Bounds for diluted mean-fields spin glass models

关于使用 Glauber 动力学对稀释自旋玻璃进行采样的研究

On sampling diluted Spin Glasses using Glauber dynamics

本文研究稀释自旋玻璃的近似采样，针对 EA 模型在随机图 $G(n,d/n)$ 上，利用块构造和路径耦合技术，证明 Glauber 动力学在特定温度范围内具有快速混合时间，并推测该范围最优。

平均场模型与非稀疏稀释平均场模型之间的自由能等价性

Free energy equivalence between mean-field models and nonsparsely diluted mean-field models

本研究严格证明了在热力学极限下，具有伯努利边分布的非稀疏稀释平均场模型，其自由能与相应平均场模型精确等价，推广了先前关于稀释Curie--Weiss模型的结果。

Spin glass models from the point of view of spin distributions

In many spin glass models, due to the symmetry among sites, any limiting joint distribution of spins under the annealed Gibbs measure admits the Aldous-Hoover representation encoded by a function $σ:[0,1]^4\to\{-1,+1\}$, and one can think of this function as a generic functional order parameter of the model. In a class of diluted models, and in the Sherrington-Kirkpatrick model, we introduce novel perturbations of the Hamiltonian that yield certain invariance and self-consistency equations for this generic functional order parameter and we use these invariance properties to obtain representati…

Hierarchical exchangeability of pure states in mean field spin glass models

The main result in this paper is motivated by the Mézard-Parisi ansatz which predicts a very special structure for the distribution of spins in diluted mean field spin glass models, such as the random K-sat model. Using the fact that one can safely assume the validity of the Ghirlanda-Guerra identities in these models, we prove hierarchical exchangeability of pure states for the asymptotic Gibbs measures, which allows us to apply a representation result for hierarchically exchangeable arrays recently proved in arXiv:1301.1259. Comparing this representation with the predictions of the Mézard-Pa…

Central limit theorem for fluctuations of linear eigenvalue statistics of large random graphs. Diluted regime

We study the linear eigenvalue statistics of large random graphs in the regimes when the mean number of edges for each vertex tends to infinity. We prove that for a rather wide class of test functions the fluctuations of linear eigenvalue statistics converges in distribution to a Gaussian random variable with zero mean and variance which coincides with "non gaussian" part of the Wigner ensemble variance.