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Linear processes in high dimensions: Phase space and critical properties

Iacopo Mastromatteo and Emmanuel Bacry

Jean-François Muzy

  • Centre de Mathématiques Appliquées, CNRS, École Polytechnique, UMR 7641, 91128 Palaiseau, France

  • Laboratoire Sciences Pour l'Environnement, CNRS, Université de Corse, UMR 6134, 20250 Corté, France

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Phys. Rev. E 91, 042142 – Published 29 April, 2015

DOI: https://doi.org/10.1103/PhysRevE.91.042142

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 (VAR) model and the multivariate Hawkes process. We analyze both deterministic and random versions of these models, showing the existence of a stable phase 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.

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Phys. Rev. E 91, 042142– Published 29 April, 2015
  • Received 18 December 2014

©2015 American Physical Society

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