between two independent copies of the spins (for the same realisation of the Gaussian couplings which allows to assert that this overlap is ``zero'' on the macroscopic scale). Basically this says that the spins in the high temperature phase behave (in some weak sense) like independent ones. Theorem 1.7 qualifies this observation further: it says that there the average length of the projection of a spin configuration on any unit vector in \(R^N\) is not macroscopically large. Theorems 1.8 and 1.9 give comparison results on the marginal of the Gibbs measures on finite with the uniform measure. The most striking results of the paper, however, are contained in the last two theorems. They concern the case \(h\neq 0\) and the validity of the so-called ``replica symmetric solution'' of the physicists. The point here is that in the small \(\overline b\) regime, for \(h\) not too large (in a region bounded by the so-called Almeida-Thouless line), the replica method yields a solution that is still relatively simple (this regime is in fact considered trivial by physicists). Here (Theorem 1.10) it is proven that there is at least a non-trivial subset of this regime in which the solution obtained by this method is indeed the correct one. An overview of some subsequent work by the author is given by him [Doc. Math., J. DMV, Extra Vol. ICM Berlin 1998, vol. I, 507-536 (1998; Zbl 0902.60089)].
scientific article; zbMATH DE number 1129463