Linear processes in high-dimension: phase space and critical properties

TL;DR

This paper studies the behavior of high-dimensional stochastic linear models, like VAR and Hawkes processes, revealing phases of stability and instability, and characterizing slow correlation decay near phase transitions.

Contribution

It provides a comprehensive analysis of phase transitions and correlation decay in high-dimensional linear stochastic models, including conditions for power-law correlations.

Findings

Existence of stable and unstable phases in high-dimensional linear models.

Slow decay of correlations near the phase transition region.

Numerical simulations confirm theoretical predictions.

Abstract

In this work we investigate the generic properties of a stochastic linear model in the regime of high-dimensionality. We consider in particular the Vector AutoRegressive model (VAR) and the multivariate Hawkes process. We analyze both deterministic and random versions of these models, showing the existence of a stable and an unstable phase. We find that along the transition region separating the two regimes, the correlations of the process decay slowly, and we characterize the conditions under which these slow correlations are expected to become power-laws. We check our findings with numerical simulations showing remarkable agreement with our predictions. We finally argue that real systems with a strong degree of self-interaction are naturally characterized by this type of slow relaxation of the correlations.