Positive Solutions for Large Random Linear Systems
Abstract
Consider a large linear system with random underlying matrix: xn = 1n + 1/(αn √βn) Mn xn, where xn 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 x n 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.
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