Positive solutions for large random linear systems,Proceedings of the American Mathematical Society

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Positive solutions for large random linear systems
Proceedings of the American Mathematical Society ( IF 0.8 ) Pub Date : 2021-03-25 , DOI: 10.1090/proc/15383
Pierre Bizeul , Jamal Najim

Abstract:Consider a large linear system where

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

$\displaystyle {\boldsymbol {x}}_n = {\boldsymbol {1}}_n +\frac 1{\alpha _n\sqrt {n}} A_n {\boldsymbol {x}}_n\, .$

We investigate the (componentwise) positivity of the solution ${\boldsymbol {x}}_n$ depending on the scaling factor $\alpha _n$ as the dimension

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.

[1] S. Allesina and S. Tang,
The stability-complexity relationship at age 40: a random matrix perspective,
Population Ecology 57 (2015), no. 1, 63-75.
[2]

中文翻译：

大型随机线性系统的正解

摘要：考虑一个大型线性系统，

该系统是一个 $n \次n$ 具有独立的实际标准高斯项的矩阵，是一个矢量，而一个未知矢量满足 ${\ boldsymbol {1}} _ n$ $n \次1$ $n \次1$ ${\ boldsymbol {x}} _ n$

$\ displaystyle {\ boldsymbol {x}} _ n = {\ boldsymbol {1}} _ n + \ frac 1 {\ alpha _n \ sqrt {n}} A_n {\ boldsymbol {x}} _ n \，。$

当维数趋于无穷大时，我们将根据比例因子调查解决方案的（分量方向）正性。我们证明了在阈值处有一个尖锐的相变：在阈值（）以下，具有负分量，概率倾向于为1，而在（）以上，所有矢量分量最终都是正的，概率倾向于为1 。我们提供了一种启发式方法来评估为正的概率。这种线性系统是作为大型Lotka-Volterra（LV）微分方程系统平衡时的解而出现的，广泛用于描述具有相互作用的大型生物群落。在正领域 ${\ boldsymbol {x}} _ n$ $\ alpha _n$

$\ alpha ^ * _ n = \ sqrt {2 \ log n}$ $\ alpha _n \ ll \ sqrt {2 \ log n}$ ${\ boldsymbol {x}} _ n$ $\ alpha _n \ gg \ sqrt {2 \ log n}$ $\ alpha ^ * _ n$ ${\ boldsymbol {x}} _ n$ ${\ boldsymbol {x}} _ n$ （在理论生态学中称为可行性），我们的结果为此类LV系统提供了一个稳定的标准，对于该LV系统而言，它是平衡解决方案。 ${\ boldsymbol {x}} _ n$

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