Universal scaling in real dimension

TL;DR

This paper investigates universal scaling behavior in non-homogeneous, long-range diluted graphs, demonstrating that the spectral dimension governs the scaling exponents, which vary continuously with the dimension, supported by numerical simulations.

Contribution

It introduces a scaling theory for universality on complex, non-homogeneous graphs controlled by spectral dimension, extending understanding beyond traditional homogeneous systems.

Findings

Universal exponents depend continuously on spectral dimension

Numerical simulations agree with theoretical predictions

Scaling behavior observed in models on complex geometries

Abstract

The concept of universality has shaped our understanding of many-body physics, but is mostly limited to homogenous systems. Here, we present a study of universality on a non-homogeneous graph, the long-range diluted graph (LRDG). Its scaling theory is controlled by a single parameter, the spectral dimension $d_{s}$, which plays the role of the relevant parameter on complex geometries. The graph under consideration allows us to tune the value of the spectral dimension continuously also to noninteger values and to find the universal exponents as continuous functions of the dimension. By means of extensive numerical simulations, we probe the scaling exponents of a simple instance of $O(\mathcal{N})$ symmetric models on the LRDG showing quantitative agreement with the theoretical prediction of universal scaling in real dimensions.