论文标题
具有非负溶液的随机线性方程式的大型系统：表征可解的阶段
Large systems of random linear equations with non-negative solutions: Characterizing the solvable and unsolvable phase
论文作者
论文摘要
线性方程式的大系统在科学中无处不在。经常，例如在考虑种群动态或化学网络时，解决方案必须是非负的。最近，已经表明，大型随机线性方程的系统表现出从一个相位的急剧过渡，其中存在非负溶液，概率是，通常找不到这样的解决方案。通过将farkas的引理与复制方法相结合的临界线分隔两个相的临界线。在这里，我们表明，相同的方法仍然可行，可以表征远离关键的两个阶段。为此，我们通过分析性地确定无法解析的阶段中系统的残余规范，并在可溶剂中的溶液鲁棒性衡量。我们的结果与数值模拟非常吻合。
Large systems of linear equations are ubiquitous in science. Quite often, e.g. when considering population dynamics or chemical networks, the solutions must be non-negative. Recently, it has been shown that large systems of random linear equations exhibit a sharp transition from a phase, where a non-negative 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.
