AI 中文总结
本文推导了非均匀随机图上伊辛模型的淬火标度极限，将其应用于贝叶斯神经网络与因果推断，得到了相关极限结果与渐近性质。
AI 中文摘要
本文在高温区域，对由图论（graphon，涵盖稠密图与稀疏图）生成的非均匀随机图上伊辛模型的线性泛函与经验自旋场，推导了淬火标度极限。首先，证明了自旋构型有限集合线性统计量的联合中心极限定理（CLT），其极限协方差由关联图论积分算子的预解式刻画。基于该结果，建立了平均磁化强度及以合适正则检验函数类为索引的自旋场的泛函中心极限定理，进一步证明了完整经验自旋场作为负 Sobolev 空间中随机广义函数的收敛性。这些标度极限可应用于贝叶斯神经网络与因果推断：对于前者，推导了输出层符号依赖伊辛模型的两层贝叶斯神经网络的无限宽度高斯过程极限；对于后者，在网络干扰下建立了平均处理效应的 Hájek 估计量的渐近正态性。
英文摘要
In this paper, we derive quenched scaling limits for linear functionals and the empirical spin field of Ising models on inhomogeneous random graphs generated by a graphon (encompassing both dense and sparse graphs), in the high-temperature regime. We first prove a joint central limit theorem (CLT) for finite collections of linear statistics of the spin configurations, where the limiting covariance is characterized by the resolvent of the associated graphon integral operator. Building on this result, we establish functional CLTs for the average magnetization and for the spin field indexed by suitable classes of regular test functions. We further prove convergence of the full empirical spin field, viewed as a random generalized function in negative Sobolev spaces. These scaling limits provide applications to both Bayesian neural networks and causal inference. Specifically, for the former, we derive infinite-width Gaussian-process limits for two-layer Bayesian neural networks with Ising-dependent output-layer signs, while for the latter, we establish the asymptotic normality of Hájek estimators for average treatment effects under network interference.
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