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Rigorous low-temperature results for the mean field \(p\)-spins interaction model. (English) Zbl 1016.82021

The title of this paper is very explicit: it does provide rigorous low-temperature results for the mean field \(p\)-spins interaction model. “Mean field” disordered models have been introduced to study the spin distribution of atoms submitted to disordered long range interaction. Objects of interests are random Gibbs measures of Ising spins on the \(N\)-dimensional hypercube. The \(p\)-spin model itself comes from a \(p\)-body random interaction, which couplings are i.i.d standard Gaussian random variables. If this model is on one hand an extension of the SK model \((p=2)\), it also provides physically relevant results for large \(p\). The SK model is well understood at high \(T\) but not much is known for low temperature and even its predicted behavior is very complicated. The \(p\)-spins predictions tends to simplify at large \(p\) and this paper provides new rigorous results for large \(p\) and low temperature, where the dependence of the Gibbs measure with the Hamiltonian is higher.
The first result (Theorem 1.1) consists in a sharp estimate of the critical inverse temperature \(\beta_p\). It is then used to formalize some physicists’ predictions on the structure of the random Gibbs measures. Replica techniques and overlaps (renormalized inner products on the skew configuration space) are introduced and it appears that the study of the dependence structure of the Gibbs measures reduces to the study of the distribution of the overlap of replicas. The author expresses physicist’s prediction and proves that for \(\beta < \beta_p\), the overlaps are essentially zero (i.e. the Gibbs measures are essentially product measures), but also that for \(\beta>\beta_p\) but not too large, the overlaps essentially takes two values, 0 and \(q\), where \(q\) is a complicated function of \(\beta\) independent of the disorder.
Once these distincts behaviors have been settled, the author devotes the rest of the paper to extract a huge amount of mathematical information on the low temperature phases, in the so-called ‘spin-glass domain’. It is first stated that for \(p\) large and \(\beta\) not much larger that \(\beta_p\), the overlaps are in general either close to zero or close to one (Theorem 1.3). This result is again in agreement with a physicist’s prediction, the breaking of ergodicity at low temperature, expressed here by the fact that Gibbs measures are sums of smaller unrelated pieces, called lumps. This picture is described rigorously in Theorems 1.4 and 1.5. The Gibbs measures have a part that is very spread out and (possibly many) parts that are supported by small sets almost orthogonal to each other. Roughly speaking, it means that almost the only way for two configurations to be correlated is to belong to the same lump. The author focuses then on the construction and distribution of these lumps and formalizes further on the physicist’s prediction and the replica method, under a mild assumption on the weights distribution of the lumps.
In a note added at the end of this paper and in another publication, the author has announced that he handled a proof of this assumption (hypothesis 1.7). Under this hypothesis, the lumps are described rigorously as pure equilibrium states and a relevant macroscopic quantity, the average quadratic magnetization, is identified.
The results are summarized in a complete and pleasant way in Section 1 and the other sections contain the different constructions and results. Some results of interest about the moment of the overlaps when \(p=4\) are given in the last section.

MSC:

82B44 Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics
82D30 Statistical mechanics of random media, disordered materials (including liquid crystals and spin glasses)
60K35 Interacting random processes; statistical mechanics type models; percolation theory
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
82B26 Phase transitions (general) in equilibrium statistical mechanics
60G15 Gaussian processes
60E15 Inequalities; stochastic orderings
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