Local Marchenko-Pastur Law for Random Bipartite Graphs
Kevin Yang†

arXiv:1704.08672v1 [math.PR] 27 Apr 2017

April 28, 2017
1

Abstract. We study the random matrix ensemble of covariance matrices arising from random (db , dw )-regular bipartite graphs on
a set of M black vertices and N white vertices, for db ≫ log4 N . We simultaneously prove that the Green’s functions of these
covariance matrices and the adjacency matrices of the underlying graphs agree with the corresponding limiting law (e.g. MarchenkoPastur law for covariance matrices) down to the optimal scale. This is an improvement from the previously known mesoscopic results.
We obtain eigenvector delocalization for the covariance matrix ensemble as consequence, as well as a weak rigidity estimate.

Contents
1. Introduction

2

1.1. Acknowledgements
1.2. Notation

3
3

2. Underlying Model and Main Results
2.1. The Random Matrix Ensemble

3
3

2.2. Random Covariance Matrices

4

2.3. The Main Result
3. Switchings on Bipartite Graphs

5
8

3.1. Switchings on Adjacency Matrices
3.2. Probability Estimates on Vertices

10
11

4. Green’s Function Analysis
4.1. Preliminary Resolvent Theory

13
13

4.2. Reductions of the Proof of Theorem 2.5
4.3. Switchings on Green’s Functions

14
16

5. The Self-Consistent Equation and Proof of Theorem 2.5
5.1. Derivation of the self-consistent equation

18
18

5.2. Analysis of the self-consistent equation
5.3. Final estimates

20
22

6. Appendix
6.1. Estimates on the Stieltjes transforms on the upper-half plane

23
23

References

24

1† Stanford University, Department of Mathematics. Email: kyang95@stanford.edu.
1

1. Introduction
Let XA denote the adjacency matrix of a random (db , dw )-biregular graph with off-diagonal blocks A, A∗ . Here, we assume
−1/2

∗
A is a matrix of size M ×N with M > N . We define the normalized empirical spectral distributions of d−1
w A A and dw
to be the following macroscopic random point masses:
X
1
µA ∗ A =
(1.1)
δd−1
(x),
w λ
N
∗

XA

λ∈σ(A A)

(1.2)

µXA =

1
M +N

X

δd−1/2 λ (x).
w

λ∈σ(XA )

It is known (see [3]) that the empirical spectral distribution of the normalized covariance matrix converges almost surely (in
the limit M, N → ∞ and d → ∞ at a suitable rate) to the Marchenko-Pastur law with parameter γ := N/M given by the
following density function:
p
(λ+ − x)(x − λ− )
̺∞ (x) dx := ̺MP (x) dx =
(1.3)
1x∈[λ− ,λ+ ] dx,
2πγx
√
where we define λ± = (1 ± γ)2 . As noted in [8], this implies that the empirical spectral distribution of the normalized
adjacency matrix converges almost surely to a linearization of the Marchenko-Pastur law given by the following density
function:

p
γ

(λ+ − E 2 )(E 2 − λ− ) E 2 ∈ [λ− , λ+ ]
(1+γ)π|E|
̺(E) =
(1.4)
.
0
E 2 6∈ [λ− , λ+ ]

We briefly remark that in the regime M = N , the linearized Marchenko-Pastur density agrees exactly with the Wigner
semicircle density, which is the limiting density for the empirical spectral distribution of random d-regular graphs on N
vertices in the limit N, d → ∞ at suitable rates. In this regime, coincidence of the empirical spectral distribution and the
limiting Wigner semicircle law was shown for intervals at the optimal scale N −1+ε in [5]. This short-scale result is crucial for
understanding eigenvalue gap and correlation statistics and showing universality of eigenvalue statistics for random regular
graphs compared to the GOE.
Moreover, the short-scale result is a drastic improvement from the order 1 result discussed above for biregular bipartite
∗
graphs. For these graphs, convergence of the empirical spectral distribution of d−1
w A A to the Marchenko-Pastur law was
shown for scales N −ε for sufficiently small ε > 0 in [8]. The techniques used in this paper included primarily analysis of
trees and ballot sequences. This result, however, is far from the optimal scale and is thus far from sufficient for showing
universality of eigenvalue statistics. The aim of this paper is remedy this problem and obtain convergence at the optimal
scale. Similar to [5], we bypass the analysis of trees and ballot sequences with a combinatorial operator on graphs known
as switchings, which are ubiquitous throughout graph theory. This will help us resample vertices in a random graph and
will be crucial in deriving a tractable self-consistent equation for the Green’s function of a random biregular bipartite graph.
However, in contrast to [8], we aim to prove convergence at the optimal scale for the ensembles of (normalized) covariance
matrices and adjacency matrices simultaneously, transferring between the analysis of each ensemble whenever convenient.
In [8], the result for adjacency matrices was derived as a consequence of the result for covariance matrices. To the author’s
knowledge, this idea is original to this paper.
For random regular graphs, universality of local bulk eigenvalue correlation statistics was shown in [4], which required
both the local law from [5] as a crucial ingredient as well as analysis of the Dyson Brownian Motion in [13]. In a similar spirit
for random biregular bipartite graphs, we prove universality of bulk eigenvalue statistics in [17] using the local law result
in this paper and an analysis of Dyson Brownian Motion for covariance matrices in [18]. Thus, this paper may be viewed as
the first in a series of three papers on random covariance matrices resembling the papers [5], [4], [13], and [12] which study
Wigner matrices.
Before we proceed with the paper, we remark that as with random regular graphs and Wigner ensembles, the covariance
matrix ensembles arising from biregular bipartite graphs is a canonical example of a covariance matrix ensemble whose
2

entries are nontrivially correlated. A wide class of covariance matrices with independent sample data entries was treated in
the papers [1], [6], [7], [15], and [16]. In these papers, local laws were derived and universality of local eigenvalue correlation
statistics were proven assuming moment conditions. Because of the nontrivial correlation structure of the sample data entries
and the lack of control of entry-wise moments, these papers and their methods cannot apply to our setting.
1.1. Acknowledgements. The author thanks H.T. Yau and Roland Bauerschmidt for suggesting the problem, referring papers, and answering the author’s questions pertaining to random regular graphs. This work was partially funded by a grant
from the Harvard College Research Program. This paper was written while the author was a student at Harvard University.
1.2. Notation. We adopt the Landau notation for big-Oh notation, and the notation a . b. We establish the notation
[[a, b]] := [a, b] ∩ Z. We let [E] denote the underlying vertex set of a graph E. For vertices v, v ′ ∈ E, we let vv ′ denote the

edge in E containing v and v ′ . For a real symmetric matrix H, we let σ(H) denote its (real) spectrum.
2. Underlying Model and Main Results

We briefly introduce the underlying graph model consisting of bipartite graphs on a prescribed vertex set.
Definition 2.1. Suppose V = {1b , 2b , . . . , Mb , 1w , . . . , Nw } is a set of labeled vertices, and suppose E is a simple graph on
V . We say the graph E is bipartite with respect to the vertex sets (V , Vb , Vw ) if V admits the following decomposition:
[
(2.1)
V = {1b , 2b , . . . , Mb } {1w , 2w , . . . , Nw } =: Vb ∪ Vw ,

such that for any vertices vi , vj ∈ Vb and vk , vℓ ∈ Vw , the edges vi vj and vk vℓ are not contained in E.
Moreover, for fixed integers db , dw > 0, we say that a bipartite graph E is (db , dw )-regular if each v ∈ Vb has db neighbors
and if each w ∈ Vw has dw neighbors.

Remark 2.2. For the remainder of this paper, we will refer to Vb as the set of black vertices and Vw as the set of white vertices.
Moreover, we will refer to a (db , dw )-regular graph simply as a biregular graph, if the parameters db , dw are assumed. In
particular, when referring to a biregular graph we assume a bipartite structure. Lastly, the set of (db , dw )-regular graphs on
the vertex sets (V , Vb , Vw ) will be denoted by Ω, where we suppress the dependence of the parameters M, N, db , dw without
the risk of confusion.
We now record the following identity which follows from counting the total number of edges in a biregular graph E:
(2.2)

M db = N dw ,

where M = Mb and N = Nw . We retain this notation for M, N for the remainder of the paper.
2.1. The Random Matrix Ensemble. We now introduce a modification of the random matrix ensemble studied in [8],
retaining the notation used in the introduction of this paper. We first note the adjacency matrix of a biregular graph is a block
matrix with vanishing diagonal, i.e. has the following algebraic form:
!
0 A
(2.3)
XA =
,
A∗ 0
where A is a matrix of size M ×N . By the biregular assumption of the underlying graph, the matrix XA exhibits the following
eigenvalue-eigenvector pair:
!
p
eb
db dw , emax = √
λmax =
(2.4)
,
αew
where eb and ew are constant ℓ2 -normalized vectors of dimension M and N respectively. By the Perron-Frobenius theorem,
the eigenvalue λmax is simple.
3

Using ideas from [5], the matrix ensemble of interest is the ensemble X = X (M, N, db , dw ) of normalized adjacency
matrices given by
0
H∗

X =

(2.5)

H
0

!

−1/2
H = dw
(A − db eb e∗w ) , .

,

Because the eigenspace corresponding to λmax is one-dimensional, standard linear algebra implies that upon a normalization
−1/2

factor of dw , the matrices X and XA will share the same eigenvalue-eigenvector pairs orthogonal to the eigenspace
corresponding to λmax . On this maximal eigenspace, the matrix X will exhibit emax as an eigenvector corresponding to the
eigenvalue λ = 0. Moreover, as noted in [8] the spectrum of X will be compactly supported in a sense we will make shortly
make precise.
To complete our discussion of the random matrix ensemble of interest, we first note that X is clearly in bijection with the
set of biregular graphs on the fixed triple (V , Vb , Vw ), which is a finite set for each fixed M, N, db , dw . Because Ω is finite,
we may impose the uniform probability measure on it.
We now define the following fundamental parameters:
α :=

(2.6)

M
dw
,
=
N
db

γ :=

1
N
=
.
α
M

where here we use the identity (2.2). As in [8], we now impose the following constraints on the parameters M, N, db , dw :
lim

(2.7)

M,N →∞

α > 1.

This assumption is not crucial as we may also relabel the vertices V if M < N in the limit. This assumption will be convenient
in our analysis of the spectral statistics, however. This completes our construction of the random matrix ensemble of interest.
2.2. Random Covariance Matrices. Strictly speaking, the random matrix ensemble studied in [8] was the ensemble X∗
consisted of the corresponding N × N covariance matrices:
X∗ = H ∗ H,

(2.8)

−1/2
H = dw
(A − db eb e∗w ).

The ensemble X introduced in this paper may be realized as a linearization of the ensemble X∗ of covariance matrices.
This will be the upshot of working instead with the matrix ensemble X whenever convenient, i.e. when studying linear
perturbations of the adjacency matrix of a biregular graph. The following result shows that when transferring between the
ensembles X and X∗ , the spectral data is preserved. This result is standard in linear algebra and the analysis of compact
operators, but we include it for completeness and since organizational purposes, as the result does not seem to be written
precisely and formally in any standard text.
Before we give the result and its (short) proof, we define the following third matrix ensemble X∗,+ of M × M covariance
matrices:
X∗,+ = HH ∗ .

(2.9)

The ensemble X∗,+ will not play any essential role in our analysis of random matrix ensembles and is included in this paper
for the sake of completeness of our results.
Proposition 2.3. Suppose H is a real-valued matrix of size M × N with M > N , and suppose X is a block matrix of the
following form:
!
0 H
(2.10)
X =
.
H∗ 0
• (I). The spectrum of X admits the following decomposition:
(2.11)

σ(X) = σ 1/2 (H ∗ H) ∪ ζ(X),
where σ 1/2 (H ∗ H) denotes the pairs of eigenvalues (±λ) such that (±λ)2 is an eigenvalue of H ∗ H. Here, ζ(X) denotes
the set of eigenvalues not in σ 1/2 (H ∗ H), all of which are 0.
4

• (II). The spectrum of HH ∗ admits the following decomposition:
σ(HH ∗ ) = σ(H ∗ H) ∪ ζ 2 (X),

(2.12)

where ζ 2 (X) denotes the set of eigenvalues not in σ(H ∗ H), all of which are 0.
• (III). Suppose λ2 ∈ σ(H ∗ H) is associated to the following ℓ2 -normalized eigenvectors:
v∗ ! H ∗ H,

(2.13)

(2.14)

v∗,+ ! HH ∗ .

Then ±λ is associated to the following ℓ2 -normalized eigenvector pair of X:
!
1 v∗,+
±λ ! √
.
2 ±v∗

• (IV). Conversely, any eigenvalue pair ±λ ∈ σ 1/2 (H ∗ H) is associated to the following ℓ2 -normalized eigenvector pair of
X:

1
±λ ! √
2

(2.15)

v∗,+
±v∗

!

,

where v∗,+ is an ℓ2 -normalized eigenvector of HH ∗ with eigenvalue λ2 and v∗ is an ℓ2 -normalized eigenvector of H ∗ H
with eigenvalue λ2 .
• (V). Suppose λ = 0 ∈ ζ 2 (X) is associated to the ℓ2 -normalized eigenvector vλ of HH ∗ . Then for some λ′ = 0 ∈ ζ(X),
the corresponding ℓ2 -normalized eigenvector is given by
!
vλ
′
λ !
(2.16)
.
0
• (VI). Conversely, suppose λ ∈ ζ(X). Then λ = 0 is associated to the following ℓ2 -normalized eigenvector of X:
!
vλ
(2.17)
,
λ !
0
where vλ is an ℓ2 -normalized eigenvector of HH ∗ with eigenvalue λ′ = 0.
Remark 2.4. We briefly note that Proposition 2.3 applies to a much more general class of covariance matrices and their
linearizations, as it does not refer to any underlying graph or graph structure.
Proof. Statements (I) – (II) are consequence of the SVD (singular value decomposition) of the matrix H and dimensioncounting. Statements (III) – (VI) follow from a direct calculation and dimension-counting.

2.3. The Main Result. We begin with notation for the Stieltjes transforms of the Marchenko-Pastur law and its linearization,
respectively:
w ̺ (x)
∞
m∞ (z) =
(2.18)
dx,
x−z
R
w ̺(x)
(2.19)
dx.
m(z) =
x−z
R

Here, we take z ∈ C+ or z ∈ C− . We also define the following perturbed Stieltjes transforms to address the ensemble X∗,+ :
(2.20)
(2.21)

w γ̺ (x) + (γ − 1)δ (x)
γ−1
∞
0
=
dx,
z
x−z
R
w γρ(x) + (γ − 1)δ (x)
γ−1
0
=
dx.
m+ (z) := γm(z) +
z
x−z

m∞,+ (z) := γm∞ (z) +

R

5

This describes the limiting spectral behavior. For the graphs themselves, we define the following Green’s functions of the
matrix ensembles X , X∗ , and X∗,+ :
G(z) = (X − z)−1 ,

(2.22)

G∗ (z) = (X∗ − z)−1 ,

(2.23)

X ∈X;

G∗,+ (z) = (X∗,+ − z)−1 ,

(2.24)

X∗ ∈ X∗ ;
X∗,+ ∈ X∗,+ .

We also define the Stieltjes transforms of each covariance matrix ensemble X∗ and X∗,+ , which may be realized as the
Stieltjes transform of the corresponding empirical spectral distribution in spirit of Proposition 2.3:
1
1
Tr G∗ (z), s∗,+ (z) =
Tr G∗,+ (z).
N
M
Lastly, for the ensemble X , we define instead the partial Stieltjes transforms which average only over the diagonal terms of
s∗ (z) =

(2.25)

a specified color (black or whtie):
sb (z) =

(2.26)

M
1 X
Gii (z),
M i=1

sw (z) =

N
1 X
Gkk (z).
N
k=1

We now introduce domains in the complex plane on which we study the Green’s functions of each matrix ensemble. These
domains are engineered to avoid the singularities in the Green’s functions near the origin, and in the case of the linearized
Marchenko-Pastur law, the edge of the support. To this end, we establish notation for the following subsets of the complex
plane for any fixed ε > 0:
Uε,± := {z = E + iη : |E| > ε, η > 0} ,

(2.27)

Uε := Uε,+ ∪ Uε,− .

(2.28)

We will also need to define the following control parameters:
D := db ∧

(2.29)

N2
,
d3b

1
1
Φ(z) := √
+√ ,
Nη
D
"

(2.30)

Fz (r) = F (r) :=

(2.31)

We now present the main result of this paper.

1
1+ p
(λ+ − z)(z − λ− )

! #
r

∧

√

r.

Theorem 2.5. Suppose ξ = ξN is a parameter chosen such that the following growth conditions on D and η hold:
ξ log ξ ≫ log2 N,

(2.32)

|η| ≫

ξ2
,
N

D ≫ η2 .

Then for any fixed ε > 0, we have the following estimates with probability at least 1−e−ξ log ξ , uniformly over all z = E+iη ∈ Uε

with η satisfying the growth condition in (2.32):
(2.33)

max |[G∗ (z)]ii − m∞ (z)| = O (Fz (ξΦ)) , max |[G∗ (z)]ij | = O
i

i6=j



ξΦ(z 2 )
z



.

Similarly, for any fixed ε > 0, we have the following estimates with probability at least 1−e−ξ log ξ , uniformly over all parameters
z = E + iη ∈ Uε with η satisfying the growth condition in (2.32):


ξΦ(z 2 )
(2.34)
.
max |[G∗,+ (z)]ii − m∞,+ (z)| = O (Fz (ξΦ)) , max |[G∗,+ (z)]ij | = O
i
i6=j
z
Conditioning on the estimates (2.33) and (2.34), respectively, uniformly over z = E + iη ∈ Uε with η satisfying the growth

condition in (2.32), we have
(2.35)

|s∗ (z) − m∞ (z)| = O (Fz (ξΦ)) ,

|s∗,+ (z) − m∞ (z)| = O (Fz (ξΦ)) .
6

Conditioning on the estimates (2.33), uniformly over all z = E + iη with η satisfying the growth condition in (2.32), we have
(2.36)
(2.37)


max |[G(z)]kk − m(z)| = O zFz2 (ξΦ(z 2 )) ,

k>M

max |[G(z)]kℓ | = O (ξΦ) .

M<k<ℓ

Moreover, conditioning on the estimates (2.34), for any fixed ε > 0, uniformly over all z = E + iη ∈ Uε with η satisfying the
growth condition in (2.32), we have

(2.38)
(2.39)


max |[G(z)]ii − m+ (z)| = O zFz2 (ξΦ(z 2 )) ,
i6M

max |[G(z)]ij | = O(ξΦ).

i<j6M

Conditioning on (2.33) and (2.34), we have the following estimates uniformly over all z = E + iη with η satisfying the growth
condition in (2.32):
(2.40)
(2.41)


|sb (z) − m+ (z)| = O zFz2 (ξΦ(z 2 )) ,

|sw (z) − m(z)| = O zFz2 (ξΦ(z 2 )) .

Lastly, the estimates (2.33) and (2.35) hold without the condition |E| > ε if α > 1. The estimates (2.38) and (2.39) hold without

the condition |E| > ε if α = 1.

Remark 2.6. We briefly remark on the repulsion assumption |E| > ε in Theorem 2.5. The removal of this assumption discussed
at the end of the statement of Theorem 2.5 is a direct consequence of studying the dependence of the singularities of the Green’s
functions and Stieltjes transforms at the origin with respect to the structural parameter α. For example, the presence of a
singularity of m∞ at the origin occurs exactly when α = 1. Moreover, the singularities in the Stieltjes transforms of matrices
and the singularities of m∞,+ at the origin cancel each other out, allowing for a regularization at the origin.
Remark 2.7. We last remark that if α = 1, the covariance matrices X∗ and X∗,+ are equal in law. This comes from symmetry
of the bipartite graph between the two vertex sets Vb and Vw , i.e. the graph statistics are unchanged upon relabeling the
graph. This allows us to remove the assumption |E| > ε > 0 for certain estimates in Theorem 2.5 in the regime α = 1.
We now discuss important consequences of Theorem 2.5, the first of which is the following result on eigenvector delocalization, i.e. an estimate on the ℓ∞ -norm of an eigenvector in terms of its ℓ2 -norm. The proof of this delocalization result will
be delegated to a later section after we study in more detail the spectral data of covariance matrices and their linearizations.
Corollary 2.8. (Eigenvector Delocalization).
Assume the setting of Theorem 2.5, and suppose u is an eigenvector of X∗ with eigenvalue λ. Then with probability at least
1 − e−ξ log ξ , we have
(2.42)

kukℓ∞ = O




ξ
√ kukℓ2 .
N

We briefly remark that the eigenvector delocalization fails for the larger covariance matrix X∗,+ .
Proof. First, we note in the case u ∈ Span(eb ), the result is true trivially. Moreover, by Proposition 2.3, it suffices to prove

the claim for eigenvectors of the linearization X, replacing the ℓ∞ -norm by a supremum over indices k > M .
We now take for granted |zm∞ (z 2 )| = O(1) uniformly for z = E + iη ∈ C+ ; this follows from an elementary analysis

of the Stieltjes transform discussed in the appendix of this paper. This allows us to obtain the following string of inequalities
7

with probability at least 1 − e−ξ log ξ and any index k > M :
|u(k)|2 6

(2.43)

X

vβ 6=u

2

η 2 |vβ (k)|
(λβ − λ)2 + η 2

(2.44)

= η Im[G(λ + iη)]kk

(2.45)

6 η zm∞ (z 2 ) + O(η

(2.46)

6 2η,

p

ξΦ)

where we used the local law for the linearization X to estimate the second line. This completes the derivation of the eigenvector delocalization.

We conclude this preliminary discussion concerning consequences of Theorem 2.5 with the following weak rigidity estimates. We briefly remark that it relies heavily upon the Helffer-Sjostrand formula and functional calculus, and beyond these
tools, the local law in Theorem 2.5. To state the result, we first introduce the following definition.
Definition 2.9. For each i ∈ [[1, N ]], we define the i-th classical location, denoted γi , by the following quantile formula:
wγi
i
=
̺∞ (E) dE,
N
−∞

(2.47)

where we recall ̺∞ denotes the density function of the Marchenko-Pastur law.
The following consequence of Theorem 2.5 will compare the classical location γi to the i-th eigenvalue λi of the covariance
matrix X∗ , where the ordering on the eigenvalues is the increasing order.
Corollary 2.10. For any fixed κ > 0 and index i ∈ [[κN, (1 − κ)N ]], we have, with probability at least 1 − e−ξ log ξ ,
 2 
ξ
|λi − γi | = O
(2.48)
.
D1/4
For details of the proof, we refer to Section 5 in [4] and Section 7 in [13].
We now give an outline for the derivation of the local law. The proof will roughly consist of the following three steps:
• (I). The first step will be to adapt the methods in [5] to define and study a method of resampling biregular graphs in
Ω. The resampling will be generated by local operations on a given graph known as switchings, which we will define
more precisely in a later section. The local nature of the resampling method will help us derive equations exploiting
the probabilistic stability of the Green’s function under these switchings.
• (II). The second step will be to study the Green’s functions of the three matrix ensemble simultaneously. This includes
both a preliminary analysis and a further analysis using the switching dynamics established in the previous step.
In particular, we derive an approximate self-consistent equation for the diagonal entries of the Green’s function and
study its stability properties. As in [5], this will help us compare the diagonal of the Green’s function to the associated
Stieltjes transform. The equation in [5], however, contains a constant leading-order coefficient whereas for covariance
matrices the leading-order coefficient is nonconstant. We adapt the methods suitably to handle this nonlinearity.
3. Switchings on Bipartite Graphs
We begin by introducing notation necessary to define switchings on biregular graphs. Switchings will be local operations
on the biregular graphs, so we will establish notation for vertices and edges containing said vertices as follows.
Notation 3.1. A generic vertex in Vb (resp. Vw ) will be denoted by vb (resp. vw ).
b
For a fixed graph E ∈ Ω, we will denote the edges in E containing vb by {eb,µ }dµ=1
. Moreover, for a fixed edge eb,µ containing

vb , we will denote the neighboring vertex by vb,µ , so that eb,µ = vb vb,µ .
w
. For a fixed edge ew,ν containing vw , we will denote the
Similarly, the edges in E containing vw will be denoted by {ew,ν }dν=1
neighboring vertex by vw,ν .

8

For a fixed vertex vb ∈ Vb , we establish the notation for the set of edges not containing vb :
(3.1)

Uvb := {edges e ∈ E : vb 6∈ e} .

Similarly for a fixed vertex vw ∈ Vw , we define the following set of edges not containing vw :
(3.2)

Uvw := {edges e ∈ E : vw 6∈ e} .

We may now begin to define a switching on a generic graph E ∈ Ω. To this end, we fix a black vertex vb ∈ Vb and an edge

ev,µ for some µ ∈ [[1, db ]]. We define the following space of subgraphs of E:
(3.3)

Svb ,µ,E := {S ⊂ E : S = {eb,µ , pb,µ , qb,µ }, pb,µ 6= qb,µ ∈ Uvb } .

In words, the set Svb ,µ,E is the set of graphs consisting of the edges eb,µ and any two distinct edges pb,µ and qb,µ , neither of
which contains the vertex vb . Similarly, we may define for a fixed white vertex vw ∈ Vw and edge ew,ν , for some ν ∈ [[1, dw ]],
the same set of graphs:

(3.4)

Svw ,µ,E := {S ⊂ E : S = {ew,ν , pw,ν , qw,ν } : pw,ν 6= qw,ν ∈ Uvw } .

Notation 3.2. A generic graph in Svb ,µ,E will be denoted by Sb,µ . A generic graph in Svw ,ν,E will be denoted by Sw,ν .
The set Svb ,µ,E contains the edge-local data along which switchings on graphs will be defined. To make this precise, we
need to introduce the following indicator functions. First, we define the following configuration vectors for fixed vertices
vb ∈ Vb and vw ∈ Vw :
(3.5)

d

d

w
Svw := (Sw,ν )ν=1
.

b
Svb := (Sb,µ )µ=1
,

With this notation, we define the following indicator functions that detect graph properties in Sb,µ and Sw,ν .
(3.6)
(3.7)

I(Sb,µ ) = 1 (|[Sb,µ ]| = 6) ,
Y
J(Svb , µ) =
1 ([Sb,µ ] ∩ [Sb,µ′ ] = {vb }) ,
µ′ 6=µ

(3.8)

W (Svb ) = {µ : I(Sb,µ )J(Svb , µ) = 1} .

For white vertices vw ∈ Vw , the functions I, J and W retain the same definition upon replacing b with w and µ with ν.
e which will make the switchings systematic from the perspective of
We now define the augmented probability spaces Ω

Markovian dynamics. For a fixed black vertex vb ∈ Vb and a fixed white vertex vw ∈ Vw , we define the following augmented
space:
)
(
db
dw
Y
Y
e = (E, Sv , Svw ) , E ∈ Ω, Sv ∈
(3.9)
Ω
Svb ,µ,E , Svw ∈
Svw ,ν,E .
b
b
µ=1

ν=1

e To this end we define switchings on configuration vectors
We now precisely define switchings by defining dynamics on Ω.
Svb and Svw ; we first focus on the configuration vectors for black vertices.

Fix a label µ and consider a component Sb,µ of a uniformly sampled configuration vector Svb . Precisely, the components
of Svb are sampled jointly uniformly and independently from Svb ,µ,E , where E ∈ Ω is uniform over all µ and sampled
uniformly. We now define the following map:
(3.10)

Tb :

db
Y

µ=1

Svb ,µ,E −→

db
Y

Svb ,µ,E ′

µ=1

where E ′ ∈ Ω is possibly different from E. The map is given as follows: for any µ, we define the map Tb,µ

S
µ 6∈ W (Svb )
b,µ
(3.11)
Tb,µ (Sb,µ ) =
.
(S , s ) µ ∈ W (S )
b,µ

9

b,µ

vb

We define the graph (Sb,µ , sb,µ ) as follows; this is where we now introduce randomness into the dynamics Tb . Suppose
µ ∈ W (Svb ), in which case Sb,µ is 1-regular and bipartite with respect to the vertex sets ([Svb ], V1 , V2 ). Consider the set of
1-regular bipartite graphs with respect to the vertex set ([Sb,µ ], V1 , V2 ). In words, this is the set of 1-regular graphs on [Sb,µ ]
such that, upon replacing Sb,µ with any such graph, the global graph E remains biregular. We now define (Sb,µ , sb,µ ) to be
drawn from this set uniformly at random conditioning on the event (Sb,µ , sb,µ ) 6= Sb,µ . Lastly, we define the following global
dynamics:

Tb =

(3.12)

db
Y

Tb,µ ,

µ=1

where the product is taken as composition. We note this product is independent of the order of composition; this is a consequence of the definition of the functions I, J and W . For white vertices vw ∈ Vw , we define the map Tw by replacing all
black indices b and white indices w.
e because we are allowed to change the underlying graph E when varyWe note that the maps Tb and Tw define maps on Ω,
e this is the utility of the almost-product representation of Ω.
e This allows us to finally define switchings
ing over the space Ω;

of a biregular graph.

Definition 3.3. For a fixed black vertex vb ∈ Vb and a fixed label µ ∈ [[1, db ]], the local switching at vb along µ is the map
Tb,µ . The global switching is the map Tb .
Similarly, for a fixed white vertex vw ∈ Vw and a fixed label ν ∈ [[1, dw ]], the local switching at vw along ν is the map

Tw,ν . The global switching at vw is the map Tw .

Remark 3.4. We note our construction, technically, implies the mappings Tb,µ and Tw,ν are random mappings on the auge Via this construction, we obtain a probability measure Ω
e induced by the uniform measure and a uniform
mented space Ω.

sampling of switchings. To obtain an honest mapping on the original space Ω, we may instead construct deterministic mappings by averaging over the random switchings. For precise details, we cite [5].
3.1. Switchings on Adjacency Matrices. We now aim to translate the combinatorics of graph switchings into analysis of
adjacency matrices. Suppose E ∈ Ω is a biregular graph with adjacency matrix A. We will fix the following notation.
Notation 3.5. For an edge e = ij on the vertex set V , we let ∆ij denote the adjacency matrix of the graph on V consisting only
of the edge e. In particular, ∆ij is the matrix whose entries are given by
(3.13)

(∆ij )kℓ = δik δjℓ + δiℓ δjk .

In the context of switchings on biregular graphs, the matrices ∆ij are perturbations of adjacency matrices. This is made
precise in the following definition.
Definition 3.6. Fix a black vertex vb ∈ Vb and a label µ ∈ [[1, db ]]. A local switching of A, denoted Tb,µ , at vb along µ is
given by the following formula:

(3.14)

Tb,µ (A) = A − Sb,µ + (Sb,µ , sb,µ ),

where Sb,µ is a component of a random, uniformly sampled configuration vector Svb . A global switching of A, denoted Tb ,
is the composition of the random mappings Tb,µ .
Similarly, we may define local switchings and global switchings of adjacency matrices for white vertices by replacing the
black subscript b with the white subscript w, and replacing the label µ with ν.
Clearly, a local or global switching of an adjacency matrix is the adjacency matrix corresponding to a local or global
switching of the underlying graph. To realize the matrices ∆ij as perturbations, we will rewrite the formula defining Tb,µ
as follows. As usual, we carry out the discussion for black vertices vb ∈ Vb , though the details for white vertices vb follow
analogously.

10

First, we recall the following notation for a component Sb,µ ∈ Svb ,µ,E of a configuration vector Svb :
Sb,µ := {eb,µ , pb,µ , qb,µ } ,

(3.15)

subject to the constraint that Sb,µ contains three distinct edges.
Notation 3.7. We will denote the vertices of pb,µ by ab,p,µ ∈ Vb and aw,p,µ ∈ Vw . Similarly, we will denote the vertices of qb,µ

by ab,q,µ ∈ Vb and aw,q,µ ∈ Vw .

With this notation, we may rewrite the random mapping Tb,µ as follows:


Tb,µ (A) = A − ∆vb ,vb,µ + ∆ab,p,µ aw,p,µ + ∆ab,q,µ aw,q,µ + ∆vb ,x + ∆ab,p,µ ,y + ∆ab,q,µ ,z ,
(3.16)

where we recall eb,µ = vb vb,µ . Here, the variables x, y, z are three distinct vertices sampled from the set of white vertices
{vb,µ , aw,p,µ , aw,q,µ } conditioning on the following constraint on ordered triples:
(x, y, z) 6= (vb,µ , aw,p,µ , aw,q,µ ).

(3.17)

3.2. Probability Estimates on Vertices. In this discussion, we obtain estimates on the distribution of graph vertices after
performing switchings. The main estimates here show that the vertices are approximately uniformly distributed, which we
make precise in the following definition.
Definition 3.8. Suppose S is a finite set and X is an S-valued random variable. We say (the distribution of) X is approximately uniform if the following bound on total variation holds:


X
1
1
(3.18)
.
P (X = s) −
6 O √
|S|
dw D
s∈S

e These σ-algebras will allow us to focus on edge-local features of graphs
We now introduce the following σ-algebras on Ω.

E ∈ Ω upon conditioning on the global data E.

Definition 3.9. For a fixed label µ ∈ [[0, db ]], we define the following σ-algebras:
Fµ := σ (E, (Sb,1 , sb,1 ), . . . , (Sb,µ , sb,µ )) ,


Gµ := σ E (Sb,µ′ , sb,µ′ )µ′ 6=µ .

(3.19)
(3.20)

We similarly define the σ-algebras Fν and Gν for ν ∈ [[1, dw ]] for white vertices.

In particular, conditioning on F0 corresponds to conditioning on the graph E only. The last piece of probabilistic data we
introduce is the following notation, which will allow us to compare i.i.d. switchings on biregular graphs.
e denotes a
Notation 3.10. Suppose X is a random variable on the graph data E, {(Sb,µ , sb,µ )}µ , {(Sw,ν , sw,ν )}ν . Then X
e {(Seb,µ , seb,µ )µ , {(Sew,ν , sew,ν )}, where the tildes on the graph data denote i.i.d. resamplings.
random variable on the variables E,

Notation 3.11. For notational simplicity, by pµ , qµ or pν , qν , we will refer to either pb,µ , qb,µ or pw,ν , qw,ν , respectively, whenever
the discussion applies to both situations.

We now focus on obtaining an estimate on the distribution of the pair of edges (pµ , qµ ), and similarly for (pν , qν ). As
with all results concerning switchings from here on, details of proofs resemble those of Section 6 in [5], so we omit details
whenever redundant.
Lemma 3.12. Conditioned on Gµ , the pair (pµ , qµ ) is approximately uniform, i.e., for any bounded symmetric function F , we
have
(3.21)

EGµ F (pµ , qµ ) =



X
1
1
kF
k
F
(p,
q)
+
O
∞ .
(N dw )2
N
p,q∈E
11

Similarly, for any bounded function F , we have
EGµ ,qµ F (pµ ) =

(3.22)



1 X
1
F (p) + O
kF k∞ .
N dw
N
p∈E

Proof. Assume we resample about vb ∈ Vb ; the case for vw ∈ Vw follows analogously. By definition, we have
X
1
EGµ F (pµ , qµ ) =
(3.23)
F (p),
(N dw − dw )(N dw − dw − 1)
p∈Evb

where Evb is the set of edges in E that are not incident to vb . Then, (3.21) follows from the following estimate


1
1
1
(3.24)
+
O
=
(N dw − dw )(N dw − dw − 1)
(N dw )2
N 3 d2w

as well as the estimate |Evb | 6 (N dw )2 , and lastly the estimate |EvCb | 6 N dw . This last upper bound follows combinatorially;
for details, see the proof of Lemma 6.2 in [5]. The estimate (3.22) follows from a similar argument.

Because an edge is uniquely determined by its vertices in the graph, we automatically deduce from Lemma 3.12 the following approximately uniform estimate for resampled vertices as well.
Corollary 3.13. Conditioned on Gµ , the pair (pµ (b), qµ (b)) (resp. (pµ (w), qµ (w))) is approximately uniform.
Similarly, conditioned on Gµ and qµ (b) (resp. qµ (w)), the random variable pµ (b) (resp. pµ (w)) is approximately uniform.
Proof. This follows immediately upon applying Lemma 3.12 to the function F (pµ , qµ ) = f (pµ (b), qµ (b)).



To fully exploit the resampling dynamics, we need a lower bound on the probability that a local switching Sb,µ , sb,µ around
a vertex vb ∈ Vb does not leave the graph fixed. In particular, we need an estimate for the probability of the event µ ∈ W (Svb )
where here µ is fixed and the set W is viewed as random. As discussed in [5], to provide an estimate, the naive approach
to estimating this probability conditioning on Gµ fails in an exceptional set. Precisely, suppose the µ-th neighbor vb,µ of vb
lives in Sb,µ′ for some µ′ 6= µ. In this case, almost surely, we have [Sb,µ ] ∩ [Sb,µ′ ] is nontrivial. It turns out this is the only

obstruction, so we aim to show that vb,µ ∈ [Sb,µ′ ] occurs with low probability for any µ′ 6= µ.
Formally, we define the following indicator random variable which detects this exceptional set:
Y
h(Svb , µ) =
(3.25)
1 (vb,µ ∈ Sb,µ′ ) .
µ′ 6=µ

Thus, the estimates we need are given in the following result.
Lemma 3.14. For any neighbor index µ, we have
PGµ [I(Sb,µ )J(Svb , µ) = h(Svb , µ)] > 1 − O

(3.26)
Moreover, we have

PF0 [h(Svb , µ) = 1] > 1 − O

(3.27)



db
N





db
N



.

.

Proof. We first note that (3.26) follows immediately conditioning on h = 0. In particular, the first lower bound (3.26) follows
from a combinatorial analysis of the underlying graph using the following union bound:
PGµ ,h=1 [I(Sb,µ )J(Svb , µ) = 0] 6 PGµ [I(Sb,µ ) = 0]
+ PGµ ,h=1 [J(Svb , µ) = 0] .

(3.28)
Similarly, (3.27) follows from the union bound
(3.29)

PF0 [h(Svb , µ) = 0] 6

X

µ′ 6=µ

For details, we refer back to [5].

PF0 [vb,µ ∈ [Sµ′ ]] .


12

f , W are i.i.d. copies of
We conclude this section with an estimate that compares independent resamplings. Recall that W

the random variable W (Svb ). The following result bounds the fluctuation in W (Svb ) from independent resamplings.
Lemma 3.15. Almost surely, we know
(3.30)



f = O(1),
# W ∆W

where the implied constant is independent of N . Moreover, we also have
 
h
i
f 6= ∅ 6 O db .
PGµ W ∆W
(3.31)
N

The proof follows the argument concerning Lemma 6.3 in [5] almost identically, so we omit it. We now present the final

estimate on adjacency matrices comparing switched matrices upon i.i.d. switchings in the sense of matrix perturbations. This
will allow us to perform and control resamplings of biregular graphs, in particular using the resolvent perturbation identity.
Lemma 3.16. Under the setting of the resampling dynamics, we have
(3.32)

e − A = Tb,µ (A) − Teb,µ (A)
e
A

with probability at least 1 − O(db /N ). Almost surely, we have

O(1)

(3.33)

e−A =
A

X

∆xy ,

x,y=1

such that either, conditioning on Gµ , the random indices x, y are approximately uniform in the corresponding set Vb or Vw or,
conditioning on Gµ , pµ , peµ , at least one of the random indices x, y is approximately uniform in the appropriate vertex set.
Lastly, the statement remains true upon switching instead at a white vertex vw along an edge label ν.
Proof. The result follows from unfolding Lemma 3.15 and the following deterministic identity:
i
h
e − A = 1 f Teb,µ (E) − E − 1µ∈W [Tb,µ (A) − A]
A
µ∈W
X
(3.34)
± [Tb,µ (A) − A] ,
+
f ∆W
µ′ ∈W

f contains the indexing label µ′ .
where the sign corresponds to which of the random sets W or W



4. Green’s Function Analysis

4.1. Preliminary Resolvent Theory. Here we record the following fundamental identities for studying the Green’s functions of adjacency matrices in the ensemble X . These identities are standard and follow from standard linear algebra. First,
because these identities hold for Green’s functions of any real symmetric matrix, we fix the following notation.
Notation 4.1. Suppose F is a function of the Green’s function or matrix entries of matrices belonging to any one of the matrix
ensembles X, X∗ , or X∗,+ . Then we establish the notation F⋆ to be the function obtained when restricted to the matrix ensemble
X⋆ , where we take ⋆ to be blank or ⋆ = ∗ or ⋆ = ∗, +.
Lemma 4.2. (Resolvent Identity)
Suppose A and B are invertible matrices. Then we have
(4.1)

A−1 − B −1 = A−1 (B − A)B −1 .

e denote real symmetric or complex Hermitian matrices with Green’s functions G(z) and G(z),
e
In particular, if H and H
respectively, for z 6∈ R, then
h 
 i
e
e −H G
e (z).
G(z) − G(z)
= G H
(4.2)
e
As an immediate consequence by letting G(z)
= G(z), we deduce the following off-diagonal averaging identity.
13

Corollary 4.3. (Ward Identity)
Suppose H is a real symmetric matrix of size N with Green’s function G(z). Then for any fixed row index i ∈ [[1, N ]],
(4.3)

N
X

k=1

|Gik (E + iη)|2 =

Im Gii (E + iη)
.
η

In particular, we obtain the following a priori estimate for any matrix index (i, j):
(4.4)

|Gij (E + iη)| 6

1
,
η

and thus for any matrix index (i, j), the function Gij (z) is locally Lipschitz with constant η −2 .
The third preliminary result we give is the following representation of the Green’s function G(z; H) in terms of the
spectral data of H which is also an important result in compact operator and PDE theory. This spectral representation will
be indispensable for exploiting the rich spectral correspondence among covariance matrices and their linearizations.
Lemma 4.4. (Spectral Representation)
Suppose H is a real-symmetric or complex-Hermitian matrix with eigenvalue-eigenvector pairs {(λα , uα )}α , and let G(z)

denote its Green’s function. Then for any matrix index (i, j), we have
(4.5)

Gij (z) =

N
X
uα (i)uα (j)
,
λα − z
α=1

where the overline notation denotes the complex conjugate of the vector entry. In particular, the Green’s function is complex
Hermitian.
We conclude this preliminary discussion of the Green’s function G(z; H) with the following local regularity result concerning a maximal Green’s function. The proof of this result may be found as Lemma 2.1 in [5]. To state it, we now define
the maximal Green’s functions of interest, which may be viewed as control parameters for the sake of this paper:


Γ(E + iη) = max |Gij (z)| ∨ 1,
i,j

Γ∗ (E + iη) = sup Γ(E + iη ′ ).
η ′ >η

Lemma 4.5. For any z = E + iη ∈ C+ , the function Γ(z) is locally Lipschitz continuous in η with the following bound on its
almost-everywhere derivative:
(4.6)

|∂η Γ(z)| 6

Γ(z)
.
η

In particular, for any κ > 1 and z = E + iη ∈ C+ , we have

η
Γ E+i
(4.7)
6 κΓ(E + iη).
κ

4.2. Reductions of the Proof of Theorem 2.5. We now return to the setting of biregular bipartite graphs, i.e. the ensembles
X , X∗ , and X∗,+ . We begin with the following consequence of Lemma 4.4, which relates the Green’s function entries of
matrices from each of the three matrix ensembles of interest.
Lemma 4.6. Suppose X is a block matrix of the form (2.10), and suppose i, j ∈ [[1, M + N ]] are indices chosen such that either

i, j 6 M or i, j > M . Then, for any z = E + iη ∈ C+ , we have

zG (z 2 ) i, j 6 M
∗,+
Gij (z) =
(4.8)
.
zG (z 2 )
i, j > M
∗
14

Proof. For simplicity, we suppose X is real symmetric as the proof for complex Hermitian matrices is similar. First suppose
i, j 6 M . By the spectral representation in (4.5) and Proposition 2.3, we obtain


X uα (i)uα (j)
X
1 uα (i)uα (j)
uα (i)uα (j)
√
√
Gij (z) =
(4.9)
,
=
+
λα − z
2
λα − z
− λα − z
α
λ∈σ(HH ∗ )
where the last equality holds by abuse of notation for eigenvectors of the covariance matrix HH ∗ versus the linearization X.
This completes the derivation for the case i, j 6 M . The proof for the case i, j > M follows by the exact same calculation, but
instead taking a summation over σ(H ∗ H) and noting the eigenvector terms uα (i)uα (j) vanish for λα ∈ ζ(X) by Statements
(V) and (VI) in Proposition 2.3.

Lemma 4.6 now gives the first reduction of the proof of Theorem 2.5.
Lemma 4.7. Assuming the setting of Theorem 4.1, then the following two estimates are equivalent:
• (I). For any fixed ε > 0, we have with probability at least 1 − e−ξ log ξ , uniformly over z = E + iη ∈ Uε with η ≫ ξ 2 /N ,
max |[G∗ (z)]ii − m∞ (z)| = O (Fz (ξΦ)) , max |[G∗ (z)]ij | = O(ξΦ).

(4.10)

i

i6=j

• (II). For any fixed ε > 0, we have with probability at least 1 − e−ξ log ξ , uniformly over z = E + iη ∈ Uε with η ≫ ξ 2 /N ,

max [G(z)]kk − zm∞ (z 2 ) = O zFz2 (ξΦ(z 2 )) ,

max |[G(z)]kℓ | = O zξΦ(z 2 ) .

(4.11)

k>M

(4.12)

M<k<ℓ

Similarly, the above equivalence holds replacing G∗ with G∗,+ and taking the maximums over i 6 M and i, j 6 M .
From Lemma 4.6, we also deduce the next reduction.
Lemma 4.8. Assuming the setting of Theorem 4.1, the following estimates are equivalent for any z ∈ C+ :

sb (z) − zm∞,+ (z 2 ) = O zFz2 (ξΦ(z 2 )) ,

sw (z) − zm∞ (z 2 ) = O zFz2 (ξΦ(z 2 )) .

(4.13)
(4.14)

Similarly, the following estimates are equivalent for any z ∈ C+ :
(4.15)

|s∗ (z) − m∞ (z)| = O (Fz (ξΦ)) ,

(4.16)

|s∗,+ (z) − m∞,+ (z)| = O (Fz (ξΦ)) .

We briefly note that Lemma 4.8 improves upon Lemma 4.7 in that it removes the restriction |E| > ε on the energy.

We are now in a position to make our final reduction of the proof of the local laws in Theorem 4.1, which exploits the first
reduction in Lemma 4.7 and thus allows us to focus on the covariance matrices X∗ and X∗,+ . The reduction will depend on
the following result, for which we need to define the following spectral domain.
DN,δ,ξ = {z = E + iη : |E| 6 N δ , ξ 2 /N 6 η 6 N }.

(4.17)

Proposition 4.9. Suppose ξ, ζ > 0 and D ≫ ξ 2 . If, for a fixed z ∈ DN,δ,ξ , we have
P (Γ∗⋆ (z) = O(1)) > 1 − e−ζ ,

(4.18)

then, with probability at least 1 − e−(ξ log ξ)∧ζ+O(log N ) , we have
(4.19)

max |[G⋆ (z)]ii − m(z)| = O(Fz (ξΦ(z))), max |[G⋆ (z)]ij | = O
i

Here, ⋆ can take the values ⋆ = ∗ and ⋆ = ∗, +.

i6=j

15



ξΦ(z 2 )
z



.

To deduce Theorem 2.5 from Proposition 4.9, we follow the exactly the argument used to deduce Theorem 1.1 from Proposition 2.2 in [5]. The only thing we need to check to apply the same argument are the following bounds:
m∞ (z), m∞,+ (z) 6 C,

(4.20)

for some constant C = O(1). These estimates are proven in the appendix of this paper. We may also extend this argument
to remove the energy repulsion assumption |E| > ε, which we precisely state in the following lemma.
Lemma 4.10. Suppose Proposition 4.9 holds. Then the estimates (4.10) and (4.11) hold without the assumption |E| > ε. Consequently, the estimates (4.14) and (4.15) hold without the assumption |E| > ε. Moreover, if α > 1, then the estimate (4.7) holds

without the assumption |E| > ε.

Proof. We may again apply the iteration scheme used in proving Theorem 1.1 from Proposition 2.2 in [5]. Here, we need to
check the estimate m(z) = O(1) and that in the regime α > 1, we have the estimate m∞ (z) = O(1). These are similarly
derived in the appendix of this paper.



Thus, contingent on estimates derived in the appendix, to prove Theorem 2.5 it suffices to prove Proposition 4.9. This will
be the focus for remainder of the paper. In particular, we may now work with an explicitly smaller domain DN,δ,ξ and an a
priori estimate on the maximal Green’s function.
4.3. Switchings on Green’s Functions. The main result in this subsection consists of the following estimates comparing
Green’s function entries to index-wise averages. These estimates will be fundamental to controlling terms that show up
naturally in the derivation of the self-consistent equation. Before we can state the result, we need to first introduce a notion
of high probability used throughout the remainder of this paper.
Definition 4.11. Fix a parameter t = tN ≫ log N , and a probability space Ω. We say an event Ξ ⊂ Ω holds with t-high

probability, or t-HP for short, if
(4.21)

P ΞC



6 e−t+O(log N ) .

As suggested in Theorem 4.1, we will take the parameter t = (ξ log ξ) ∧ ζ. We now state the main estimates.
Lemma 4.12. Fix a vertex i = vb ∈ Vb and an edge label µ ∈ [[1, db ]]. Suppose z = E + iη ∈ C+ satisfies the following
constraints for a fixed ε > 0:

|E| > ε,

(4.22)

η &

1
.
N

Suppose further that Γ = O(1) holds with t-HP. Then for all fixed indices j, ℓ, r we have
!
M+N


1 X
−1/2
−1/2
EF0 Gvb,µ j −
Φ),
sw Gij + O(dw
Gkj = − EF0 dw
N
k=M+1
"
!#
M+N


1 X
−1/2
−1/2
EF0 Gℓr Gvb,µ j −
sw Gij + O(dw
= − EF0 Gℓr dw
Φ).
Gkj
N
k=M+1

Similarly, fix a vertex k = vw ∈ Vw and µ ∈ [[1, dw ]], and suppose z ∈ C+ satisfies the constraints (4.22). Further suppose
Γ = O(1) with t-HP. Then for all fixed indices j, ℓ, r, we have
!
M


1 X
−1/2
−1/2
Φ),
sb Gkj + O(dw
Gij = − EF0 dw
EF0 Gvw,µ j −
M i=1
"
!#
M


1 X
−1/2
−1/2
Φ).
EF0 Gℓr Gvw,µ j −
sb Gkj + O(dw
= − EF0 Gℓr dw
Gij
M i=1
16

Before we provide a proof of this result, we will need an auxiliary estimate comparing Green’s function entries from i.i.d.
samples of graphs. To state this auxiliary estimate, we define first the conditional maximal Green’s functions for any fixed
edge label µ ∈ [[1, db ]] or ν ∈ [[1, dw ]]:
Γµ = Γµ (z) := kΓ(z)kL∞ (Gµ ) ,

(4.23)

where the notation L∞ (Gµ ) in the norm denotes the L∞ -norm conditioning on the σ-algebra Gµ .
Lemma 4.13. For any fixed indices i, j ∈ [[1, M + N ]] and any label µ ∈ [[1, db ]], we have
−1/2
eij = O(dw
Gij − G
Γµ Γ).

(4.24)

Moreover, suppose x, y are random variables such that, conditioned on Gµ and x, the random variable y is approximately uniform.
Then we have
2

EGµ |Gxy | = O(Γ4µ Φ2 ).

(4.25)

The estimate also holds for any fixed label ν ∈ [[1, dw ]] of a white vertex.
The proof of this result follows from the proof of Lemma 3.9 in [5] so we omit it. We now record the following consequence
of Lemma 4.13 and proceed to the proof of Lemma 4.12.
Corollary 4.14. In the setting of Lemma 4.13, we have


−1/2
Γ = Γµ + O dw
Γµ Γ .

(4.26)
Proof. (of Lemma 4.12).

We prove the first estimate only for vb ∈ Vb ; the proof of the second estimate and the estimates for vw ∈ Vw are analogous.
In expectation, we first note
e ve j .
EF0 Gvb,µ j = EF0 G
b,µ

Moreover, because G(z) is independent of the random variable veb,µ , and because, conditioned on Gµ , the random variable

veb,µ ∈ Vw is approximately uniform, we also know
EF0

1
N

M+N
X

k=M+1

Gkj

!

−1/2

= EF0 Gveb,µ j + O(db

Φ).

Thus, it suffices to compute


e ve j .
− EF0 Gveb,µ j − G
b,µ

By the resolvent identity, we have the following equation holding in expectation:




X
e ve j = EF0 
eℓj  .
e − X)kℓ G
EF0 −Gveb,µ j − G
(4.27)
Gveb,µ k (X
b,µ
k,ℓ

−1/2

Unfolding the high-probability equation (11.15) in Lemma 11.9, we have, with probability at least 1 − O(dw
on F0 ,


e − X = d−1/2 ∆v ve − ∆v v + Σb + d−1/2 ∆ve v − ∆v v + Σ∗ .
(4.28)
X
b,µ b
b
w
b b,µ
w
b,µ b
b b,µ

Φ) conditioned

Here, we recall that vb,µ (resp. veb,µ ) is the vertex adjacent to vb in Svb ,µ (resp. Sevb ,µ ) after resampling. Also, Σb is the matrix
given by a sum of terms ±∆xy where one of the following two conditions holds:

• Conditioned on Gµ , peµ , the random variable x is approximately uniform, or;
• Conditioned on Gµ , the random variable y is approximately uniform.
17

Thus, upon unfolding the RHS of (4.27), we see one term is given by, in expectation,
h
i
h
i
−1/2
e ij = EF0 d−1/2 sG
eij + O(d−1/2 Φ)
Gveb,µ evb,µ G
EF0 dw
w
w
i
h
−1/2
−1/2
Φ),
sGij + O(dw
= EF0 dw

where the first equality holds because veβ,µ is approximately uniform by Corollary 11.6 and the second holds since for any
e ij conditioned on Gµ . In particular, we have
fixed indices i, j, we have Gij ∼ G
i


h


e ij − Gij
e ij = EF0 Gve ev Gij + EF0 Gve ve
G
EF0 Gveb,µ evb,µ G
b,µ b,µ
b,µ b,µ



1
= EF0 Gveb,µ evb,µ Gij + O √
,
D

where the second equality holds by Lemma 4.13. Thus, it suffices to bound the remaining terms in (4.27). By (4.28), it suffices
e yj . By the second result in Lemma 4.13 and the Schwarz inequality, in the case
to estimate the expectation of terms Gveb,µ x G

where y is approximately uniform conditioning on Gµ , we have, with high probability,

e yj |2 + O(D−1/2 ) 6 O(Φ).
e yj 6 EF0 |G
EF0 Gveb,µ x G

Thus, by the assumption Γ = O(1), Lemma 4.12 follows after accumulating the finitely many events all holding with probability at least 1 − O(d−1/2 Φ).



5. The Self-Consistent Eqation and Proof of Theorem 2.5
5.1. Derivation of the self-consistent equation. We begin by introducing the following two pieces of notation for a
e = (Zek )k∈[[M+1,M+N ]] :
random vector Z = (Zi )i∈[[1,M]] and Z

(5.1)

E(i) Z =

M
1 X
Zi ,
M i=1

E(k) Ze =

1
N

M+N
X

k=M+1

ek .
Z

We now record the following concentration estimate which will allow us to take expectations and exploit the estimates in
Lemma lemma:conditionalexpectationscederivation. To do so, we introduce the following notation.
Definition 5.1. Suppose X is an L1 -random variable, and suppose σ(·) is a σ-algebra which X is measurable with respect
to. We define the σ(·)-fluctuation of X to be the following centered random variable:
Xσ(·) := X − Eσ(·) X.

(5.2)

Proposition 5.2. Suppose that z = E + iη ∈ DN,δ,ξ ∩ Uε , that ζ > 0, and that Γ = O(1) with probability at least 1 − e−ζ .
Fix k = O(1) and pairs of indices Ik = {(i1 , j1 ), . . . , (ik , jk )}. Define the random variable
XIk (z) = Gi1 j1 . . . Gik jk .
Then, for any ξ = ξ(N ) satisfying ξ → ∞ as N → ∞, we have the following pointwise concentration estimate:


(5.3)
P (XIk (z))F0 = O(ξΦ) > 1 − e−[(ξ log ξ)∧ζ]+O(log N ) .
The proof of Proposition 5.2 follows from the proof of Proposition 4.1 in [5] so we omit it.

We now consider the matrix equation HG = zG + Id and compute the diagonal entries of both sides. The (i, i)-entry of
the RHS is clearly given by zGii + 1. We now study the LHS, considering the (k, k)-entry for k ∈ [[M, M + 1]]. By matrix
multiplication we have
(5.4)

(HG)kk =

M
X

Hki Gik =

−1/2
dw

i=1

(5.5)

−1/2
= dw


dw 
M X
X
1
δi,vb,ν −
Gik
M
i=1 ν=1
dw
X

ν=1
18


Gvb,ν k − E(i) Gik ,

where we used the relation M db = N dw . Appealing to Lemma 4.12 we deduce the following identity:
!
dw 
X
1
−1
EF0 (HG)kk = − EF0
(5.6)
sb Gii + O(dw Φ)
= − EF0 sb Gii + O(Φ).
dw
ν=1
Taking an expectation conditioning on F0 in the matrix equation HG = zG + Id, we see
(5.7)

1 + z EF0 Gkk = − EF0 sb Gkk + O (Φ) .

Using Proposition 5.2 to account for the F0 -fluctuation of the Green’s function terms, we ultimately deduce a stability equation for the diagonal (k, k)-entries of G, with k > M . We may run a similar calculation for indices i ∈ [[1, M ]] and derive
the following system of equations:
(5.8)

1 + (z + γsw )Gii = O ((1 + |z|)ξΦ) ,

(5.9)

1 + (z + sb )Gkk = O ((1 + |z|)ξΦ) .

Although this system is a priori coupled, we now appeal to Lemma 4.6 to decouple the equations. More precisely, we deduce
the following system of decoupled equations:


1−γ
(5.10)
Gii = O ((1 + |z|)ξΦ) ,
1 + z + sb +
z


γ−1
1 + z + γsw +
(5.11)
Gkk = O ((1 + |z|)ξΦ) .
z
From here, we may proceed in two fashions. First, we may use Lemma 7.3 to deduce stability equations for the Green’s
functions G∗ and G∗,+ , relating the diagonal entries of these Green’s functions to the Stieltjes transforms s∗ and s∗,+ . On the
other hand, we may also average over the diagonal entries and deduce self-consistent equations for the Stieltjes transforms
sb , sw and s∗ , s∗,+ . We summarize these estimates in the following proposition.
Proposition 5.3. Suppose Γ = O(1) with t-HP, and let z = E + iη ∈ Uε satisfy η ≫ N −1 . Then for any i ∈ [[1, M ]] and
k ∈ [[M + 1, M + N ]], we have the following equations uniformly over such z with t-HP:


1−γ
(5.12)
Gii = O((1 + |z|)ξΦ),
1 + z + sb +
z


γ−1
1 + z + γsw +
(5.13)
Gkk = O((1 + |z|)ξΦ),
z
(5.14)

1 + (z + 1 − γ + zs∗,+ ) [G∗,+ ]ii = O((1 + |z|1/2 )ξΦ) = O((1 + |z|)ξΦ),

(5.15)

1 + (z + γ − 1 + γzs∗ ) [G∗ ]kk = O((1 + |z|1/2 )ξΦ) = O((1 + |z|)ξΦ).

Moreover, we have the following averaged equations uniformly over such z with t-HP:


1−γ
(5.16)
sb = O((1 + |z|)ξΦ),
1 + z + sb +
z


γ−1
1 + z + γsw +
(5.17)
sw = O((1 + |z|)ξΦ),
z
(5.18)
(5.19)

1 + (z + 1 − γ + zs∗,+ ) s∗,+ = O((1 + |z|1/2 ξΦ) = O((1 + |z|)ξΦ),
1 + (z + γ − 1 + γzs∗ ) s∗ = O((1 + |z|1/2 )ξΦ) = O((1 + |z|)ξΦ).

Proof. It remains to upgrade the self-consistent equations (5.12) – (5.19) to hold over all such z = E + iη with t-HP. To this
end, we appeal to the Lipschitz continuity of the Green’s function entries on a sufficiently dense lattice as in the proof of
Lemma 8.5.


19

5.2. Analysis of the self-consistent equation. From a direct calculation, we note the Stieltjes transform m∞ of the
Marchenko-Pastur law with parameter γ 6 1 is given by the following self-consistent equation:
(5.20)

γzm2∞ (z) + (γ + z − 1) m∞ (z) + 1 = 0.

For the augmented Stieltjes transform m∞,+ , we may similarly deduce a self-consistent equation. In our analysis, we will
be concerned with providing full details for the Stieltjes transform m∞ only, as the estimate for the augmented transform
m∞,+ will follow from the estimate on m∞ . We now note Proposition 5.3 implies the Stieltjes transform s∗ solves the same
self-consistent equation with an error of o(1) throughout the domain DN,δ,ξ ∩ Uε , with t-HP. Our goal will be to use the
stability of the self-consistent equation (5.20) under o(1) perturbations to compare s∗ and m∞ . This is the content of the
following result.
Proposition 5.4. Let m : C+ → C+ be the unique solution to the following equation:
(5.21)

γzm2 + (γ + z − 1)m + 1 = 0.

Suppose s : C+ → C+ is continuous and let
(5.22)

R := γzs2 + (γ + z − 1)s + 1.

Fix an energy E ∈ R \ [−ε, ε] for ε > 0 small and scales η0 < C(E) and η∞ 6 N , where C(E) = OE (1) is a constant to be
determined. Suppose we have
(5.23)

|R(E + iη)| 6 (1 + |z|)r(E + iη)

for a nonincreasing function r : [η0 , η∞ ] → [0, 1]. Then for all z = E + iη for η ∈ [η0 , η∞ ], we have the following estimate for
sufficiently large N :

|m − s| = O(F (r)).

(5.24)
Here, the constant C(E) is determined by
(5.25)

Im



1−γ
E + iη



> 3α1/2 ε−1/2

for all η 6 C(E).
Before we proceed with the proof of Proposition 5.4, we introduce the following notation.
Notation 5.5. We denote the solutions to the equation (5.20) by m± , where m+ maps the upper-half plane to itself, and m−
maps the upper-half plane to the lower-half plane.
Moreover, we define the following error functions:
(5.26)

v± = |m± − s| .

Having established this notation, because m+ takes values in the upper-half plane, we deduce the following upper bound
on the values taken by the imaginary part of m− as follows:


1−γ
Im(m− (z)) 6 − Im
(5.27)
< −3α1/2 ε−1/2
E + iη
for scales η < C(E). We now proceed to derive an a priori estimate on the error functions v± .
Lemma 5.6. Under the assumptions and setting of Proposition 5.4, we have
(5.28)

|v+ | ∧ |v− | 6 3α1/2 ε−1/2 F (r).
20

Proof. We appeal to the following inequality which holds for any branch of the complex square root
parameters w, ζ for which the square root is defined:
p
p
p
√
√
|ζ|
| w + ζ − w| ∧ | w + ζ + w| 6 p
(5.29)
∧ |ζ|.
|w|

√

· and any complex

In particular, this implies the following string of inequalities:

!
p
|4γzR|
p
∧ |4γzR|
|(γ + z − 1)2 − 4γz|
√
2|R|
6 p
∧ εαR
|(γ + z − 1)2 − 4γz|
2ε1/2 (1 + |z|)r(E + iη) p
6 p
∧ εα(1 + |z|)r(E + iη)
|(γ + z − 1)2 − 4γz|

1
|v+ | ∧ |v− | 6
2γz

(5.30)
(5.31)
(5.32)

where the second inequality follows from the assumption |z| > |E| > ε and the last bound follows if we choose ε 6 1. But

this is bounded by 3α1/2 ε−1/2 F (r) for any r ∈ [0, 1].



Proof. (of Proposition 5.4).
We consider two different regimes. First consider the regime where |m+ − m− | > (1 + |z|)r(η). Precisely, this is the

regime defined by η > C(E) and the energy-dependent constant D(E) such that
(1 + |z|)r(η) <

(5.33)

|(γ + z − 1)2 − 4γz|
;
D(E)

the constant D(E) will be determined later. We note by Lemma 5.6, in this regime it suffices to prove the following bound:
|v− | > |v− | ∧ |v+ |.

(5.34)

We now choose an energy-dependent constant κ(E) such that for all η ∈ [C(E), η∞ ], we have the bound
α1/2 (1 + |z|)r(η) 6 κ(E)(1 + |E + iC(E)|).

(5.35)

Moreover, note |(γ + z − 1)2 − 4γz| is increasing in η, as seen by translating z = w + 1 − γ and computing
|(γ + z − 1)2 − 4γz| = |w2 − 4γw + X| = |w(w − 4γ) + X|,
where X ∈ R. Because r(η) is non-increasing in η, for all z = E + iη with η ∈ [C(E), η∞ ], we have
(1 + |z|)r(η) <

(5.36)

κ(E)
|(γ + z − 1)2 − 4γz|.
D(E)

We first compute a uniform lower bound on the difference term as follows:
|v+ − v− | =

|(γ + z − 1)2 − 4γz|
2γ|z|

1
>
2γ|E + iη∞ |
> 0.

D(E)
(1 + |z|)r(η)
p
∧
κ(E) |(γ + z − 1)2 − 4γz|

s

D(E) p
α(1 + |z|)r(η)
κ(E)

!

By continuity of s and the estimate Lemma 5.6, choosing D(E) large enough as a function of κ(E), E, η∞ , ε, it suffices to
prove the estimate (5.24) for some η ∈ [C(E), η∞ ]. But this follows from Lemma 5.6; in particular, we have at η = C(E) and
N sufficiently large,

|v− | > | Im(s) − Im(m− )| > | Im(m− )| > 3ε−1/2 > 3ε−1/2 F (r) > |v+ | ∧ |v− |.
Thus, we have |v+ | = |v+ | ∧ |v− | in this first regime, implying the stability estimate (5.24).
Now, we take the regime where the a priori estimate

(5.37)

|(γ + z − 1)2 − 4γz| = O((1 + |z|)r(η))
21

holds. Thus, we know

p
|(γ + z − 1)2 − 4γz|
|v− | 6 |v+ | +
= |v+ | + O
2ε
= |v+ | + O(F (r)),

(1 + |z|)r(η)

p
|(γ + z − 1)2 − 4γz|

p
∧ (1 + |z|)r

!

implying the estimate in the second regime as well.



We conclude the discussion of the self-consistent equation by noting that Lemma 4.6 allows us to deduce the following
local law for the Stieltjes transform s∗,+ :
|s∗,+ − m∞,+ | = O(F (r)).

(5.38)

This estimate holds in the regime z = E + iη for η ∈ [η0 , η∞ ].
5.3. Final estimates. Fix an index k ∈ [[1, N ]], and for notational convenience, for this calculation only, we let G denote the
Green’s function G∗ . Consider the following approximate stability equation for the diagonal entry Gkk :
1 + (z + γ − 1 + γzs∗ )Gkk = O ((1 + |z|)ξΦ) .

(5.39)

We now appeal to the following estimate which will allow us to study the stability of this equation upon the replacement
s∗ → m∞ that holds with t-HP:
|zGkk | = O(1).

(5.40)
Indeed, if η ≫ 1, we have

|zGkk | 6

(5.41)

E + iη
η

= O(1).

If E ≫ 1, we appeal to the spectral representation of Gkk and deduce
|zGkk | 6 C

(5.42)

1 X |uλ (k)|2 × E
N
E − λ + iη
λ

C X
E
6
N
E − λ + iη

(5.43)

λ

= O(1),

(5.44)

where we used the uniform a priori bound λ = O(1). If η, E . 1, then we appeal to the a priori bound Γ = O(1) with t-HP
and the trivial bound z = O(1). Thus, by Proposition 5.4, we have
(5.45)

1 + (z + γ − 1 + γzm∞ )Gkk = O ((1 + |z|)ξΦ) + O(F (ξΦ)).

On the other hand, the self-consistent equation (5.20) implies the Green’s function term on the LHS may be written as
−Gkk /m∞ . Moreover, because m∞ = O(1) uniformly on the domain Uε , we establish the following estimate that holds
over all z ∈ DN,δ,ξ ∩ Uε with t-HP:

(5.46)

m∞ − Gkk = O (F (ξΦ)) ,

where we use the estimate (1 + |z|)m∞ = O(1). This completes the proof of the local law along the diagonal of G = G∗ .
To derive the estimate for the off-diagonal entries, we appeal to the Green’s function G(z) = (X −z)−1 of the linearization

X. Note this is no longer the Green’s function G∗ of the covariance matrix X∗ . In particular, we appeal to the following entrywise representation of a matrix equation (for indices i, j > M ):
(5.47)

Gij (HG)ii − Gii (HG)ij = Gij .

As in the derivation of the stability equations in Proposition 5.3, by Lemma 4.12 the expectation of the LHS is given by
h
h
i
i
−1/2
−1/2
sGii − EF0 Gii dw
EF0 Gij dw
sGij + O(Φ) = O(Φ).
22

Thus, at the cost of a concentration estimate in Proposition 5.2, we deduce
|Gij | = O(ξΦ),

(5.48)

which yields the estimate for the off-diagonal entries. By Lemma 7.3, this gives the desired estimate for the off-diagonal
entries of the Green’s function G∗ with t-HP. An analogous calculation proves the desired entry-wise estimates for the
Green’s function G∗,+ , which completes the proof of Theorem 4.1.

6. Appendix
6.1. Estimates on the Stieltjes transforms on the upper-half plane. Here, we want to control the growth of the Stieltjes
transforms on the upper-half plane. This is summarized in the following lemma.
Lemma 6.1. Uniformly over z ∈ C+ , we have
m(z) = O(1).

(6.1)

We proceed by considering the two regimes γ = 1 and γ < 1, which we refer to as the square regime and rectangular
regime, respectively.
Square Regime. If γ = 1, we rewrite the density ̺(E) as
√
4 − E2
̺γ=1 (E) =
1E∈[−2,2] ,
2π

(6.2)

which is the well-studied semicircle density, whose Stieltjes transform is given by
m(z) =

(6.3)

−z +

√
z2 − 4
.
2

For a reference on the semicircle law and its Stieltjes transform, we cite [2], [4], [5], [9], [10], and [13]. We note the branch
√
of the square root is taken so that z 2 − 4 ∼ z for large z, in which case the bound (6.1) follows immediately.
Rectangular Regime. Fix constants Λ > 0 and ε > 0 to be determined. Suppose |E| ∈ [ε, Λ]. By the representation
m(z) = zm∞ (z 2 ) and the explicit formula for m∞ (z) as given in (??), the bound (6.1) follows immediately in this energy
regime, where the implied constant in (6.1) may be taken independent of η.
Suppose now that |E| > Λ. Again by the representation m(z) = zm∞ (z 2 ), we have


p
|m(z)| = O −z 2 + i (λ+ − z 2 )(z 2 − λ− )
(6.4)


p
(6.5)
= O −z 2 + (z 2 − λ+ )(z 2 − λ− )
= O(1),

(6.6)

p
since the square root is, again, chosen so that z 4 + O(z 2 ) ∼ z 2 for large z.
Lastly, suppose |E| < ε. By definition of m(z) as the Stieltjes transform of ̺, we obtain
(6.7)

|m(z)| =

w
E 2 ∈[λ± ]

p
γ
(λ+ − E 2 )(E 2 − λ− ) dE = O
(1 + γ)π|E|(E − ε)

where the implied constant depends only on fixed data γ, λ± . Choosing ε =

p
λ− − ε

!

,

p
λ− /100 > 0, we obtain the desired bound.

We note this choice of ε is positive if and only if α > 1. This completes the proof of Lemma 6.1.
23

1



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24

