Spectral renormalization group theory on networks

TL;DR

This paper develops a spectral renormalization group framework for analyzing phase transitions in materials modeled by arbitrary networks, linking spectral properties of the graph to critical behavior.

Contribution

It introduces a novel renormalization scheme based on graph Laplacian eigenvectors for studying scalar fields on networks, connecting spectral density to critical exponents.

Findings

Critical exponents depend on the spectral density of the graph.

The method applies to both equilibrium and non-equilibrium phenomena.

Provides a new tool for analyzing amorphous materials modeled as networks.

Abstract

Discrete amorphous materials are best described in terms of arbitrary networks which can be embedded in three dimensional space. Investigating the thermodynamic equilibrium as well as non-equilibrium behavior of such materials around second order phase transitions call for special techniques. We set up a renormalization group scheme by expanding an arbitrary scalar field living on the nodes of an arbitrary network, in terms of the eigenvectors of the normalized graph Laplacian. The renormalization transformation involves, as usual, the integration over the more "rapidly varying" components of the field, corresponding to eigenvectors with larger eigenvalues, and then rescaling. The critical exponents depend on the particular graph through the spectral density of the eigenvalues.