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Universal scaling limits for spin networks via martingale methods

Simone Franchini

TL;DR

The paper addresses the universal scaling limits of fully connected spin networks with pairwise interactions, including disordered models such as the Sherrington–Kirkpatrick (SK) model. It develops a martingale-based large-deviations framework and a blanket (layer) representation that partitions the network into layers and separates core and interface contributions. It shows that in the fully connected limit the interface-dominated IO model admits a universal scaling limit expressible in terms of magnetisation eigenstates, parameterised by the magnetisation density $m$ and an $L^2$-dimensional field $h$, with connections to REM universality and Parisi-type variational principles. The results yield concrete Curie–Weiss and SK examples and provide a versatile framework bridging statistical physics with Bayesian/Active Inference formalisms and neural architectures.

Abstract

We use simple martingale methods to construct a large deviation theory of spin systems with pairwise interactions. As an application, we show that the fully connected case obeys a universal scaling limit that is just a product of magnetisation eigenstates.

Universal scaling limits for spin networks via martingale methods

TL;DR

The paper addresses the universal scaling limits of fully connected spin networks with pairwise interactions, including disordered models such as the Sherrington–Kirkpatrick (SK) model. It develops a martingale-based large-deviations framework and a blanket (layer) representation that partitions the network into layers and separates core and interface contributions. It shows that in the fully connected limit the interface-dominated IO model admits a universal scaling limit expressible in terms of magnetisation eigenstates, parameterised by the magnetisation density and an -dimensional field , with connections to REM universality and Parisi-type variational principles. The results yield concrete Curie–Weiss and SK examples and provide a versatile framework bridging statistical physics with Bayesian/Active Inference formalisms and neural architectures.

Abstract

We use simple martingale methods to construct a large deviation theory of spin systems with pairwise interactions. As an application, we show that the fully connected case obeys a universal scaling limit that is just a product of magnetisation eigenstates.

Paper Structure

This paper contains 11 sections, 110 equations, 1 figure.

Table of Contents

  1. The general model
  2. Thermodynamic limit
  3. Free energy principle
  4. Blanket (or Layer) representation
  5. Blankets and conditional expectations
  6. Blanket internal structure
  7. Fully connected models
  8. General theory of the IO model
  9. Free energy functional
  10. Example 1: Curie--Weiss model
  11. Example 2: Sherrington--Kirckpatrick model

Figures (1)

  • Figure 1: A comparison between the notion of a Markov blanket as is usually intended (e.g., by Pearl Judea_Pearl and Friston Friston) versus the “ layer” representation of FranchiniSPA2021FranchiniSPA2023FranchiniRSBwR2023. As one can see from the figures, the second case is just the interaction picture of the first: the blanket is made of vertices while the layer is made of edges. For this paper we will not make the distinction and refer to both as Markov blankets. This is done to facilitate the understanding with other fields that already use this concept.