Delocalized region of bipartite Erdős-Rényi graphs

Core Contribution

The paper proves eigenvector delocalization for the adjacency matrix of sparse bipartite Erdős–Rényi graphs G(n,p,d/√(np)) in the critical regime d ~ b·log(n+p). It extends the Alt–Ducatez–Knowles program for ordinary Erdős–Rényi graphs to the bipartite (sparse rectangular) setting. The main results are:

The technical novelty has two parts. First, the vector-valued self-consistent equation is reduced to two coupled scalar equations for typical vertices. This gives a stability estimate (Lemma 4.8) that stays usable at the spectral edges, including the inner edge, which has no analogue in the non-bipartite case. Second, the paper gives explicit Stieltjes-transform and measure formulas for the "bipartite Kesten–McKay-type" approximants m_{i,β}, including their point masses.

Methodological Rigor

The proof architecture follows the established, trusted route of Alt–Ducatez–Knowles:

1. Schur complement and a typical/atypical vertex split.

2. A conditional local law on the event that the degrees satisfy bounds.

3. Bootstrapping down in Im z.

4. Bounded approximants of the diagonal resolvent entries, which give delocalization.

The threshold computations in Theorem 2.2 are simple Poisson/Bennett tail bounds with a union bound, and they are clean and credible. Several key steps are imported from [1] with "analogous" or "omitted" proofs (Lemma 5.1, Theorem 5.2), so rigor partly rests on adaptation. There are some typographical inconsistencies, for example the exponent 1−b(…) versus the stated condition, and 1/α+1 versus 1/α−1 in one display. The scalar stability argument in Lemma 4.8 is the genuinely new piece and looks sound, if terse. The error exponents in the local law, such as (log n/d²)^{1/12}, are non-optimal but sufficient for delocalization.

Potential Impact

This is a solid but specialized contribution to random matrix theory and sparse random graphs. It relates to:

The explicit optimal thresholds and the edge-capable local law are reusable. In particular, the inner-edge treatment and the framework for general sparse rectangular matrices should be useful to others. Direct practical applications are limited, since this is pure probability. The main audience is the sparse RMT community, which is small but active.

Timeliness & Relevance

This is a very current topic. Singular values of sparse rectangular matrices (Dumitriu et al., 2025) and singular vectors (Dumitriu–Wang, 2026) have only just been studied. A concurrent preprint [10] proves bulk delocalization in a weaker range, d ≥ C√(log n)·loglog n, and semilocalization outside the support. This paper complements it by covering the edges with optimal thresholds. The overlap does reduce the exclusivity of the bulk result. Together the two papers close out the delocalization and semilocalization picture for bipartite graphs.

Strengths

Limitations

Other observations

The work is purely theoretical, so reproducibility in the empirical sense is moot. It is verifiable by experts, though the proofs lean on external lemmas. Citation prospects are moderate and concentrated within sparse random matrix theory.