Skip to Main Content

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.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.


Positive solutions for large random linear systems
HTML articles powered by AMS MathViewer

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


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.

  • S. Allesina and S. Tang, The stability–complexity relationship at age 40: a random matrix perspective, Population Ecology 57 (2015), no. 1, 63–75.
  • Clive W. Anderson, Stuart G. Coles, and Jürg Hüsler, Maxima of Poisson-like variables and related triangular arrays, Ann. Appl. Probab. 7 (1997), no. 4, 953–971. MR 1484793, DOI 10.1214/aoap/1043862420
  • Zhidong Bai and Jack W. Silverstein, Spectral analysis of large dimensional random matrices, 2nd ed., Springer Series in Statistics, Springer, New York, 2010. MR 2567175, DOI 10.1007/978-1-4419-0661-8
  • Stéphane Boucheron, Gábor Lugosi, and Pascal Massart, Concentration inequalities, Oxford University Press, Oxford, 2013. A nonasymptotic theory of independence; With a foreword by Michel Ledoux. MR 3185193, DOI 10.1093/acprof:oso/9780199535255.001.0001
  • M. Dougoud, L. Vinckenbosch, R. P Rohr, L-F. Bersier, and C. Mazza, The feasibility of equilibria in large ecosystems: A primary but neglected concept in the complexity-stability debate, PLoS Computational Biology 14 (2018), no. 2, e1005988.
  • M. R. Gardner and W. R. Ashby, Connectance of large dynamic (cybernetic) systems: critical values for stability, Nature 228 (1970), no. 5273, 784.
  • Stuart Geman, The spectral radius of large random matrices, Ann. Probab. 14 (1986), no. 4, 1318–1328. MR 866352
  • Stuart Geman and Chii-Ruey Hwang, A chaos hypothesis for some large systems of random equations, Z. Wahrsch. Verw. Gebiete 60 (1982), no. 3, 291–314. MR 664419, DOI 10.1007/BF00535717
  • T. Gibbs, J. Grilli, T. Rogers, and S. Allesina, Effect of population abundances on the stability of large random ecosystems, Phys. Rev. E 98 (2018), no. 2, 022410.
  • Roger A. Horn and Charles R. Johnson, Matrix analysis, 2nd ed., Cambridge University Press, Cambridge, 2013. MR 2978290
  • Tosio Kato, Perturbation theory for linear operators, Die Grundlehren der mathematischen Wissenschaften, Band 132, Springer-Verlag New York, Inc., New York, 1966. MR 0203473
  • M. Lidbetter, G. Lindgren, and Kh. Rot⋅sen, Èkstremumy sluchaĭnykh posledovatel′nosteĭ i protsessov, “Mir”, Moscow, 1989 (Russian). Translated from the English by V. P. Nosko; Translation edited and with a preface by Yu. K. Belyaev. MR 1004671
  • Michel Ledoux, The concentration of measure phenomenon, Mathematical Surveys and Monographs, vol. 89, American Mathematical Society, Providence, RI, 2001. MR 1849347, DOI 10.1090/surv/089
  • R. M. May, Will a large complex system be stable?, Nature 238 (1972), no. 5364, 413.
  • L. Stone, The feasibility and stability of large complex biological networks: a random matrix approach, Scientific Reports 8 (2018), no. 1, 8246.
Similar Articles
Additional Information
  • Pierre Bizeul
  • Affiliation: Institut de Mathématiques de Jussieu, UMR 7586, Sorbonne Universités, 4, place Jussieu, 75005 Paris, France
  • Email:
  • 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:
  • 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
  • © Copyright 2021 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 149 (2021), 2333-2348
  • MSC (2020): Primary 15B52, 60G70; Secondary 60B20, 92D40
  • DOI:
  • MathSciNet review: 4246786