Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Fine asymptotics of the magnetization of the annealed dilute Curie-Weiss model

Fabian Apostel, Hanna Döring, Kristina Schubert

Abstract

We consider the dilute Curie-Weiss model of size $N$, which is a generalization of the classical Curie-Weiss model where the dependency structure between the spins is not encoded by the complete graph but via the (directed) Erdős-Rényi graph on $N$ vertices in which every edge appears independently with probability $p(N)$. In the high temperature with external magnetic field regime ($0<β<1,h\in\mathbb{R}$) we prove for $p^{3}N^{2}\to\infty$ sharp cumulant bounds for the magnetization for the annealed Gibbs measure implying a central limit theorem with rate, a moderate deviation principle, a concentration inequality, a normal approximation bound with Cramér correction and mod-Gaussian convergence.

Fine asymptotics of the magnetization of the annealed dilute Curie-Weiss model

Abstract

We consider the dilute Curie-Weiss model of size , which is a generalization of the classical Curie-Weiss model where the dependency structure between the spins is not encoded by the complete graph but via the (directed) Erdős-Rényi graph on vertices in which every edge appears independently with probability . In the high temperature with external magnetic field regime () we prove for sharp cumulant bounds for the magnetization for the annealed Gibbs measure implying a central limit theorem with rate, a moderate deviation principle, a concentration inequality, a normal approximation bound with Cramér correction and mod-Gaussian convergence.
Paper Structure (7 sections, 12 theorems, 139 equations)

This paper contains 7 sections, 12 theorems, 139 equations.

Table of Contents

  1. Introduction
  2. Description of the model
  3. Main results
  4. Technical Preparation
  5. Statulevičius condition for the annealed dilute Curie-Weiss model
  6. Proof of Theorem \ref{['Statuleviciusthmcw']}
  7. Uniform asymptotics of the finite-volume pressure -- Proof of Proposition \ref{['asymptoticprop']}

Key Result

Theorem 1.5

Suppose that $h,\beta\in\mathbb{R}$ with $0<\beta<1$ and $p^{3}N^{2}\to\infty$ as $N\to\infty$. Then, we find that $m_N:=\frac{M_N-\mathbb{E}\left[M_N\right]}{\sqrt{\mathbb{V}\left[M_N\right]}}$ under the measure $\mu_{N,p,\beta,h}$ for $N$ big enough satisfies the Statulevičius condition. In partic where $\tilde{C}:=\max(C,1)$. Here we denote by $\mathbb{E}$ and $\mathbb{V}$ the expectation and t

Theorems & Definitions (35)

  • Remark 1.1
  • Definition 1.2
  • Definition 1.3
  • Remark 1.4
  • Theorem 1.5: Statulevičius condition for the magnetization for the annealed Curie-Weiss model
  • Remark 1.6
  • Corollary 1.7
  • Lemma 2.1
  • proof
  • Remark 2.2
  • ...and 25 more