The rank of sparse random matrices

TL;DR

This work determines the asymptotic rank of sparse random matrices over arbitrary fields with prescribed row/column degrees, showing that $\frac{\rk(\boldsymbol{A})}{n} \to 1-\max_{\alpha\in[0,1]}\Phi(\alpha)$ in probability, where $\Phi$ depends only on the degree distributions via their generating functions $D$ and $K$. The authors introduce a novel perturbation $A[\theta]$ that eliminates short linear relations and couple matrix sizes through an Aizenman-Sims-Starr scheme to derive a rigorous lower bound matching the cavity-method prediction, while also providing an alternative upper bound via interpolation. A central insight is that the rank is governed by large-scale dependency structures encoded in the degree distributions, with the 2-core bound providing a sharp threshold in many, but not all, cases; tightness is established under several natural degree-structure regimes, and counterexamples are discussed. The results have direct implications for the rate of low-density parity-check codes and demonstrate the limits of non-rigorous cavity-method predictions in general degree distributions. Overall, the paper delivers a rigorous, field-independent rank formula for sparse random matrices and clarifies the landscape where the 2-core bound is or is not tight, with broad methodological impact on random factor graphs and coding theory.

Abstract

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.

Figures (2)

• Figure 1: Left: the function $\Delta\mapsto2-\max_{\alpha\in[0,1]} \exp(-\Delta\exp(\Delta(\alpha-1)))+(1+(1-\alpha)\Delta)\exp(\Delta(\alpha-1))$ for Example \ref{['Ex_1']}. Middle: the function $d\mapsto1-\max_{\alpha\in[0,1]}\exp(-d\alpha^{k-1})-d(1-k\alpha^{k-1}+(k-1)\alpha^k)/k$ from Example \ref{['Ex_XOR']} with $k=3$. Right: the function $\Phi(x)$ from \ref{['eqEx16Phi']} for Example \ref{['Ex_3']}.
• Figure 2: Left: sketch of the component of $x_i$ at $t=0$; the check $p_i$ is present iff $i\leq\boldsymbol{\theta}$. Right: sketch of the factor graph $\boldsymbol{G}_\varepsilon(t)$ for $0<t<{\boldsymbol{m}}_{\varepsilon,n}$, with the $a_{i,j}$ coloured black and the other colours as in the left figure.

Theorems & Definitions (116)

• Theorem 1.1
• Theorem 1.2
• Remark 1.3
• Theorem 1.4
• Remark 1.5
• Example 1.6: the identity matrix
• Example 1.7: the adjacency matrix of random bipartite graphs
• Example 1.8: fixed row sums
• Example 1.9: non-exact 2-core bound
• Proposition 1.10