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Fluctuations of functions of sparse Erdős-Rényi graphs

Hok-Yin Chu

TL;DR

The paper analyzes fluctuations of diagonal entries $f_{ii}(A)$ for smooth test functions of the normalized adjacency matrix $A$ of sparse Erdős–Rényi graphs, focusing on scales $\eta_*$ from mesoscopic to global. By translating $f_{ii}(A)$ into a Green function framework and applying a cumulant expansion to the sparse ensemble, it proves a central limit theorem with a variance that decouples into two independent Gaussian components on different scales. The variance consists of a GOE-type term $\frac{2}{N}\mathrm{Var}(f(S))$ and a fourth-cumulant term $N\mathcal{C}_4(H_{12})\bigl(\mathbb{E}[f(S)(1-S^2)]\bigr)^2$, yielding a phase transition when $\eta_*/N$ and $\eta_*^2/q^2$ balance, i.e. at $\eta_*\asymp p$; the constants involve the semicircle density $\varrho_{sc}$ and the standard semicircular random variable $S$.

Abstract

Let $A$ be the (rescaled) adjacency matrix of the Erdős-Rényi graphs $\cal G(N,p)$. For $N^{-1+τ} \leqslant p\leqslant N^{-τ}$, we study the fluctuation of $f(A)_{ii}$ on the global and mesoscopic spectral scales. We show that the distribution of $f(A)_{ii}$ is asymptotically the sum of two independent Gaussian random variables on different scales, where a phase transition occurs on the spectral scale $p$.

Fluctuations of functions of sparse Erdős-Rényi graphs

TL;DR

The paper analyzes fluctuations of diagonal entries for smooth test functions of the normalized adjacency matrix of sparse Erdős–Rényi graphs, focusing on scales from mesoscopic to global. By translating into a Green function framework and applying a cumulant expansion to the sparse ensemble, it proves a central limit theorem with a variance that decouples into two independent Gaussian components on different scales. The variance consists of a GOE-type term and a fourth-cumulant term , yielding a phase transition when and balance, i.e. at ; the constants involve the semicircle density and the standard semicircular random variable .

Abstract

Let be the (rescaled) adjacency matrix of the Erdős-Rényi graphs . For , we study the fluctuation of on the global and mesoscopic spectral scales. We show that the distribution of is asymptotically the sum of two independent Gaussian random variables on different scales, where a phase transition occurs on the spectral scale .

Paper Structure

This paper contains 7 sections, 11 theorems, 87 equations.

Table of Contents

  1. Introduction
  2. Acknowledgment
  3. Preliminaries
  4. Proof of Theorem \ref{['thm:1']}
  5. Proof of Lemma \ref{['lemma:cumu']}
  6. Proof of Proposition \ref{['prop:char']}
  7. Evaluation of Integrals

Key Result

Theorem 1.2

Let $f$ be as above and $N_{\mathbb R}(0,1)$ be a standard Gaussian random variable. Moreover, let $S$ denote the random variable with density $\varrho_{sc}(x)\mathrel{\vcenter{\hbox{.}\hbox{.}}}= \frac{1}{2\pi}\sqrt{(4-x^2)_+}$. Then as $N \to \infty$, where $V_{ii}(f)$ is defined as and

Theorems & Definitions (17)

  • Definition 1.1: Sparse matrix
  • Theorem 1.2: Convergence of general test functions on $A$
  • Remark 1.3
  • Lemma 2.1: Cumulant expansion
  • Lemma 2.2
  • Lemma 2.3: Helffer-Sjöstrand formula Davies
  • Definition 2.4: Stochastic domination
  • Theorem 2.5: Local semicircle law for A
  • Corollary 2.6
  • proof
  • ...and 7 more