Positive solutions for large random linear systems

Consider a large linear system where $A_n$ is $n\times n$ matrix with independent real standard Gaussian entries, $\boldsymbol{1}_n$ 1$ vector of ones and unknown the $\boldsymbol{x}_n$ satisfying $$ \boldsymbol{x}_n = \boldsymbol{1}_n +\frac 1{\alpha_n\sqrt{n}} A_n \boldsymbol{x}_n\, . $$ We investigate (componentwise) positivity solution depending on scaling factor $\alpha_n$ as dimension $n$ goes to $\infty$. We prove that there sharp phase transition at threshold $\alpha^*_n =\sqrt{2\log n}$: below ($\alpha_n\ll \sqrt{2\log n}$), has negative components probability tending 1 while above ($\alpha_n\gg all vector's are eventually positive 1. At critical $\alpha^*_n$, we provide heuristics evaluate positive. Such systems arise solutions equilibrium Lotka-Volterra differential equations, widely used describe biological communities interactions such foodwebs for instance. In domaine $\boldsymbol{x}_n$, when $\alpha_n\gg n}$, establish equations whose precisely $\x_n$ stable in sense its jacobian $$ {\mathcal J}(\boldsymbol{x}_n) \mathrm{diag}(\boldsymbol{x}_n)\left(-I_n + \frac {A_n}{\alpha_n\sqrt{n}}\right) $$ has eigenvalues part one. Our results shed new light complement understanding feasibility stability issues interaction.