Pierluigi Contucci, Shannon Starr
Using a correlation inequality of Contucci and Lebowitz for spin glasses, we demonstrate existence of the thermodynamic limit for short-ranged spin glasses, under weaker hypotheses than previously available, namely without the assumption of the annealed bound.
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We study a multi-species spin glass system where the density of each species is kept fixed at increasing volumes. The model reduces to the Sherrington-Kirkpatrick one for the single species case. The existence of the thermodynamic limit is proved for all densities values under a convexity condition on the interaction. The thermodynamic properties of the model are investigated and the annealed, the replica symmetric and the replica symmetry breaking bounds are proved using Guerra's scheme. The annealed approximation is proved to be exact under a high temperature condition. We show that the repl…
Strong Topological Trivialization of Multi-Species Spherical Spin Glasses
本文研究多种类球形自旋玻璃的临界点景观，确定了退火平凡化相边界并证明其等价于淬火强拓扑平凡化，进而推导出低温下Langevin动力学的对数混合时间，方法基于Kac-Rice公式和变分问题求解。
We study aspects of the thermodynamics of quantum versions of spin glasses. By means of the Lie-Trotter formula for exponential sums of operators, we adapt methods used to analyze classical spin glass models to answer analogous questions about quantum models.
This paper characterizes the annealed complexity of bipartite spherical spin glasses, both pure and mixed. This means we give exact variational formulas for the asymptotics of the expected numbers of critical points and of local minima. This problem was initially considered by [Auffinger, Chen 2014], who gave upper and lower bounds on this complexity. We find two surprising connections between pure bipartite and pure single-species spin glasses, which were studied by [Auffinger, Ben Arous, Černý 2013]. First, the local minima of any pure bipartite model lie primarily in a low-energy band, simi…
We evaluate the high temperature limit of the free energy of spin glasses on the hypercube with Hamiltonian $H_N(σ) = σ^T J σ$, where the coupling matrix $J$ is drawn from certain symmetric orthogonally invariant ensembles. Our derivation relates the annealed free energy of these models to a spherical integral, and expresses the limit of the free energy in terms of the limiting spectral measure of the coupling matrix $J$. As an application, we derive the limiting free energy of the Random Orthogonal Model (ROM) at high temperatures, which confirms non-rigorous calculations of Marinari et al. (…
A Gaussian Convexity for Logarithmic Moment Generating Functions with Applications in Spin Glasses
本文证明了对数矩生成函数关于参数λ的凸性，并由此改进了高斯凸函数的下偏差不等式，给出了SK自旋玻璃模型中DFM猜想的定量界，并证明了负副本退火自由能的可微性。
We consider two non-mean-field models of structural glasses built on a hierarchical lattice. First, we consider a hierarchical version of the random energy model (HREM), and we prove the existence of the thermodynamic limit and self-averaging of the free energy. Furthermore, we prove that the infinite-volume entropy is positive in a high-temperature region bounded from below, thus providing an upper bound on the Kauzmann critical temperature. In addition, we show how to improve this bound by leveraging the hierarchical structure of the model. Finally, we introduce a hierarchical version of the…
The Thermodynamic Limit of Short-Range Spin Glasses with Periodic Boundary Conditions
本文证明周期边界条件下最近邻Edwards-Anderson Ising模型在离散环面上的淬火热力学极限存在，且自由能密度几乎必然自平均，仅需有限一阶矩假设，并给出有限体积修正的定量界。
In this paper we study the properties of the quenched pressure of a multi-layer spin-glass model (a deep Boltzmann Machine in artificial intelligence jargon) whose pairwise interactions are allowed between spins lying in adjacent layers and not inside the same layer nor among layers at distance larger than one. We prove a theorem that bounds the quenched pressure of such a K-layer machine in terms of K Sherrington-Kirkpatrick spin glasses and use it to investigate its annealed region. The replica-symmetric approximation of the quenched pressure is identified and its relation to the annealed on…
We prove a correlation type inequality for spin systems with quenched symmetric random interactions. This gives monotonicity of the pressure with respect to the strength of the interaction for a class of spin glass models. Consequences include existence of the thermodynamic limit for the pressure and bounds on the surface pressure. We also describe other conjectured inequalities for such systems.
We consider the free energy difference restricted to a finite volume for certain pairs of incongruent thermodynamic states (if they exist) in the Edwards-Anderson Ising spin glass at nonzero temperature. We prove that the variance of this quantity with respect to the couplings grows proportionally to the volume in any dimension greater than or equal to two. As an illustration of potential applications, we use this result to restrict the possible structure of Gibbs states in two dimensions.
We comment on recent numerical experiments by G.Hed and E.Domany [cond-mat/0608535v2] on the quenched equilibrium state of the Edwards-Anderson spin glass model. The rigorous proof of overlap identities related to replica equivalence shows that the observed violations of those identities on finite size systems must vanish in the thermodynamic limit. See also the successive version cond-mat/0608535v4
We study the free energy of a mean-field spin glass whose coupling distribution has power law tails. Under the assumption that the couplings have infinite variance and finite mean, we show that the thermodynamic limit of the quenched free energy exists, and that the free energy is self-averaging.
We provide a very simple proof for the existence of the thermodynamic limit for the quenched specific pressure for classical and quantum disordered systems on a $d$-dimensional lattice, including spin glasses. We develop a method which relies simply on Jensen's inequality and which works for any disorder distribution with the only condition (stability) that the quenched specific pressure is bounded.
We prove a duality principle that connects the thermodynamic limits of the free energies of the Hamiltonians and their squared interactions. Under the main assumption that the limiting free energy is concave in the squared temperature parameter, we show that this relation is valid in a large class of disordered systems. In particular, when applied to mean field spin glasses, this duality provides an interpretation of the Parisi formula as an inverted variational principle, establishing a prediction of Guerra.
We analyze the coupled non-linear integro-differential equations whose solutions is the thermodynamical limit of the empirical correlation and response functions in the Langevin dynamics for spherical p-spin disordered mean-field models. We provide a mathematically rigorous derivation of their FDT solution (for the high temperature regime) and of certain key properties of this solution, which are in agreement with earlier derivations based on physical grounds.
Overlap Structure and Free Energy Fluctuations in Short-Range Spin Glasses
针对短程自旋玻璃低温相的纯态混合问题，构造受边重叠范围约束的新型亚稳态，并证明任意维度下混合 Gibbs 态内纯态间自由能差涨落方差随体积线性发散。
Local independence in mean-field spin glasses
本文提出多空腔位点的严格空腔方法，证明平均场自旋玻璃中局部独立性与薄壳、重叠集中等复本对称条件等价，并统一涵盖 SK 与 Gardner 模型。
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