Hypergraph Counting and Fluctuations of Mixed p-Spin Glass Models at High Temperature

Abstract

Fluctuation problems in mean-field spin glass theory enjoy a rich history. In this paper, we investigate this problem in the high-temperature regime by analyzing the fluctuation of the partition function in the general mixed p-spin glass models under the weak external field assumption: . By extending the cluster expansion approach to this generic setting, we convert the fluctuation problem as a hypergraph counting problem and thus obtain a new multiple-transition phenomenon in terms of the fluctuation of the partition function. All our fluctuation results hold up to an explicit second moment threshold \(\beta _f\), which is away from the static phase transition point \(\beta _c\). Combining this with multivariate Stein’s method for exchangeable pairs, we also obtain an explicit convergence rate under appropriate moment assumptions on the general symmetric disorder coupling. Our results have several further implications. First, our approach works for both even and odd pure p-spin models. The leading cluster structures in the odd p case are different and more involved than in the even p case. This gives a combinatorial explanation for the folklore that odd p-spin is usually more complicated than even p. Second, in the mixed p-spin setting, the cluster structures differ depending on the relation between the minimum effective even and odd p-spins: \(p_e\) and \(p_o\), respectively. As an example, at \(h=0\), there are three sub-regimes: \(p_e<p_o, p_o<p_e<2p_o, p_e\geqslant 2p_o\), wherein the first and third ones, the mixed p-spin model behaves essentially like a pure p-spin model, and only in the second regime, it is more like a mixture. This presents another criterion for classifying mean-field spin glass models compared to the work of Auffinger and Ben Arous (Ann. Probab. 41 (2013), no. 6, 4214–4247), where the idea is based on complexity computations for spherical models. Third, our framework naturally implies a multi-scale fluctuation phenomenon conjectured in the work of Bovier and Schertzer (Probab. Theory Related Fields 189 (2024), no. 3–4, 771–810), in the \(h=0\) case for pure p-spin models. Our results suggest that this phenomenon holds for general mixed p-spin models under weak external fields, including \(h=0\). We also discuss the extension to general multi-species mixed p-spin models.