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

Area of Science:

• Mathematics
• Statistical Physics
• Computational Science

Background:

• Large systems of linear equations are fundamental in scientific modeling, often requiring nonnegative solutions for applications like population dynamics and chemical networks.
• A critical transition has been identified in random linear systems, separating regions where nonnegative solutions exist with certainty from those where they are typically absent.

Purpose of the Study:

• To extend the characterization of phases in random linear systems beyond the critical line.
• To analytically determine measures of solution existence and robustness in both solvable and unsolvable phases.

Main Methods:

• Utilizing Farkas' lemma and the replica method, previously employed to find the critical line.
• Applying these methods to analyze the system's behavior away from the critical point.

Main Results:

• Analytical determination of the residual norm in the unsolvable phase.
• Quantification of a robustness measure for solutions in the solvable phase.
• High agreement between analytical results and numerical simulations.

Conclusions:

• The replica method and Farkas' lemma are effective for characterizing random linear systems even away from criticality.
• The study provides analytical tools to understand the existence and robustness of nonnegative solutions in large-scale scientific computations.