Consider a large linear system where ${\displaystyle A_n}$ is a ${\displaystyle n\times n}$ matrix with independent real standard Gaussian entries, ${\displaystyle 1_n}$ is a ${\displaystyle n\times 1}$
vector of ones and with unknown the vector ${\displaystyle x_n}$ satisfying ${\displaystyle x_n=1_n+\frac{1}{{\alpha }_n\sqrt{n}}{A}_n{x}_n}$
We investigate the (componentwise) positivity of the solution ${\displaystyle x_n}$ depending on the scaling factor ${\displaystyle {\alpha }_n}$ as the dimension ${\displaystyle n}$ goes to ${\displaystyle \infty }$.
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 ${\displaystyle x_n}$, we establish that the Lotka-Volterra system of differential equations whose solution at equilibrium is precisely ${\displaystyle x_n}$ is stable in the sense that its jacobian ${\displaystyle 𝒥(x_n)=\text{diag}(x_n)\left(-I_n+\frac{1}{{\alpha }_n\sqrt{n}}{A}_n{x}_n\right)}$