The rank of sparse random matrices. Issue 1 (23rd April 2022) - Coja‐Oghlan, Amin; Ergür, Alperen A.; Gao, Pu; Hetterich, Samuel; Rolvien, Maurice
Abstract: We determine the asymptotic normalized rank of a random matrix A $$ \boldsymbol{A} $$ over an arbitrary field with prescribed numbers of nonzero entries in each row and column. As an application we obtain a formula for the rate of low‐density parity check codes. This formula vindicates a conjecture of Lelarge (2013). The proofs are based on coupling arguments and a novel random perturbation, applicable to any matrix, that diminishes the number of short linear relations.