POSITIVE SOLUTIONS FOR LARGE RANDOM LINEAR SYSTEMS

Consider a large linear system where A(n) is an n x n matrix with independent real standard Gaussian entries, 1(n) is an n x 1 vector of ones and with unknown the n x 1 vector x(n) satisfying x(n) = 1(n) + 1/alpha(n)root n A(n)x(n). We investigate the (componentwise) positivity of the solution x(n) depending on the scaling factor alpha(n) as the dimension n goes to infinity. We prove that there is a sharp phase transition at the threshold alpha(n)* = root 2 log n: below the threshold (alpha(n) << root 2 log n), x(n) has negative components with probability tending to 1 while above (alpha(n) >> root 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 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 x(n) (a property known as feasibility in theoretical ecology), our results provide a stability criterion for such LV systems for which x(n), is the solution at equilibrium.