Universality in the one-dimensional chain of phase-coupled oscillators

TL;DR

This paper applies a renormalization group method to analyze synchronization phenomena in a one-dimensional chain of phase-coupled oscillators, revealing universal scaling laws and a universality class in the weak randomness regime.

Contribution

It demonstrates the effectiveness of RG in predicting synchronization and universal behavior in weakly random oscillator chains, extending its applicability beyond strong randomness.

Findings

RG accurately predicts cluster formation in weak randomness.

Identifies a universality class with a specific critical exponent.

Shows universal scaling laws for small coupling regimes.

Abstract

We apply a recently developed renormalization group (RG) method to study synchronization in a one-dimensional chain of phase-coupled oscillators in the regime of weak randomness. The RG predicts how oscillators with randomly distributed frequencies and couplings form frequency-synchronized clusters. Although the RG was originally intended for strong randomness, i.e. for distributions with long tails, we find good agreement with numerical simulations even in the regime of weak randomness. We use the RG flow to derive how the correlation length scales with the width of the coupling distribution in the limit of large coupling. This leads to the identification of a universality class of distributions with the same critical exponent $\nu$. We also find universal scaling for small coupling. Finally, we show that the RG flow is characterized by a universal approach to the unsynchronized fixed point, which provides physical insight into low-frequency clusters.