Singular vectors of sparse random rectangular matrices: Transition between delocalized and semilocalized phases
Abstract
Consider the bipartite Erdős--Rényi graph 𝔾(n,m,d/√(mn)) at the critical sparsity d = blog(N) for some b > 0 with N = n + m, and constant aspect ratio γ= n/m ≥ 1. We show that the spectrum of its centered and rescaled bi-adjacency matrix, whose limiting law is the Marčenko--Pastur (MP) distribution, is split into two phases: in the bulk, singular vectors are completely delocalized, while near the edges, singular vectors are semilocalized, with their mass concentrating on small disjoint balls, centered at vertices with extreme degrees. As b decreases, we identify sharp thresholds for the emergence of outlier singular values. For explicit bR⋆ and bL⋆ as functions of γ, we prove that when b > bR⋆, there is no outlier outside the bulk; when bL⋆< b < bR⋆, only right outliers emerge; and when b < bL⋆, outliers are present on both sides, all with high probability. This removes an extra technical condition from our previous work arxiv:2508.01456, providing a complete characterization of the parameter space. In sparser regimes, in terms of left outliers, we provide some nontrivial generating mechanisms, demonstrating that local graph structure plays a key role in understanding extreme singular values. Our proofs rely on two technical inputs: (1) an entrywise local law in the MP bulk, where the deterministic equivalent of the resolvent depends on both variance profiles and normalized vertex degrees, different from the Gaussian setting; (2) spectral radius upper bounds for nonbacktracking matrices constructed from regularized random graphs, recently developed in arXiv: 2609.26615. Our results reveal fruitful phenomena for sparse random rectangular matrices at criticality, which are beyond the universality class of their Gaussian counterparts.
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BibTeX
@misc{dumitriu2026singular,
title = {{Singular vectors of sparse random rectangular matrices: Transition between delocalized and semilocalized phases}},
author = {Ioana Dumitriu and Hai-Xiao Wang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.22658},
eprint = {2610.22658},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.22658},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Singular vectors of sparse random rectangular matrices: Transition between delocalized and semilocalized phases
AU - Ioana Dumitriu
AU - Hai-Xiao Wang
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2610.22658
UR - https://arxiv.org/abs/2610.22658
AB - Consider the bipartite Erdős--Rényi graph 𝔾(n,m,d/√(mn)) at the critical sparsity d = blog(N) for some b > 0 with N = n + m, and constant aspect ratio γ= n/m ≥ 1. We show that the spectrum of its centered and rescaled bi-adjacency matrix, whose limiting law is the Marčenko--Pastur (MP) distribution, is split into two phases: in the bulk, singular vectors are completely delocalized, while near the edges, singular vectors are semilocalized, with their mass concentrating on small disjoint balls, centered at vertices with extreme degrees. As b decreases, we identify sharp thresholds for the emergence of outlier singular values. For explicit b_R^⋆ and b_L^⋆ as functions of γ, we prove that when b > b_R^⋆, there is no outlier outside the bulk; when b_L^⋆< b < b_R^⋆, only right outliers emerge; and when b < b_L^⋆, outliers are present on both sides, all with high probability. This removes an extra technical condition from our previous work arxiv:2508.01456, providing a complete characterization of the parameter space. In sparser regimes, in terms of left outliers, we provide some nontrivial generating mechanisms, demonstrating that local graph structure plays a key role in understanding extreme singular values. Our proofs rely on two technical inputs: (1) an entrywise local law in the MP bulk, where the deterministic equivalent of the resolvent depends on both variance profiles and normalized vertex degrees, different from the Gaussian setting; (2) spectral radius upper bounds for nonbacktracking matrices constructed from regularized random graphs, recently developed in arXiv: 2609.26615. Our results reveal fruitful phenomena for sparse random rectangular matrices at criticality, which are beyond the universality class of their Gaussian counterparts.
ER -