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Spectral properties of the Laplacian of Scale-Free Percolation models

Rajat Subhra Hazra, Nandan Malhotra

TL;DR

This work analyzes the centred Laplacian spectrum of a scale-free spatial random graph on a torus, where vertex weights follow a Pareto law and long-range connections are controlled by distance with exponent $\alpha$. Using a Gaussianisation–moment method pipeline and free-probability techniques, the authors prove the existence of a deterministic limiting spectral distribution $\nu_{\tau}$ in the dense regime $0<\alpha<1$ (with $\tau>3$), and identify it as the law of $T_W^{1/2} T_s T_W^{1/2} + \sqrt{\mathbb{E}[W]}\, T_W^{1/4} T_g T_W^{1/4}$, where $T_W$ has Pareto law, $T_s$ is semicircular, and $T_g$ is Gaussian; freeness relations specify the non-commutative structure. The limit is shown to be independent of $\alpha$ and reduces to a semicircle–Gaussian free additive convolution in the degenerate-weight case. The paper also provides a detailed Gaussianisation–variance-profile reduction, truncation control, and decoupling scheme, culminating in a robust moment-method argument and an operator-valued limit description. Simulations illustrate the predicted limiting behavior for concrete parameters, highlighting the practical relevance for diffusion and spectral analysis on heterogeneous, spatially embedded networks.

Abstract

We consider scale-free percolation on a discrete torus $\mathbf{V}_N$ of size $N$. Conditionally on an i.i.d. sequence of Pareto weights $(W_i)_{i\in \mathbf{V}_N}$ with tail exponent $τ-1>0$, we connect any two points $i$ and $j$ on the torus with probability $$p_{ij}= \frac{W_iW_j}{\|i-j\|^α} \wedge 1$$ for some parameter $α>0$. We focus on the (centred) Laplacian operator of this random graph and study its empirical spectral distribution. We explicitly identify the limiting distribution when $α<1$ and $τ>3$, in terms of the spectral distribution of some non-commutative unbounded operators.

Spectral properties of the Laplacian of Scale-Free Percolation models

TL;DR

This work analyzes the centred Laplacian spectrum of a scale-free spatial random graph on a torus, where vertex weights follow a Pareto law and long-range connections are controlled by distance with exponent . Using a Gaussianisation–moment method pipeline and free-probability techniques, the authors prove the existence of a deterministic limiting spectral distribution in the dense regime (with ), and identify it as the law of , where has Pareto law, is semicircular, and is Gaussian; freeness relations specify the non-commutative structure. The limit is shown to be independent of and reduces to a semicircle–Gaussian free additive convolution in the degenerate-weight case. The paper also provides a detailed Gaussianisation–variance-profile reduction, truncation control, and decoupling scheme, culminating in a robust moment-method argument and an operator-valued limit description. Simulations illustrate the predicted limiting behavior for concrete parameters, highlighting the practical relevance for diffusion and spectral analysis on heterogeneous, spatially embedded networks.

Abstract

We consider scale-free percolation on a discrete torus of size . Conditionally on an i.i.d. sequence of Pareto weights with tail exponent , we connect any two points and on the torus with probability for some parameter . We focus on the (centred) Laplacian operator of this random graph and study its empirical spectral distribution. We explicitly identify the limiting distribution when and , in terms of the spectral distribution of some non-commutative unbounded operators.

Paper Structure

This paper contains 24 sections, 17 theorems, 170 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Our contribution:
  3. Outline of the article:
  4. Setup and main results
  5. Setup
  6. Main Results
  7. Discussion and simulations.
  8. Notation
  9. Gaussianisation: Setup for the proof of Theorem \ref{['theorem:Laplacian limiting measure']}
  10. Gaussianisation
  11. Simplification of the variance profile.
  12. Truncation
  13. Decoupling
  14. Moment method: Proof of Theorem \ref{['theorem:Laplacian limiting measure']}
  15. Step 1.
  16. ...and 9 more sections

Key Result

Theorem 2.1

Consider the random graph $\mathbb{G}_N$ on $\mathbf{V}_N$ with connection probabilities given by connection_proba with parameters $\tau>3$ and $0<\alpha< 1$. Let $\mathop{\mathrm{ESD}}\nolimits(\mathbf{\Delta}_N^\circ)$ be the empirical spectral distribution of $\mathbf{\Delta}_N^\circ$ defined in

Figures (2)

  • Figure 3: Modifying the graph $G_{\tilde{\pi}}$ to construct $\tilde{G}_{\tilde{\pi}}$. Here, we pick two vertices $s_1,s_2\in V_{\tilde{\pi}}$, with $\tilde{n}_{s_1}=3, \tilde{n}_{s_2}=4$.
  • Figure :

Theorems & Definitions (30)

  • Theorem 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Theorem 2.5
  • Lemma 3.1: Gaussianisation of $\mathbf{\Delta}_N$
  • Lemma 3.2
  • proof : Proof of Lemma \ref{['lemma:mean zero gaussianisation']}
  • Lemma 3.3
  • proof
  • ...and 20 more