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Existence of Full Replica Symmetry Breaking for the Sherrington-Kirkpatrick Model at Low Temperature

Yuxin Zhou

TL;DR

This work resolves a long-standing question about the Sherrington-Kirkpatrick model by proving the existence of full replica symmetry breaking (FRSB) just below the low-temperature threshold. The authors establish a precise structure for the Parisi measure $\mu_{\beta}$ near the critical temperature: it is supported on an interval $[0,\upsilon_{\beta}]$ with a single jump at the right endpoint, i.e., $\mu_{\beta}=\nu_{\beta}+(1-m)\delta_{\upsilon_{\beta}}$, where $\nu_{\beta}$ has full support on $[0,\upsilon_{\beta})$ and a smooth density. The proof combines the Auffinger-Chen Parisi-measure criterion with new auxiliary analytic tools ($\Gamma_{\mu}$, $G_{\mu}$, $F_{\mu}$) and a reverse Gaussian integration by parts to rule out all non-FRSB configurations in three case analyses near the critical temperature $\beta=1/\sqrt{2}$. This result confirms physics predictions about the FRSB phase in Ising spin glasses and has implications for algorithms that rely on the Parisi measure structure, showing that FRSB persists slightly beyond the high-temperature regime. The approach also provides a framework for further exploring the FRSB regime at lower temperatures in the SK model and related mixed $p$-spin models.

Abstract

We prove the existence of full replica symmetry breaking (FRSB) for the Sherrington-Kirkpatrick (SK) model at low temperature. More specifically, we prove that slightly beyond the critical temperature, the Parisi measure for the SK model is supported on an interval starting at the origin and only has one jump discontinuity at the right endpoint.

Existence of Full Replica Symmetry Breaking for the Sherrington-Kirkpatrick Model at Low Temperature

TL;DR

This work resolves a long-standing question about the Sherrington-Kirkpatrick model by proving the existence of full replica symmetry breaking (FRSB) just below the low-temperature threshold. The authors establish a precise structure for the Parisi measure near the critical temperature: it is supported on an interval with a single jump at the right endpoint, i.e., , where has full support on and a smooth density. The proof combines the Auffinger-Chen Parisi-measure criterion with new auxiliary analytic tools (, , ) and a reverse Gaussian integration by parts to rule out all non-FRSB configurations in three case analyses near the critical temperature . This result confirms physics predictions about the FRSB phase in Ising spin glasses and has implications for algorithms that rely on the Parisi measure structure, showing that FRSB persists slightly beyond the high-temperature regime. The approach also provides a framework for further exploring the FRSB regime at lower temperatures in the SK model and related mixed -spin models.

Abstract

We prove the existence of full replica symmetry breaking (FRSB) for the Sherrington-Kirkpatrick (SK) model at low temperature. More specifically, we prove that slightly beyond the critical temperature, the Parisi measure for the SK model is supported on an interval starting at the origin and only has one jump discontinuity at the right endpoint.

Paper Structure

This paper contains 14 sections, 93 equations, 2 figures.

Table of Contents

  1. Introduction and main results
  2. Background: The Ising spin glass model and Parisi measures
  3. Main Results
  4. Earlier related works
  5. More discussion about FRSB phase of the SK model
  6. Proof ideas
  7. Proof outline of Theorem \ref{['mainthm']}
  8. Properties of Parisi Measures
  9. Proof Outline of Theorem \ref{['mainthm']}
  10. Case I: $(q,q') \cap$ supp .$\mu_{\beta}=\emptyset$, $q\leq\mathcal{K}$, $q'-q>\frac{A}{3B}$ and $q,q' \in$ supp .$\mu_{\beta}$.
  11. Case II: $(q,q') \cap$ supp .$\mu_{\beta}=\emptyset$, $q\leq\mathcal{K}$, $q'-q\leq \frac{A}{3B}$ and $q,q' \in$ supp .$\mu_{\beta}$.
  12. Case III: $[\mathcal{K},1]\cap$ supp .$\mu_{\beta}\not=\emptyset$.
  13. Proof of Corollary \ref{['coromain']}
  14. Proof of Theorems \ref{['thmFderiv']} and \ref{['Fm']}

Figures (2)

  • Figure 1: Phase transitions of $\mu_{\beta}$ with respect to $\beta$ for the SK model. The phase in black are previous results ALR for $0<\beta\leq \frac{1}{\sqrt{2}}$. The phase in blue is our main results for $\frac{1}{\sqrt{2}}<\beta\leq\frac{1}{\sqrt{2}}+\eta$ in Theorem \ref{['mainthm']} and the phase in grey remains unknown.
  • Figure 2: Distributions $\alpha_{\mu_\beta}$ of Parisi measures $\mu_\beta$ for the SK model. The figures from left to right are $\alpha_{\mu_\beta}$ for $0< \beta \leq \frac{1}{\sqrt{2}}$, $\beta \rightarrow ( \frac{1}{\sqrt{2}})^+$ and $\frac{1}{\sqrt{2}}< \beta \leq \frac{1}{\sqrt{2}}+ \eta$, respectively.

Theorems & Definitions (5)

  • proof : Proof of Corollary \ref{['coroFderiv']}
  • proof : Proof of Corollary \ref{['coromain']}
  • proof : Proof of Theorem \ref{['Fm']}
  • proof : Proof of Theorem \ref{['thmFderiv']}
  • proof : Proof of Lemma \ref{['lemmaGIP']}