Eigenvalue Distribution of Max-Plus Random Matrices

Random matrix theory (RMT) plays a crucial role in various scientific and engineering disciplines, providing powerful tools to model and analyze systems with inherent randomness. Extending this theory beyond the traditional plus-times semiring is essential for addressing problems where standard arithmetic operations are insufficient, s uch a $s g$ raphs o rdeep neural networks (DNNs) which make use the of the max operation. This paper simulates and models the limiting eigenvalue distribution of random matrices over the max-plus semiring for Gaussian and Exponential entry distributions. Methods to compute maxplus eigenvalues are discussed. The Gumbel distribution proves to be a promising candidate for the limiting distribution, with the Gumbel parameters closely related to those of the Gumbel distribution of the entry distribution. A curve-fitter was used to find o ptimal G umbel p arameters w hen fiting to the empirical max-plus eigenvalue distributions.