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

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 https://doi.org/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** - 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** - 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 https://doi.org/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 sluchaĭ nykh 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** - 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.

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2020):
15B52,
60G70,
60B20,
92D40

Retrieve articles in all journals with MSC (2020): 15B52, 60G70, 60B20, 92D40

Additional Information

**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

Article copyright:
© Copyright 2021
American Mathematical Society