# Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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## Positive solutions for large random linear systemsHTML articles powered by AMS MathViewer

by Pierre Bizeul and Jamal Najim
Proc. Amer. Math. Soc. 149 (2021), 2333-2348 Request permission

## Abstract:

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