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A simplified Parisi Ansatz II: REM universality

Simone Franchini

TL;DR

This work develops a replica-free framework for analyzing the Sherrington–Kirkpatrick model by partitioning the system into layers and approximating interfaces with a Random Energy Model (REM). It establishes REM universality for Gaussian noise across all temperatures by introducing a J-field and a renormalized J' field, and then connects finite-temperature Gibbs averages to an REM-PPP description with a temperature-dependent interpolation parameter λ. The paper provides detailed thermodynamic-limit analysis, including quantization of field fluctuations, magnetization-eigenstate structure, and the entropy of overlaps, and it shows how REM-like behavior emerges beyond the low-temperature limit. These results offer a replica-free route to the full Parisi functional and illuminate deep connections between REM universality and spin-glass phase structure, with potential implications for neural networks and related disordered systems.

Abstract

In a previous work [A simplified Parisi Ansatz, Franchini, S., Commun. Theor. Phys., 73, 055601 (2021)] we introduced a simple method to compute the Random Overlap Structure of Aizenmann, Simm and Stars and the full RSB Parisi formula for the Sherrington-Kirkpatrick Model without using replica theory. The method consists in partitioning the system into smaller sub-systems that we call layers, and iterate the Bayes rule. A central ansatz in our derivation was that these layers could be approximated by Random Energy Models of the Derrida type. In this paper we analyze the properties of the interface in detail, and show the equivalence with the Random Energy Model at any temperature.

A simplified Parisi Ansatz II: REM universality

TL;DR

This work develops a replica-free framework for analyzing the Sherrington–Kirkpatrick model by partitioning the system into layers and approximating interfaces with a Random Energy Model (REM). It establishes REM universality for Gaussian noise across all temperatures by introducing a J-field and a renormalized J' field, and then connects finite-temperature Gibbs averages to an REM-PPP description with a temperature-dependent interpolation parameter λ. The paper provides detailed thermodynamic-limit analysis, including quantization of field fluctuations, magnetization-eigenstate structure, and the entropy of overlaps, and it shows how REM-like behavior emerges beyond the low-temperature limit. These results offer a replica-free route to the full Parisi functional and illuminate deep connections between REM universality and spin-glass phase structure, with potential implications for neural networks and related disordered systems.

Abstract

In a previous work [A simplified Parisi Ansatz, Franchini, S., Commun. Theor. Phys., 73, 055601 (2021)] we introduced a simple method to compute the Random Overlap Structure of Aizenmann, Simm and Stars and the full RSB Parisi formula for the Sherrington-Kirkpatrick Model without using replica theory. The method consists in partitioning the system into smaller sub-systems that we call layers, and iterate the Bayes rule. A central ansatz in our derivation was that these layers could be approximated by Random Energy Models of the Derrida type. In this paper we analyze the properties of the interface in detail, and show the equivalence with the Random Energy Model at any temperature.
Paper Structure (26 sections, 149 equations, 1 figure)

This paper contains 26 sections, 149 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Summary
  3. The interface model
  4. Thermodynamic limit
  5. REM universality
  6. Thermodynamic limit
  7. Field fluctuations
  8. Uniform distribution
  9. Half-Gaussian distribution
  10. Series analysis of the Half-Gaussian
  11. Convergence to the ground state
  12. Eigenstates of magnetization
  13. Lattice gas
  14. Entropy of the overlap
  15. Urn methods
  16. ...and 11 more sections

Figures (1)

  • Figure 1: Two configurations $X$ (first row) and $Y_{0}$ (second row) extracted from $\mathcal{V}\left(\epsilon\right)$, reordered in such way that both $X$ and $X\cap Y_{0}$ are compact sets. The last row shows the partition into the disjoint non-overlapping components $X'$, $Y'_{0}$ and the common component $X\cap Y_{0}$ projected on $V$ (last row).