Consider a large linear system where is a matrix with independent real standard Gaussian entries, is a vector of ones and with unknown the vector satisfying We investigate the (componentwise) positivity of the solution depending on the scaling factor as the dimension goes to . 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 , we establish that the Lotka-Volterra system of differential equations whose solution at equilibrium is precisely is stable in the sense that its jacobian
Positive solutions for large random linear systems