# On non-negative solutions to large systems of random linear equations

## TL;DR

This paper investigates the conditions under which large random linear systems have non-negative solutions, identifying a sharp threshold based on the ratio of unknowns to equations, with implications for various models.

## Contribution

It analytically determines the non-negative solution threshold for large random systems and connects it to known models like perceptron storage and resource competition.

## Key findings

- Identifies a sharp transition point for solution existence.
- Derives the threshold as a function of statistical properties.
- Validates results with numerical simulations.

## Abstract

Systems of random linear equations may or may not have solutions with all components being non-negative. The question is, e.g., of relevance when the unknowns are concentrations or population sizes. In the present paper we show that if such systems are large the transition between these two possibilities occurs at a sharp value of the ratio between the number of unknowns and the number of equations. We analytically determine this threshold as a function of the statistical properties of the random parameters and show its agreement with numerical simulations. We also make contact with two special cases that have been studied before: the storage problem of a perceptron and the resource competition model of MacArthur.