Recently, sharp matrix concentration inequalities~\cite{BBvH23,BvH24} were developed using the theory of free probability. In this work, we design polynomial time deterministic algorithms to construct outcomes that satisfy the guarantees of these inequalities. As direct consequences, we obtain polynomial time deterministic algorithms for the matrix Spencer problem~\cite{BJM23} and for constructing near-Ramanujan graphs. Our proofs show that the concepts and techniques in free probability are useful not only for mathematical analyses but also for efficient computations.

翻译：近期，利用自由概率论理论发展了尖锐的矩阵集中不等式~\cite{BBvH23,BvH24}。在本研究中，我们设计了多项式时间确定性算法，以构造满足这些不等式保证的结果。作为直接推论，我们获得了解决矩阵Spencer问题~\cite{BJM23}以及构造近拉马努金图的多项式时间确定性算法。我们的证明表明，自由概率论中的概念和技术不仅对数学分析有用，也对高效计算具有重要价值。