Positive Solutions for Large Random Linear Systems

摘要

Consider a large linear system with random underlying matrix:egin{equation*}{{mathbf{x}}_n} = {{mathbf{1}}_n} + rac{1}{{{lpha _n}sqrt {{eta _n}} }}{M_n}{{mathbf{x}}_n},end{equation*}where is the unknown, 1n is a vector of ones, Mn is a random matrix and αn, βn are scaling parameters to be specified. We investigate the componentwise positivity of the solution xn depending on the scaling factors, as the dimensions of the system grow to infinity.We consider 2 models of interest: The case where matrix Mn has independent and identically distributed standard Gaussian random variables, and a sparse case with a growing number of vanishing entries.In each case, there exists a phase transition for the scaling parameters below which there is no positive solution to the system with growing probability and above which there is a positive solution with growing probability.These questions arise from feasibility and stability issues for large biological communities with interactions.