Positive Solutions for Large Random Linear Systems

Pierre Bizeul (1) , Maxime Clenet (2) , Jamal Najim (2)

1 IMJ-PRG (UMR_7586) - Institut de Mathématiques de Jussieu - Paris Rive Gauche
2 LIGM - Laboratoire d'Informatique Gaspard-Monge

Consider a large linear system with random underlying matrix: x_n = 1_n + 1/(α_n √β_n) M_n x_n, where x_n is the unknown, 1_n is a vector of ones, M_n is a random matrix and α_n, β_n are scaling parameters to be specified. We investigate the componentwise positivity of the solution x n depending on the scaling factors, as the dimensions of the system grow to infinity. We consider 2 models of interest: The case where matrix M_n has independent and identically distributed standard Gaussian random variables, and a sparse case with a growing number of vanishing entries.In each case, there exists a phase transition for the scaling parameters below which there is no positive solution to the system with growing probability and above which there is a positive solution with growing probability.These questions arise from feasibility and stability issues for large biological communities with interactions.

Mots clés: Linear equation; Large Random Matrices; Extreme values; Lotka-Volterra equations; Feasibility and stability in foodwebs

Domaines: Probabilités [math.PR]; Biodiversité et Ecologie

Pierre Bizeul, Maxime Clenet, Jamal Najim. Positive Solutions for Large Random Linear Systems. ICASSP 2020 - 2020 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), May 2020, Barcelona, Spain. pp.8777-8781, ⟨10.1109/ICASSP40776.2020.9053593⟩. ⟨hal-03093082⟩