Spin Covariance Fluctuations in the SK Model at High Temperature

Based on {H}, it is well known that the rescaled two point correlation functions \[ √N σ_i ; σ_j = √N ( σ_i σ_j - σ_i σ_j) \] 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→ ∞. Inspired by recent results from {ABSY, BSXY, BXY}, we provide a novel proof of the distributional convergence which is based on expanding σᵢ ; σⱼ into a sum over suitable weights of self-avoiding paths from vertex i to j. Compared to {H}, our key observation is that the path representation of σᵢ ; σⱼ provides a direct explanation of the specific form of the limiting distribution of √N σᵢ ; σⱼ.