Large systems of random linear equations with nonnegative solutions: Characterizing the solvable and the unsolvable phase

Physical Review. E
Stefan Landmann, Andreas Engel

Abstract

Large systems of linear equations are ubiquitous in science. Quite often, e.g., when considering population dynamics or chemical networks, the solutions must be nonnegative. Recently, it has been shown that large systems of random linear equations exhibit a sharp transition from a phase, where a nonnegative solution exists with probability one, to one where typically no such solution may be found. The critical line separating the two phases was determined by combining Farkas' lemma with the replica method. Here we show that the same methods remain viable to characterize the two phases away from criticality. To this end we analytically determine the residual norm of the system in the unsolvable phase and a suitable measure of robustness of solutions in the solvable one. Our results are in very good agreement with numerical simulations.

References

Dec 7, 1979·Journal of Theoretical Biology·J Schnakenberg
Aug 18, 1972·Nature·R M May
Jul 17, 2014·The Journal of Chemical Physics·Matteo Polettini, Massimiliano Esposito
Feb 12, 2017·Physical Review Letters·Mikhail Tikhonov, Remi Monasson

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