Fall 2020

The Rank of Sparse Random Matrices

Thursday, Sep. 24, 2020 10:35 am11:00 am PDT

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Jane Gao, University of Waterloo

We determine the rank of a random matrix 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 coupling arguments and a novel random perturbation, applicable to any matrix, that diminishes the number of short linear relations.