Gumbel central limit theorem for max-min and min-max

Area of Science:

• Statistical physics
• Random matrix theory
• Extreme value theory

Background:

• Max-min and min-max computations are crucial in science and engineering.
• Challenges arise with large matrices and incomplete data.
• Existing methods struggle with real-world data limitations.

Purpose of the Study:

• To establish statistical limit laws for max-min and min-max of large random matrices.
• To provide a computational framework for challenging matrix computations.
• To bridge random-matrix theory and extreme-value theory.

Main Methods:

• Applied a statistical-physics approach.
• Developed limit laws analogous to the central limit theorem.
• Intertwined random-matrix theory and extreme-value theory.

Main Results:

• Established limit laws for max-min and min-max of large random matrices.
• Demonstrated that Gumbel statistics emerge universally, regardless of matrix entry distribution.
• Coupled matrix dimensions geometrically.

Conclusions:

• The derived limit laws offer general and universal statistical properties for matrix computations.
• These findings are practical and expected to have broad applications in physical sciences and beyond.
• The study provides a robust theoretical framework for analyzing large random matrices.