Spin Covariance Fluctuations in the SK Model at High Temperature

Based on \cite{H}, it is well known that the rescaled two point correlation functions $$\sqrt{N}⟨{\sigma }_i;{\sigma }_j⟩=\sqrt{N}(⟨{\sigma }_i{\sigma }_j⟩-⟨{\sigma }_i⟩⟨{\sigma }_j⟩)$$ in the Sherrington-Kirkpatrick spin glass model with non-zero external field admit at sufficiently high temperature an explicit non-Gaussian distributional limit as $N\to \mathrm{\infty }$. Inspired by recent results from \cite{ABSY, BSXY, BXY}, we provide a novel proof of the distributional convergence which is based on expanding $⟨{\sigma }_i;{\sigma }_j⟩$ into a sum over suitable weights of self-avoiding paths from vertex $i$ to $j$. Compared to \cite{H}, our key observation is that the path representation of $⟨{\sigma }_i;{\sigma }_j⟩$ provides a direct explanation of the specific form of the limiting distribution of $\sqrt{N}⟨{\sigma }_i;{\sigma }_j⟩$.