Fractal fluctuations at mixed-order transitions in interdependent networks

TL;DR

This paper investigates the fractal nature of order parameter fluctuations near mixed-order phase transitions in interdependent networks, revealing fractal fluctuations with specific dimensions and exponents despite compact order parameter structures.

Contribution

It uncovers that order parameter fluctuations are fractal near mixed-order transitions and characterizes their self-similarity through new fractal dimension and correlation length exponent formulas.

Findings

Order parameter fluctuations are fractal up to a diverging correlation length.

Fractal dimension of fluctuations is $d_f'=3d/4$.

Correlation length exponent is $ u'=2/d$.

Abstract

We study the geometrical features of the order parameter's fluctuations near the critical point of mixed-order phase transitions in randomly interdependent spatial networks. In contrast to continuous transitions, where the structure of the order parameter at criticality is fractal, in mixed-order transitions the structure of the order parameter is known to be compact. Remarkably, we find that although being compact, the fluctuations of the order parameter close to mixed-order transitions are fractal up to a well-defined correlation length $\xi'$, which diverges when approaching the critical threshold. We characterize the self-similar nature of these critical fluctuations through their fractal dimension, $d_f'=3d/4$, and correlation length exponent, $\nu'=2/d$, where $d$ is the dimension of the system. By means of percolation and magnetization, we demonstrate that $d_f'$ and $\nu'$ are independent on the symmetry of the underlying process for any $d$ of the underlying networks.