The rank of sparse random matrices

Abstract

Generalising prior work on the rank of random matrices over finite fields [Coja-Oghlan and Gao 2018], we determine the rank of a random matrix A with prescribed numbers of non-zero entries in each row and column over any field F. The rank formula turns out to be independent of both the field and the distribution of the non-zero matrix entries. The proofs are based on a blend of algebraic and probabilistic methods inspired by ideas from mathematical physics.MSC: 05C80, 60B20, 94B05Amin Coja-Oghlan's research received support under DFG CO 646/3. Alperen A. Ergür partially funded by Einstein Foundation, Berlin.

Recent progress in combinatorial random matrix theory

“…In [18], Coja-Oghlan, Ergür, Gao, Hetterich, and Rolvien studied random matrices with prescribed number of non-zeroes in each row and column and achieved an asymptotically sharp estimate for the rank; see [18] for details.…”

Rank deficiency of random matrices

“…Finally, we mention that in recent years, there have been several other works on the (co)rank of random matrices (e.g. [4,3]). However, the focus of these works is on the determination of the typical (co)rank of various models, which is completely different from our goal of (sharply) determining the rate of the probability of having corank at least k for matrices which are of full rank with high probability.…”

Rank of the vertex-edge incidence matrix of $r$-out hypergraphs

“…This is closer to a model of random matrices considered in Cooper, Frieze and Pegden, [6], where the columns are chosen independently from all columns with s ones. A more general paper by Coja-Oghlan et al, [4], gives the limiting rank in this latter model for a wide range of assumptions on the distribution of non-zero entries in the rows and columns. The fundamental difference between the r-out model of random matrices, and those of [4], [6] is the presence of an n × n identity matrix as a sub-matrix in the without replacement case.…”

“…The finite co-rank given in Theorem 1 can be contrasted with results for the edge-vertex incidence matrix of random hypergraphs, ( [4], [6]), where the expected co-rank is linear in the number of vertices n, and the probability of a full rank matrix is exponentially small.…”