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Positive solutions for large random linear systems

Abstract : Consider a large linear system where $A_n$ is a $n\times n$ matrix with independent real standard Gaussian entries, $\boldsymbol{1}_n$ is a $n\times 1$ vector of ones and with unknown the $n\times 1$ vector $\boldsymbol{x}_n$ satisfying $\boldsymbol{x}_n = \boldsymbol{1}_n +\frac 1{\alpha_n\sqrt{n}} A_n \boldsymbol{x}_n\, .$ We investigate the (componentwise) positivity of the solution $\boldsymbol{x}_n$ depending on the scaling factor $\alpha_n$ as the dimension $n$ goes to $\infty$. 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 systems of differential equations, widely used to describe large biological communities with interactions such as foodwebs for instance. In the domaine of positivity of the solution $\boldsymbol{x}_n$, that is when $\alpha_n\gg \sqrt{2\log n}$, we establish that the Lotka-Volterra system of differential equations whose solution at equilibrium is precisely $\x_n$ is stable in the sense that its jacobian ${\mathcal J}(\boldsymbol{x}_n) = \mathrm{diag}(\boldsymbol{x}_n)\left(-I_n + \frac {A_n}{\alpha_n\sqrt{n}}\right)$ has all its eigenvalues with negative real part with probability tending to one. Our results shed a new light and complement the understanding of feasibility and stability issues for large biological communities with interaction.
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Submitted on : Monday, January 4, 2021 - 4:33:05 PM
Last modification on : Thursday, September 29, 2022 - 10:47:11 AM

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Pierre Bizeul, Jamal Najim. Positive solutions for large random linear systems. Proceedings of the American Mathematical Society, In press, pp.1. ⟨10.1090/proc/15383⟩. ⟨hal-02093170v2⟩

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