   ISSN 1088-6826(online) ISSN 0002-9939(print)

Positive solutions for large random linear systems

Authors: Pierre Bizeul and Jamal Najim
Journal: Proc. Amer. Math. Soc. 149 (2021), 2333-2348
MSC (2020): Primary 15B52, 60G70; Secondary 60B20, 92D40
DOI: https://doi.org/10.1090/proc/15383
Published electronically: March 25, 2021
MathSciNet review: 4246786
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Consider a large linear system where $A_n$ is an $n\times n$ matrix with independent real standard Gaussian entries, ${\boldsymbol {1}}_n$ is an $n\times 1$ vector of ones and with unknown the $n\times 1$ vector ${\boldsymbol {x}}_n$ satisfying \begin{equation*} {\boldsymbol {x}}_n = {\boldsymbol {1}}_n +\frac 1{\alpha _n\sqrt {n}} A_n {\boldsymbol {x}}_n . \end{equation*} We investigate the (componentwise) positivity of the solution ${\boldsymbol {x}}_n$ depending on the scaling factor $\alpha _n$ as the dimension $n$ goes to infinity. We prove that there is a sharp phase transition at the threshold $\alpha ^*_n =\sqrt {2\log n}$: below the threshold ($\alpha _n\ll \sqrt {2\log n}$), ${\boldsymbol {x}}_n$ has negative components with probability tending to 1 while above ($\alpha _n\gg \sqrt {2\log n}$), all the vector’s components are eventually positive with probability tending to 1. At the critical scaling $\alpha ^*_n$, we provide a heuristics to evaluate the probability that ${\boldsymbol {x}}_n$ is positive.

Such linear systems arise as solutions at equilibrium of large Lotka-Volterra (LV) systems of differential equations, widely used to describe large biological communities with interactions. In the domain of positivity of ${\boldsymbol {x}}_n$ (a property known as feasibility in theoretical ecology), our results provide a stability criterion for such LV systems for which ${\boldsymbol {x}}_n$ is the solution at equilibrium.

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Pierre Bizeul
Affiliation: Institut de Mathématiques de Jussieu, UMR 7586, Sorbonne Universités, 4, place Jussieu, 75005 Paris, France
Email: pierre.bizeul@imj-prg.fr

Jamal Najim
Affiliation: Laboratoire d’Informatique Gaspard Monge, UMR 8049, CNRS & Université Gustave Eiffel, 5, Boulevard Descartes, Champs sur Marne, 77454 Marne-la-Vallée Cedex 2, France
MR Author ID: 691013
Email: najim@univ-mlv.fr

Received by editor(s): April 8, 2019
Received by editor(s) in revised form: February 26, 2020, and September 18, 2020
Published electronically: March 25, 2021
Additional Notes: The second author was supported by Labex Bézout, French ANR grant ANR-17-CE40-0003 and CNRS Project 80 Prime - KARATE
Communicated by: Zhen-Qing Chen