Rigorous explanation for self-organized criticality in complex systems

Ascertain, at a rigorous mathematical level, the mechanism by which many real complex systems self-organize at or close to a critical point, clarifying why such systems tend to evolve toward criticality.

Background

The paper studies a tick-by-tick agent-based model for price formation where buy and sell orders are modeled as Hawkes processes with mean-field interaction. In the large-scale limit at a critical parameter regime, the aggregated dynamics converge to a stochastic volatility model exhibiting leverage effect and superlinear mean reversion, features supported by empirical evidence.

In discussing criticality, the authors connect their results to the broader paradigm of self-organized criticality (SOC), noting that Hawkes-process-based models naturally suggest looking near critical points. They highlight that, despite empirical and theoretical indications across disciplines, a rigorous understanding of why many real complex systems self-organize at or near criticality is still lacking, framing this as an open question.