The rank of sparse random matrices

Abstract

We determine the asymptotic normalized rank of a random matrix 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.

The $k$-XORSAT Threshold Revisited

“…The techniques developed in [4] were extended to more general random matrix models with identically distributed rows [6]; the main result of that paper also implies the k-XORSAT threshold, but the proof is rather complicated. Additionally, for a still more general model of random matrices over general (not necessarily finite) fields an asymptotic formula for the normalised rank was obtain via the Aizenman-Sims-Starr scheme [7]. Furthermore, an independent result yields the asymptotic rank of the random matrix A over F 2 , albeit without obtaining the precise full rank threshold [10].…”

“…Furthermore, an independent result yields the asymptotic rank of the random matrix A over F 2 , albeit without obtaining the precise full rank threshold [10]. Here we employ the pinning technique from [7] (Lemma 3), which is an adaptation of the more general pinning method for discrete probability distributions developed in [24,28].…”

“…Adding a few rows to a matrix, the randomised pinning operation mostly removes 'short linear relations'. The operation, devised in this form in [7], actually works on any matrix, not just on the random matrix A. Hence, let A be any F q -matrix of size M × N .…”

“…The purpose of this operation is to diminish the number of short relations. To be precise, following [7] we call a set J ⊆ [N ] of columns a relation of A if there exists a vector y ∈ F M q such that supp(y A) = j ∈ [N ] : (y A) j = 0 is a non-empty subset of J. In other words, the non-zero entries of the linear combination y A = 0 of the rows of A are confined to J.…”

The $k$-XORSAT threshold revisited

“…The techniques developed in [4] were extended to more general random matrix models with identically distributed rows [6]; the main result of that paper also implies the k-XORSAT threshold, but the proof is rather complicated. Additionally, for a still more general model of random matrices over general (not necessarily finite) fields an asymptotic formula for the normalised rank was obtain via the Aizenman-Sims-Starr scheme [7].…”