Amin Coja-Oghlan - The rank of sparse random matrices

We determine the rank of a random matrix A over an arbitrary field with prescribed numbers of non-zero 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 the Aizenman-Sims-Starr scheme and a novel random perturbation, applicable to any matrix, that likely diminishes the number of short linear relations.
40 years of Replica Symmetry Breaking
10-13 September 2019, Rome