Superboolean rank and the size of the largest triangular submatrix of a random matrix

TL;DR

This paper investigates the asymptotic behavior of the largest permuted triangular submatrix in a random matrix, which is crucial for understanding matrix rank in combinatorial algebraic contexts.

Contribution

It provides new insights into the asymptotic size of the largest permuted triangular submatrix in large random matrices, linking matrix structure to rank determination.

Findings

Asymptotic characterization of the largest permuted triangular submatrix size

Connections between submatrix structure and matrix rank

Results applicable to combinatorial algebraic problems

Abstract

We explore the size of the largest (permuted) triangular submatrix of a random matrix, and more precisely its asymptotical behavior as the size of the ambient matrix tends to infinity. The importance of such permuted triangular submatrices arises when dealing with certain combinatorial algebraic settings in which these submatrices determine the rank of the ambient matrix, and thus attract a special attention.