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Further results on the free energy of the random field Ising chain in the case of centered disorder

Orphée Collin

TL;DR

This work analyzes the RFIC with centered IID disorder in the strong-coupling regime ($J\to\infty$) and derives a refined expansion for the limiting free energy ${\mathcal F}(J)$ beyond the first-order term. By representing the partition function as a product of random matrices and connecting ${\mathcal F}(J)$ to the Lyapunov exponent, the authors first determine the limiting maximal energy density ${\mathcal M}(J)$ and then obtain a second-order correction ${\frac{\vartheta^2}{(2J)^2}}$ with a computable constant ${\widetilde{\kappa}}$ tied to the edge-process $Z$. They provide an explicit expression for the universal denominator constant ${\widehat{\kappa}}$ in terms of ladder heights of an associated random walk, and show that the complete expansion involves a non-universal subtraction ${\widehat{\kappa}}- {\widetilde{\kappa}}$ in the main denominator, clarifying how edge effects contribute to non-universality. The results sharpen previous first-order findings (e.g., CGGH25) by delivering explicit constants and a rigorous decomposition via ${\Gamma}$-extrema, with methods rooted in renewal theory, Brownian scaling limits, and invariant measures for reduced edge processes. This advances understanding of non-universal corrections in disordered spin chains and informs precise asymptotics for RFIC free energies.

Abstract

We study the expansion of the limiting free energy density of the random field Ising chain with centered IID disorder and homogeneous coupling parameter, when the latter goes to infinity. We extend the first order result of [Collin, 2025] to the general case of finite second moment. Furthermore, we identify the value of the unknown constant appearing in the main theorem of [Collin, Giacomin, Greenblatt, Hu, 2025], which is the first non-universal constant in the expansion of the limiting free energy density.

Further results on the free energy of the random field Ising chain in the case of centered disorder

TL;DR

This work analyzes the RFIC with centered IID disorder in the strong-coupling regime () and derives a refined expansion for the limiting free energy beyond the first-order term. By representing the partition function as a product of random matrices and connecting to the Lyapunov exponent, the authors first determine the limiting maximal energy density and then obtain a second-order correction with a computable constant tied to the edge-process . They provide an explicit expression for the universal denominator constant in terms of ladder heights of an associated random walk, and show that the complete expansion involves a non-universal subtraction in the main denominator, clarifying how edge effects contribute to non-universality. The results sharpen previous first-order findings (e.g., CGGH25) by delivering explicit constants and a rigorous decomposition via -extrema, with methods rooted in renewal theory, Brownian scaling limits, and invariant measures for reduced edge processes. This advances understanding of non-universal corrections in disordered spin chains and informs precise asymptotics for RFIC free energies.

Abstract

We study the expansion of the limiting free energy density of the random field Ising chain with centered IID disorder and homogeneous coupling parameter, when the latter goes to infinity. We extend the first order result of [Collin, 2025] to the general case of finite second moment. Furthermore, we identify the value of the unknown constant appearing in the main theorem of [Collin, Giacomin, Greenblatt, Hu, 2025], which is the first non-universal constant in the expansion of the limiting free energy density.

Paper Structure

This paper contains 22 sections, 14 theorems, 176 equations.

Table of Contents

  1. Introduction
  2. Overview
  3. Definitions and basic facts
  4. Theorems
  5. Results concerning the limiting maximal energy density
  6. Result concerning the free energy
  7. Consequences and links with the existing literature
  8. Structure of the paper
  9. Maximal energy configuration
  10. Process of $\Gamma$-extrema associated to the random walk $S$
  11. Statistics of the $\Gamma$-stretches for large $\Gamma$ and proof of Theorem \ref{['th:first_order_calM']}
  12. Environment around a $\Gamma$-extremum
  13. Proof of Theorem \ref{['th:dev_calF']}
  14. Heuristic
  15. Lower bound
  16. ...and 7 more sections

Key Result

Lemma 1

Assume that $\mu$ has a finite first moment. For fixed $J\in(0, \infty)$ and $(g,d)\in\{+1, -1\}^2$, for almost every realization of the disorder sequence $h$ (i.e., $\mathbb{P}$-almost surely), there is convergence as $N\to\infty$ of the free energy density and of the maximal energy density towards deterministic quantities denoted respectively ${\mathcal{F}}(J)$ and $\mathcal{M}(J)$, which depe

Theorems & Definitions (33)

  • Lemma 1
  • proof
  • Remark 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Corollary 6
  • Corollary 7
  • Remark 8
  • proof
  • ...and 23 more