Critical exponents in zero dimensions

TL;DR

This paper introduces zero-dimensional models to exactly calculate critical exponents near instability onset, revealing anomalous and multiscaling behaviors influenced by colored multiplicative noise.

Contribution

It provides exact calculations of critical exponents for a family of models, highlighting novel anomalous and multiscaling phenomena.

Findings

Exact critical exponents for all moments calculated

Identification of anomalous exponents deviating from mean-field predictions

Demonstration of multiscaling behavior in the models

Abstract

In the vicinity of the onset of an instability, we investigate the effect of colored multiplicative noise on the scaling of the moments of the unstable mode amplitude. We introduce a family of zero dimensional models for which we can calculate the exact value of the critical exponents $\beta_m$ for all the moments. The results are obtained through asymptotic expansions that use the distance to onset as a small parameter. The examined family displays a variety of behaviors of the critical exponents that includes anomalous exponents: exponents that differ from the deterministic (mean-field) prediction, and multiscaling: non-linear dependence of the exponents on the order of the moment.