Stability of large complex systems with heterogeneous relaxation dynamics – Archive ouverte HAL

Pierre Mergny 1, 2 Satya N. Majumdar 1

Pierre Mergny, Satya N. Majumdar. Stability of large complex systems with heterogeneous relaxation dynamics. J.Stat.Mech., 2021, 2112 (12), pp.123301. ⟨10.1088/1742-5468/ac3b47⟩. ⟨hal-03532432⟩

We study the probability of stability of a large complex system of size N within the framework of a generalized May model, which assumes a linear dynamics of each population size n
$_{i}$ (with respect to its equilibrium value): . The a
$_{i}$ > 0’s are the intrinsic decay rates, J
$_{ij}$ is a real symmetric (N × N) Gaussian random matrix and measures the strength of pairwise interaction between different species. Our goal is to study how inhomogeneities in the intrinsic damping rates a
$_{i}$ affect the stability of this dynamical system. As the interaction strength T increases, the system undergoes a phase transition from a stable phase to an unstable phase at a critical value T = T
$_{c}$. We reinterpret the probability of stability in terms of the hitting time of the level b = 0 of an associated Dyson Brownian motion (DBM), starting at the initial position a
$_{i}$ and evolving in ‘time’ T. In the large N → ∞ limit, using this DBM picture, we are able to completely characterize T
$_{c}$ for arbitrary density μ(a) of the a
$_{i}$’s. For a specific flat configuration , we obtain an explicit parametric solution for the limiting (as N → ∞) spectral density for arbitrary T and σ. For finite but large N, we also compute the large deviation properties of the probability of stability on the stable side T < T $_{c}$ using a Coulomb gas representation.

  • 1. LPTMS – Laboratoire de Physique Théorique et Modèles Statistiques
  • 2. X – École polytechnique

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