Positive Solutions for Large Random Linear Systems

Consider a large linear system with random underlying matrix: x
_n
= 1
_n
+ 1/(α
_n
√β
_n
)M
_n
x
_n
, where x
_n
is the unknown, 1
_n
is a vector of ones, M
_n
is a random matrix and α
_n
, β
_n
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 M
_n
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.