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Random vectors on the spin configuration of a Curie-Weiss model on Erdos-Renyi random graphs

Dominik R. Bach

TL;DR

The paper studies the asymptotic behavior of random vectors in a diluted ferromagnetic model on directed Erdős–Rényi graphs in the regime $\beta<1$ and $Np\to\infty$. It shows that any spin-vector observable with a CW limiting law remains asymptotically distributed according to the CW law when averaged over graph realizations, extending LLN/CLT results from the Curie–Weiss model to the diluted setting. The main results establish a law of large numbers for two disjoint spin groups converging to a two-point mixture at $\pm m(\beta)$ and a central limit theorem with a covariance matrix $C$ dependent on the group fractions $\alpha_1,\alpha_2$ and temperature $\beta$, with the diluted model mirroring the CW limit in probability. The work achieves this via a technical framework of integral quotients, typical/atypical spin configurations, and moment expansions that tie BG expectations to CW benchmarks, including a detailed treatment of fluctuations in the two-group case. This provides a rigorous bridge between the diluted ER graph spin system and the well-understood Curie–Weiss behavior, with potential implications for understanding fluctuations in diluted magnetic systems and related networked spin models.

Abstract

This article is concerned with the asymptotic behaviour of random vectors in a diluted ferromagnetic model. We consider a model introduced by Bovier & Gayrard (1993) with ferromagnetic interactions on a directed Erdős-Rényi random graph. Here, directed connections between graph nodes are uniformly drawn at random with a probability p that depends on the number of nodes N and is allowed to go to zero in the limit. If $Np\longrightarrow\infty$ in this model, Bovier & Gayrard (1993) proved a law of large numbers almost surely, and Kabluchko et al. (2020) proved central limit theorems in probability. Here, we generalise these results for $β<1$ in the regime $Np\longrightarrow\infty$. We show that all those random vectors on the spin configuration that have a limiting distribution under the Curie-Weiss model converge weakly towards the same distribution under the diluted model, in probability on graph realisations. This generalises various results from the Curie-Weiss model to the diluted model. As a special case, we derive a law of large numbers and central limit theorem for two disjoint groups of spins.

Random vectors on the spin configuration of a Curie-Weiss model on Erdos-Renyi random graphs

TL;DR

The paper studies the asymptotic behavior of random vectors in a diluted ferromagnetic model on directed Erdős–Rényi graphs in the regime and . It shows that any spin-vector observable with a CW limiting law remains asymptotically distributed according to the CW law when averaged over graph realizations, extending LLN/CLT results from the Curie–Weiss model to the diluted setting. The main results establish a law of large numbers for two disjoint spin groups converging to a two-point mixture at and a central limit theorem with a covariance matrix dependent on the group fractions and temperature , with the diluted model mirroring the CW limit in probability. The work achieves this via a technical framework of integral quotients, typical/atypical spin configurations, and moment expansions that tie BG expectations to CW benchmarks, including a detailed treatment of fluctuations in the two-group case. This provides a rigorous bridge between the diluted ER graph spin system and the well-understood Curie–Weiss behavior, with potential implications for understanding fluctuations in diluted magnetic systems and related networked spin models.

Abstract

This article is concerned with the asymptotic behaviour of random vectors in a diluted ferromagnetic model. We consider a model introduced by Bovier & Gayrard (1993) with ferromagnetic interactions on a directed Erdős-Rényi random graph. Here, directed connections between graph nodes are uniformly drawn at random with a probability p that depends on the number of nodes N and is allowed to go to zero in the limit. If in this model, Bovier & Gayrard (1993) proved a law of large numbers almost surely, and Kabluchko et al. (2020) proved central limit theorems in probability. Here, we generalise these results for in the regime . We show that all those random vectors on the spin configuration that have a limiting distribution under the Curie-Weiss model converge weakly towards the same distribution under the diluted model, in probability on graph realisations. This generalises various results from the Curie-Weiss model to the diluted model. As a special case, we derive a law of large numbers and central limit theorem for two disjoint groups of spins.
Paper Structure (16 sections, 11 theorems, 25 equations)

This paper contains 16 sections, 11 theorems, 25 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Description of the model
  4. Main results
  5. Technical preparation
  6. Results from previous work
  7. Technical preparation for the proofs of proposition \ref{['lem:moment convergence-Np']}
  8. Proof of the main results
  9. Convergence of integral quotients
  10. Definitions and propositions
  11. Preparation: expanding the expectations
  12. Proof of proposition \ref{['lem:moment convergence-Np']}, part 1
  13. Proof of proposition \ref{['lem:moment convergence-Np']}, part 2
  14. Convergence of bounded integrals
  15. Convergence in distribution (theorem \ref{['thm:weak-convergence-probability-Rd']})
  16. ...and 1 more sections

Key Result

Theorem 1

Let $\beta<1$ and $Np\longrightarrow\infty$ as $N\longrightarrow\infty$, let $\left(Y_{N}\right)_{N\in\mathbb{N}}$ with $Y_{N}:\mathcal{X}_{N}\longrightarrow\mathbb{R}^{d}$, $d\in\mathbb{\mathbb{N}}$, have limiting image distribution $\mathbb{P}_{Y}^{\left(CW\right)}$ such that $\left(\mathbb{P}_{N} In shorthand notation, we write:

Theorems & Definitions (15)

  • Theorem 1
  • Theorem 2
  • Definition 3
  • Lemma 4
  • Lemma 5
  • Definition 6
  • Lemma 7
  • Lemma 8
  • Lemma 9
  • Definition 10
  • ...and 5 more