Criticality and Universality of Generalized Kuramoto Model

TL;DR

This paper investigates the critical behavior and universality classes of generalized Kuramoto oscillators in even dimensions, revealing universal critical exponents and a new upper critical dimension through analytical and numerical methods.

Contribution

It introduces a unified framework for understanding synchronization transitions in high-dimensional Kuramoto models, identifying universal critical exponents and the role of dimensionality.

Findings

Universal critical exponents $eta=1/2$ and $ar{ u}=5/2$ for even $D$

Unconventional upper critical dimension $d_u=5$

Validation of theoretical predictions through extensive simulations

Abstract

We explore synchronization transitions in even-$D$-dimensional generalized Kuramoto oscillators on both complete graphs and $d$-dimensional lattices. In the globally coupled system, analytical expansions of the self-consistency equations, incorporating finite-size corrections, reveal universal critical exponents $\beta = 1/2$ and $\bar{\nu} = 5/2$ for all even $D$, indicating an unconventional upper critical dimension $d_u = 5$. Extensive numerical simulations across multiple $D$ confirm these theoretical predictions. For locally coupled systems, we develop a framework based on spin-wave theory and fluctuation-resolved functional network diagnostics, which captures criticality in entrainment transition. A modified Edwards-Anderson order parameter further validates the predicted exponents. This combined theoretical and numerical study uncovers a family of universality classes characterized by $D$-independent but $d$-dependent criticality, offering a unified perspective on symmetry and dimensionality in nonequilibrium synchronization phenomena.