3.1. Proof of Lemma 3.1

In a first step, we express the singular vector entries as an integral of Green functions over a random interval, which is recorded as the following lemma.

Lemma 3.2. Under the assumptions of Lemma 3.1, there exist some small constants ε, δ > 0 satisfying

\[\delta \gt 2\varepsilon ,\quad \quad \varepsilon \gt C{\varepsilon _1},\quad \quad \delta \lt {C^{ - 1}}{\varepsilon _0},\](3.5)

for some large constant C > C ₁ (recall (3.2) for C ₁) such that

\[\mathop {\lim }\limits_{N \to \infty } \mathop {\max }\limits_{l \le N_k^\delta } \mathop {\max }\limits_{\mu ,\nu } |{{\mathbb{E}}^V}\theta (N{\zeta _{{\alpha ^{'}}}}(\mu ){\zeta _{{\alpha ^{'}}}}(\nu )) - {{\mathbb{E}}^V}\theta ({N \over \pi }\int_I {\tilde G_{\mu \nu }}(z){\mathcal{X}}(E){\kern 1pt} {\rm{d}}E)| = 0,\]

where I is defined as

\[I{\kern 1pt} : = [{a_{2k - 1}} - {N^{ - 2/3 + \varepsilon }},\;{a_{2k - 1}} + {N^{ - 2/3 + \varepsilon }}]\](3.6)

when (1.11) holds

\[I{\kern 1pt} : = [{a_{2k}} - {N^{ - 2/3 + \varepsilon }},\quad \quad {a_{2k}} + {N^{ - 2/3 + \varepsilon }}]\]

when (1.12) holds. We define

\[{\mathcal X}(E){\kern 1pt} : = {\bf{1}}({\lambda _{{\alpha ^{'}} + 1}} \lt {E^ - } \le {\lambda _{{\alpha ^{'}}}}),\](3.7)

where E ^± : = E ± N^ε η. The conclusion holds if we replace X_V with X_G.

Proof. We first observe that

\[{\zeta _{{\alpha ^{'}}}}(\mu ){\zeta _{{\alpha ^{'}}}}(\nu ) = {\eta \over \pi }\int_{\mathbb{R}} {{{\zeta _{{\alpha ^{'}}}}(\mu ){\zeta _{{\alpha ^{'}}}}(\nu )} \over {{{(E - {\lambda _{{\alpha ^{'}}}})}^2} + {\eta ^2}}}{\kern 1pt} {\rm{d}}E.\]

Choose a and b such that

\[a{\kern 1pt} : = \min \{ {\lambda _{{\alpha ^{'}}}} - {N^\varepsilon }\eta ,{\lambda _{{\alpha ^{'}} + 1}} + {N^\varepsilon }\eta \} ,\quad \quad b{\kern 1pt} : = {\lambda _{{\alpha ^{'}}}} + {N^\varepsilon }\eta .\](3.8)

We also observe the elementary inequality (see the equation above Equation (6.10) of [Reference Erdös, Yau and Yin18]), for some constant C > 0,

\[\int_x^\infty {\eta \over {\pi ({y^2} + {\eta ^2})}}{\kern 1pt} {\rm{d}}y \le {{C\eta } \over {x + \eta }},\quad \quad x \gt 0.\](3.9)

By (3.3), (3.8), and (3.9), with probability 1 – N ^{–D ₁}, we have

\[{\zeta _{{\alpha ^{'}}}}(\mu ){\zeta _{{\alpha ^{'}}}}(\nu ) = {\eta \over \pi }\int_a^b {{{\zeta _{{\alpha ^{'}}}}(\mu ){\zeta _{{\alpha ^{'}}}}(\nu )} \over {{{(E - \lambda _\alpha ^{'})}^2} + {\eta ^2}}}{\kern 1pt} {\rm{d}}E + O({N^{ - 1 - \varepsilon + {\varepsilon _1}}}).\](3.10)

By (3.2), (3.3), (3.5), (3.10), and mean value theorem, we have

\[{{\mathbb{E}}^V}\theta (N{\zeta _{{\alpha ^{'}}}}(\mu ){\zeta _{{\alpha ^{'}}}}(\nu )) = {{\mathbb{E}}^V}\theta ({{N\eta } \over \pi }\int_a^b {{{\zeta _{{\alpha ^{'}}}}(\mu ){\zeta _{{\alpha ^{'}}}}(\nu )} \over {{{(E - {\lambda _{{\alpha ^{'}}}})}^2} + {\eta ^2}}}{\kern 1pt} {\rm{d}}E) + o(1).\](3.11)

Define \[\lambda _t^ \pm {\kern 1pt} : = {\lambda _t} \pm {N^\varepsilon }\eta ,{\kern 1pt} t = {\alpha ^{'}},\;{\alpha ^{'}} + 1\], and by (3.8), we have

\[\int_a^b {\kern 1pt} {\rm{d}}E = \int_{\lambda _{{\alpha ^{'}} + 1}^ + }^{\lambda _{{\alpha ^{'}}}^ + } {\kern 1pt} {\rm{d}}E + {\bf{1}}(\lambda _{{\alpha ^{'}} + 1}^ + > \lambda _{{\alpha ^{'}}}^ - )\int_{\lambda _{{\alpha ^{'}}}^ - }^{\lambda _{{\alpha ^{'}r} + 1}^ + } {\kern 1pt} {\rm{d}}E.\]

By (3.2), (3.3), (3.11), and the mean value theorem, we have

\[{{\mathbb{E}}^V}\theta (N{\zeta _{{\alpha ^{'}}}}(\mu ){\zeta _{{\alpha ^{'}}}}(\nu )) = {{\mathbb{E}}^V}\theta ({{N\eta } \over \pi }\int_{\lambda _{{\alpha ^{'}} + 1}^ + }^{\lambda _{{\alpha ^{'}}}^ + } {{{\zeta _{{\alpha ^{'}}}}(\mu ){\zeta _{{\alpha ^{'}}}}(\nu )} \over {{{(E - {\lambda _{{\alpha ^{'}}}})}^2} + {\eta ^2}}}{\kern 1pt} {\rm{d}}E) + o(1),\]

where we used (2.18) and (3.5). Next we can, without loss of generality, consider the case when (1.11) holds. By (3.3) and (3.5), we observe that, with probability \[1 - {N^{ - {D_1}}}\], we have \[\lambda _{{\alpha ^{'}}}^ + \le {a_{2k - 1}} + {N^{ - 2/3 + \varepsilon }}\] and \[\lambda _{{\alpha ^{'}} + 1}^ + \ge {a_{2k - 1}} - {N^{ - 2/3 + \varepsilon }}.\] By (2.18) and the choice of I in (3.6), we have

\[{{\mathbb{E}}^V}\theta (N{\zeta _{{\alpha ^{'}}}}(\mu ){\zeta _{{\alpha ^{'}}}}(\nu )) = {{\mathbb{E}}^V}\theta ({{N\eta } \over \pi }\int_I {{{\zeta _{{\alpha ^{'}}}}(\mu ){\zeta _{{\alpha ^{'}}}}(\nu )} \over {{{(E - {\lambda _{{\alpha ^{'}}}})}^2} + {\eta ^2}}}{\mathcal{X}}(E){\kern 1pt} {\rm{d}}E) + o(1).\]

Recall (3.1). We can split the summation as

\[{1 \over \eta }{\tilde G_{\mu \nu }}(z) = \sum\limits_{\beta \ne {\alpha ^{'}}} {{{\zeta _\beta }(\mu ){\zeta _\beta }(\nu )} \over {{{(E - {\lambda _\beta })}^2} + {\eta ^2}}} + {{{\zeta _{{\alpha ^{'}}}}(\mu ){\zeta _{{\alpha ^{'}}}}(\nu )} \over {{{(E - {\lambda _{{\alpha ^{'}}}})}^2} + {\eta ^2}}}.\](3.12)

Define \[{\mathcal A}{\kern 1pt} : = \{ \beta \ne {\alpha ^{'}}:{\lambda _\beta }\] is not in the kth bulk component}. By (3.3), with probability \[1 - {N^{ - {D_1}}}\], we have

\[|\sum\limits_{\beta \ne {\alpha ^{'}}} {{N\eta } \over \pi }\int_I {{{\zeta _\beta }(\mu ){\zeta _\beta }(\nu )} \over {{{(E - {\lambda _\beta })}^2} + {\eta ^2}}}{\kern 1pt} {\rm{d}}E| \\ \quad \quad \le {{{N^{{\varepsilon _1}}}} \over \pi }(\sum\limits_{\beta \in A} \int_I {\eta \over {{\eta ^2} + {{(E - {\lambda _\beta })}^2}}}{\kern 1pt} {\rm{d}}E + \sum\limits_{\beta \in {A^c}} \int_I {\eta \over {{\eta ^2} + {{(E - {\lambda _\beta })}^2}}}{\kern 1pt} {\rm{d}}E).\](3.13)

By Assumption 1.2, with probability \[1 - {N^{{D_1}}}\], we have

\[{{{N^{{\varepsilon _1}}}} \over \pi }\sum\limits_{\beta \in A} \int_I {\eta \over {{\eta ^2} + {{(E - {\lambda _\beta })}^2}}}{\kern 1pt} {\rm{d}}E \le {N^{{\varepsilon _1}}}\sum\limits_{\beta \in A} {N^{ - 4/3 - {\varepsilon _0} + \varepsilon }}.\](3.14)

Define

\[l(\beta ){\kern 1pt} : = \beta - \sum\limits_{t \lt k} {N_t}.\]

By (3.3), with probability \[1 - {N^{ - {D_1}}}\], for some small constant 0 < δ < 1, we have

\[{{{N^{{\varepsilon _1}}}} \over \pi }\sum\limits_{\beta \in {A^c}} \int_I {\eta \over {{{(E - {\lambda _\beta })}^2} + {\eta ^2}}}{\kern 1pt} {\rm{d}}E \le {N^{{\varepsilon _1} + \delta }} + {1 \over \pi }\sum\limits_{\beta \in {A^c},{\kern 1pt} l(\beta ) \ge N_k^\delta } \int_I {{{N^{{\varepsilon _1}}}\eta } \over {{\eta ^2} + {{(E - {\lambda _\beta })}^2}}}{\kern 1pt} {\rm{d}}E.\](3.14)

By Assumption 1.2, (1.9), (2.18), and the assumption that δ > 2ε, it is easy to check that (see [Reference Knowles and Yin24, Equation (3.12)])

\[{(E - {\lambda _\beta })^2} \ge c{({{l(\beta )} \over N})^{4/3}},\quad \quad c \gt 0\,{\rm{is}}\,{\rm{some}}\,{\rm{constant}}.\](3.16)

By (3.16), with probability \[1 - {N^{ - {D_1}}}\], we have

\[\frac{1}{\pi }\sum\limits_{\beta \in {{\mathcal A}^c},{\kern 1pt} l(\beta ) \ge N_k^\delta } \int_I \frac{{{N^{{\varepsilon _1}}}\eta }}{{{\eta ^2} + {{(E - {\lambda _\beta })}^2}}}{\kern 1pt} {\rm{d}}E \le {N^{{\varepsilon _1} - {\varepsilon _0} + \varepsilon }}\int_{{N^\delta } - 1}^N \frac{1}{{{x^{4/3}}}}{\kern 1pt} {\rm{d}}x \le {N^{ - \delta /3 + {\varepsilon _1} - {\varepsilon _0} + \varepsilon }}.\]

Recall (3.5). We can restrict ε ₁ – ε ₀ + ε < 0, so that, with probability \[1 - {N^{ - {D_1}}}\], this yields

\[\sum\limits_{\beta \in {A^c},{\kern 1pt} l(\beta ) \ge N_k^\delta } \int_I {{{N^{{\varepsilon _1}}}\eta } \over {{\eta ^2} + {{(E - {\lambda _\beta })}^2}}}{\kern 1pt} {\rm{d}}E \le {N^{ - \delta /3}}.\](3.17)

By (3.13), (3.14), (3.15), and (3.17), with probability \[1 - {N^{ - {D_1}}}\], we have

\[|\sum\limits_{\beta \ne {\alpha ^{'}}} {{N\eta } \over \pi }\int_I {{{\zeta _\beta }(\mu ){\zeta _\beta }(\nu )} \over {{{(E - {\lambda _\beta })}^2} + {\eta ^2}}}{\kern 1pt} {\rm{d}}E| \le {N^{\delta + 2{\varepsilon _1}}}.\](3.18)

By (3.2), (3.3), (3.12), (3.18), and the mean value theorem, we have

\[|{{\mathbb{E}}^V}\theta ({{N\eta } \over \pi }\int_I {{{\zeta _{{\alpha ^{'}}}}(\mu ){\zeta _{{\alpha ^{'}}}}(\nu )} \over {{{(E - {\lambda _{{\alpha ^{'}}}})}^2} + {\eta ^2}}}{\mathcal{X}}(E){\kern 1pt} {\rm{d}}E) - {{\mathbb{E}}^V}\theta ({N \over \pi }\int_I {\tilde G_{\mu \nu }}(E + i\eta ){\mathcal{X}}(E){\kern 1pt} {\rm{d}}E)| \\ \quad \quad \le {N^{{C_1}(\delta + 2{\varepsilon _1})}}{E^V}\sum\limits_{\beta \ne {\alpha ^{'}}} {{N\eta } \over \pi }\int_I {{|{\zeta _\beta }(\mu ){\zeta _\beta }(\nu )|} \over {{{(E - {\lambda _\beta })}^2} + {\eta ^2}}}X(E){\kern 1pt} {\rm{d}}E,\](3.19)

where C ₁ is defined in (3.2). To complete the proof, it suffices to estimate the right-hand side of (3.19). Similarly to (3.14), we have

\[\sum\limits_{\beta \in {\mathcal{A}}} \int_I {\eta \over {{\eta ^2} + {{(E - {\lambda _\beta })}^2}}}{\kern 1pt} {\rm{d}}E \le {N^{ - 1/3 - {\varepsilon _0} + \varepsilon }}.\](3.20)

Choose a small constant 0 < δ₁ < 1 and repeat the estimation in (3.17) to obtain

\[\sum\limits_{\beta \in {{\mathcal{A}}^c},{\kern 1pt} l(\beta ) \ge N_k^{{\delta _1}}} \int_I {\eta \over {{\eta ^2} + {{(E - {\lambda _\beta })}^2}}}{\kern 1pt} {\rm{d}}E \le {N^{ - {\delta _1}/3 + \varepsilon - {\varepsilon _0}}}.\](3.21)

Recall (1.11), (3.3), and (3.9). Using a discussion similar to that above Equation (3.14) of [Reference Knowles and Yin24], we conclude that

\[\sum\limits_{\beta \in {{\mathcal{A}}^c},{\kern 1pt} l \le l(\beta ) \le N_k^{{\delta _1}}} {{N\eta } \over \pi }{{\mathbb{E}}^V}\int_I {{|{\zeta _\beta }(\mu ){\zeta _\beta }(\nu )|} \over {{{(E - {\lambda _\beta })}^2} + {\eta ^2}}}{\mathcal{X}}(E){\kern 1pt} {\rm{d}}E \\ \quad \quad \le {{\mathbb{E}}^V}\int_{{\lambda _{{\alpha ^{'}} + 1}} + {N^\varepsilon }\eta }^\infty {{{N^{{\varepsilon _1}}}\eta } \over {{{(E - {\lambda _{{\alpha ^{'}} + 1}})}^2} + {\eta ^2}}}{\kern 1pt} {\rm{d}}E \\ \quad \quad \le {N^{ - \varepsilon + {\varepsilon _1}}},\](3.22)

where we have used the fact that \[\beta \in {{\mathcal{A}}^c}\] and \[\;l \lt l(\beta ) \le N_k^{{\delta _1}}\] imply that \[{\lambda _\beta } \le {\lambda _{{\alpha ^{'}} + 1}}\] It is notable that the above bound is independent of δ. It remains to estimate the summation of the terms when \[\beta \in {{\mathcal{A}}^c}\] and l(β) < l. For a given constant, ε′ satisfies

\[\delta \gt 2\varepsilon ',\quad \quad \varepsilon ' \gt C{\varepsilon _1},\quad \quad \delta \lt {C^{ - 1}}{\varepsilon _0}.\](3.23)

We partition \[I = {I_1} \cup {I_2}\] with \[{I_1} \cap {I_2} = \emptyset \], where

\[{I_1}: = \{ E \in I:\,{\rm{there}}\,{\rm{exists}}\,\beta ,\beta \in {A^c},\;l(\beta ) \lt l,\;|E - {\lambda _\beta }| \le {N^{{\varepsilon ^{'}}}}\eta \} .\](3.24)

By (3.3) and (3.24), using a similar discussion to that used for (3.22), we have

\[\sum\limits_{\beta \in {{\mathcal{A}}^c};\;l(\beta ) \lt l} {{N\eta } \over \pi }{{\mathbb{E}}^V}\int_{{I_2}} {{|{\zeta _\beta }(\mu ){\zeta _\beta }(\nu )|} \over {{{(E - {\lambda _\beta })}^2} + {\eta ^2}}}{\mathcal{X}}(E){\kern 1pt} {\rm{d}}E \le {N^{ - 2{\varepsilon ^{'}} + {\varepsilon _1}}}.\]

It is easy to check that on I ₁ when \[{\lambda _{{\alpha ^{'}} + 1}} \le {\lambda _{{\alpha ^{'}}}} \lt {\lambda _\beta }\], we have (see (3.15) of [Reference Knowles and Yin24])

\[{1 \over {{{(E - {\lambda _\beta })}^2} + {\eta ^2}}}{\bf{1}}({E^ - } \le {\lambda _{{\alpha ^{'}}}}) \le {{{N^{2\varepsilon }}} \over {{{({\lambda _{{\alpha ^{'}} + 1}} - {\lambda _{{\alpha ^{'}}}})}^2} + {\eta ^2}}}.\](3.25)

By Lemma 2.2, the above equation holds with probability \[1 - {N^{ - {D_1}}}\]. By (3.3), (3.25), and a discussion similar to that used in [Reference Knowles and Yin24, Equation (3.16)], we have

\[\sum\limits_{\beta \in {{\mathcal{A}}^c},{\kern 1pt} l(\beta ) \le l} {{N\eta } \over \pi }{{\mathbb{E}}^V}\int_{{I_1}} {{|{\zeta _\beta }(\mu ){\zeta _\beta }(\nu )|} \over {{{(E - {\lambda _\beta })}^2} + {\eta ^2}}}{\mathcal{X}}(E){\kern 1pt} {\rm{d}}E \le {{\mathbb{E}}^V}\int_{{I_1}} {{{N^{{\varepsilon _1} + 2\varepsilon }}{\eta ^2}} \over {{{({\lambda _{{\alpha ^{'}} + 1}} - {\lambda _{{\alpha ^{'}}}})}^2} + {\eta ^2}}}{\kern 1pt} {\rm{d}}E \\ \le {{\mathcal{E}}^V}{\bf{1}}(|{\lambda _{\alpha ' + 1}} - {\lambda _{\alpha '}}| \le {N^{ - 1/3}}{\eta ^{1/2}}) + {N^{ - {D_1} + {\varepsilon _1} + 3\varepsilon }} \\ \le {N^{ - {\varepsilon _0}/2 + 3\varepsilon }}.\](3.26)

By (3.20), (3.21), (3.22), (3.23), and (3.26), we conclude the proof of (3.19). It is clear that our proof still applies when we replace X _V with X _G □

In a second step, we write the sharp indicator function of (3.7) as a some smooth function q of \[{\tilde G_{\mu \nu }}\]. To be consistent with the proof of Lemma 3.2, we consider the bulk edge a _{2k – 1}. Define

\[{\vartheta _\eta }(x): = {\eta \over {\pi ({x^2} + {\eta ^2})}} = {1 \over \pi }\rm {Im}{1 \over {x - {\rm{i}}\eta }}.\]

We define a smooth cutoff function q ≡ q _α′: ℝ → ℝ₊ as

\[q(x) = \left( {\matrix{ 1 & {{\rm{if}}|x - l| \le {\textstyle{1 \over 3}},} \cr 0 & {{\rm{if}}|x - l| \ge {\textstyle{2 \over 3}},} \cr } } \right.\](3.27)

where l is defined in (1.11). We also let Q ₁=Y*Y

Lemma 3.3. For ε given in (3.5), define

\[{\mathcal {X}_E}(x) : = {\bf{1}}({E^ - } \le x \le {E_U}),\](3.28)

where \[{E_U} : = {a_{2k - 1}} + 2{N^{ - 2/3 + \varepsilon }}.\], and define \[\tilde \eta : = {N^{ - 2/3 - 9{\varepsilon _0}}},\] where ε ₀ is defined in (3.4). Then

\[\mathop {\lim }\limits_{N \to \infty } \mathop {\max }\limits_{l \le N_k^\delta } \mathop {\max }\limits_{\mu ,\nu } \biggl|{E^V}\theta \biggl(N {\zeta _{\alpha '}}(\mu )\zeta _{\alpha '}(\nu )\biggl) - {E^V}\theta \biggl({N \over \pi }\int_I {\tilde G_{\mu \nu }}(z)q[Tr({\mathcal {X}_E}*{\vartheta _{\tilde \eta }})({Q_1})] {\rm{d}}E\biggl)\biggr| = 0,\]

where I is defined in (3.6) and ‘*’ is the convolution operator.

Proof. For any E ₁ < E ₂, denote the number of eigenvalues of Q ₁ in [E ₁, E ₂] by

\[{\mathcal {N}}({E_1},{E_2}) : = \# \{ j:{E_1} \le {\lambda _j} \le {E_2}\} .\](3.29)

Recall (3.6) and (3.7). It is easy to check that, with probability \[1 - {N^{ - {D_1}}}\], we have

\[\begin{align} & N\int_I {\tilde G_{\mu \nu }}(z)\mathcal {X}(E) {\rm{d}}E = N\int_I {\tilde G_{\mu \nu }}(z){\bf{1}}(N({E^ - },{E_U}) = l) {\rm{d}}E \\& \quad\quad\quad\quad\quad\quad\quad\quad\quad = N\int_I {\tilde G_{\mu \nu }}(z)q[\rm {Tr}{\mathcal {X}_E}({Q_1})] {\rm{d}}E,\end{align}\](3.30)

where, for the second equality, we used (2.18) and Assumption 1.2. We use the following Lemma to estimate (3.29) by its delta approximation smoothed on the scale \[\tilde \eta \]. The proof is given in the supplementary material [Reference Ding14].

Lemma 3.4

For \[t = {N^{ - 2/3 - 3{\varepsilon _0}}},\] there exists some constant C, and with probability \[1 - {N^{ - {D_1}}}\], for any E satisfying

\[|{E^ - } - {a_{2k - 1}}| \le {3 \over 2}{N^{ - 2/3 + \varepsilon }},\]

we have

\[|{\rm {Tr}}{\mathcal {X}_E}({Q_1}) - {\rm {Tr}}({\mathcal {X}_E}*{\vartheta _{\tilde \eta }})({Q_1})| \le C({N^{ - 2{\varepsilon _0}}} + {\mathcal {N}}({E^ - } - t,\;{E^ - } + t)).\](3.31)

By Equation (A.7) of [Reference Knowles and Yin26], for any z ∈ D(τ) defined in (2.1), we have

\[{\rm Im} m(z) \sim \left\{ {\matrix{ {{\eta \over {\sqrt {\kappa + \eta } }},} & {E \notin {\rm supp} (\rho ),} \cr {\sqrt {\kappa + \eta } ,} & {E \in {\rm supp} (\rho ),} \cr } } \right.\](3.32)

where κ := |E – a _{2k – 1}. When μ = v, with probability \[1 - {N^{ - {D_1}}}\], we have

\[\begin{align} & \mathop {\sup }\limits_{E \in I} |{\tilde G_{\mu \mu }}(E + {\rm{i}}\eta )| = \mathop {\sup }\limits_{E \in I} |Im{G_{\mu \mu }}(z)| \\ & \quad\quad\quad\quad\quad\quad\quad \le \mathop {\sup }\limits_{E \in I} (Im|{G_{\mu \mu }}(z) - {m_{}}(z)| + |Im{m_{}}(z)|) \\ & \quad\quad\quad\quad\quad\quad\quad \le {N^{ - 1/3 + {\varepsilon _0} + 2\varepsilon }},\end{align}\]

where we have used (2.17) and (3.32). When μ ≠ ν, we use the identity

\[{\tilde G_{\mu \nu }} = \eta \sum\limits_{k = M + 1}^{M + N} {G_{\mu k}}{\overline G _{\nu k}}.\]

By (2.17) and (3.32), with probability \[1 - {N^{ - {D_1}}}\], we have \[\mathop {\sup }\nolimits_{E \in I} |{\tilde G_{\mu \nu }}(z)| \le {N^{ - 1/3 + {\varepsilon _0} + 2\varepsilon }}\]. Therefore, for E ∈ I, with probability \[1 - {N^{ - {D_1}}}\], we have

\[\mathop {\sup }\limits_{E \in I} |{\tilde G_{\mu \nu }}(E + {\rm{i}}\eta )| \le {N^{ - 1/3 + 3{\varepsilon _0}/2}}.\](3.33)

Recall (3.27). By (3.30), (3.31), (3.33), and the smoothness of q, with probability \[1 - {N^{ - {D_1}}}\], we have

\[\begin{align} & \biggl|N\int_I {\tilde G_{\mu \nu }}(z){\mathcal X}(E) {\rm{d}}E - N\int_I {\tilde G_{\mu \nu }}(z)q[Tr({X_E}*{\vartheta _{\tilde \eta }}({Q_1}))] {\rm{d}}E\biggr| \\ & \quad \le CN\sum\limits_{l(\beta ) \le N_k^\delta } \int_I |{\tilde G_{\mu \nu }}(z)|{\bf{1}}(|{E^ - } - {\lambda _\beta }| \le t) {\rm{d}}E + {N^{ - {\varepsilon _0}/4}} \\ & \quad \le C{N^{1 + \delta }}|t|\mathop {\sup }\limits_{z \in I} |{\tilde G_{\mu \nu }}(z)| + {N^{ - {\varepsilon _0}/4}}.\end{align}\](3.34)

By (3.33) and (3.34), we have

\[\biggl|N\int_I {\tilde G_{\mu \nu }}(z){\mathcal X}(E) {\rm{d}}E - N\int_I {\tilde G_{\mu \nu }}(z)q[{\rm Tr}({{\mathcal X}_E}*{\vartheta _{\tilde \eta }}({Q_1}))] {\rm{d}}E\biggr| \le C{N^{ - {\varepsilon _0}/2 + \delta }} + {N^{ - {\varepsilon _0}/4}}.\]

Using a discussion similar to that used for (3.13), by (3.2) and (3.5), we complete the proof. □

In the final step, we use the Green function comparison argument to prove the following lemma, whose proof is given in Section 3.2.

Lemma 3.5. Under the assumptions of Lemma 3.3, we have

\[\mathop {\lim }\limits_{N \to \infty } \mathop {\max }\limits_{\mu ,\nu } ({E^V} - {E^G})\theta \biggl({N \over \pi }\int_I {\tilde G_{\mu \nu }}(z)q[{\rm Tr}({X_E}*{\vartheta _{\tilde \eta }})({Q_1})] {\rm{d}}E\biggr) = 0.\]

The proof of Lemma 3.1 follows from the proof of Lemma 3.3.

3.2. The Green function comparsion argument

In this section we prove Lemma 3.5 using the Green function comparison argument. At the end of this section we discuss how we can extend Lemma 3.1 to Theorem 1.1 and Theorem 1.2. By the orthonormal properties of ξ and ζ, and (2.6), we have

\[{\tilde G_{ij}} = \eta \sum\limits_{k = 1}^M {G_{ik}}{\overline G _{jk}},\quad \quad {\tilde G_{\mu \nu }} = \eta \sum\limits_{k = M + 1}^{M + N} {G_{\mu k}}{\overline G _{\nu k}}.\](3.35)

By (2.17), with probability \[1 - {N^{ - {D_1}}}\], we have

\[|{G_{\mu \mu }}| = O(1),\;|{G_{\mu \nu }}| \le {N^{ - 1/3 + 2{\varepsilon _0}}},\quad \quad \mu \ne \nu .\](3.36)

We first drop the all diagonal terms in (3.35).

Lemma 3.6. Recall that E _U = a _{2k – 1} + 2N ^{– 2/3 + ε} and \[\tilde \eta = {N^{ - 2/3 - 9{\varepsilon _0}}}\]. We have

\[{E^V}\theta \biggl[{N \over \pi }\int_I {\tilde G_{\mu \nu }}(z)q[{\rm Tr}({{\mathcal X}_E}*{\vartheta _{\tilde \eta }})({Q_1})] {\rm{d}}E \biggr] - {E^V}\theta \biggl[\int_I x(E)q(y(E)) {\rm{d}}E\biggr] = o(1),\](3.37)

where

\[x(E) : = {{N\eta } \over \pi }\sum\limits_{k = M + 1, k \ne \mu ,\nu }^{M + N} {X_{\mu \nu ,k}}(E + {\rm{i}}\eta ),\quad \quad y(E) : = {{\tilde \eta } \over \pi }\int_{{E^ - }}^{{E_U}} \sum\limits_k \sum\limits_{\beta \ne k} {X_{\beta \beta ,k}}(E + {\rm{i}}\tilde \eta ) {\rm{d}}E,\](3.38)

and \[{X_{\mu \nu ,k}} : = {G_{\mu k}}{\overline G _{\nu k}}\]. The conclusion holds if we replace X _V with X _G.

Proof. We first observe that, by (3.36), with probability \[1 - {N^{ - {D_1}}}\], we have

\[|x(E)| \le {N^{2/3 + 3{\varepsilon _0}}},\](3.39)

which implies that

\[\int_I |x(E)| {\rm{d}}E \le {N^{4{\varepsilon _0}}}.\](3.40)

By (3.35) and (3.36), with probability \[1 - {N^{ - {D_1}}}\], we have

\[|{N \over \pi }{\tilde G_{\mu \nu }}(E + {\rm{i}}\eta ) - x(E)| = {{N\eta } \over \pi }|{G_{\mu \mu }}{\overline G _{\nu \mu }} + {G_{\mu \nu }}{\overline G _{\nu \nu }}| \\ \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \le N\eta ({\bf{1}}(\mu = \nu ) + {N^{ - 1/3 + 2{\varepsilon _0}}}{\bf{1}}(\mu \ne \nu )).\](3.41)

By Equations (5.11) and (6.42) of [Reference Ding and Yang16], we have

\[{\rm Tr}({{\mathcal X}_E}*{\vartheta _{\tilde \eta }}({Q_1})) = {N \over \pi }\int_{{E^ - }}^{{E_U}} Im\,{m_2}(w + {\rm{i}}\tilde \eta ) {\rm{d}}w,\quad \quad \sum\limits_{\mu \nu } |{G_{\mu \nu }}(w + {\rm{i}}\tilde \eta {)|^2} = {{N\,Im\,{m_2}(w + {\rm{i}}\tilde \eta )} \over {\tilde \eta }}.\](3.42)

Therefore, we have

\[{\rm Tr}({{\mathcal X}_E}*{\vartheta _{\tilde \eta }}({Q_1})) - y(E) = {{\tilde \eta } \over \pi }\int_{{E^ - }}^{{E_U}} \sum\limits_{\beta = M + 1}^{M + N} |{G_{\beta \beta }}{|^2} {\rm{d}}w.\](3.43)

By (3.43), the mean value theorem, and the fact that q is smooth enough, we have

\[|q[{\rm Tr}({{\mathcal X}_E}*{\vartheta _{\tilde \eta }})({Q_1})] - q[y(E)]| \le {N^{ - 1/3 - 7{\varepsilon _0}}}.\](3.44)

Therefore, by the mean value theorem, (3.2), (3.5), (3.39), (3.40), (3.41), and (3.44), we complete the proof. □

To prove Lemma 3.5, by (3.37), it suffices to prove that

\[[{{\mathbb E}^V} - {{\mathbb E}^G}]\theta (\int_I x(E)q(y(E)) {\rm{d}}E) = o(1).\](3.45)

We use the Green function comparison argument to prove (3.45), where we follow the basic approach of [Reference Ding and Yang16, Section 6] and [Reference Knowles and Yin24, Section 3.1]. Define a bijective ordering map Φ on the index set, where

\[\Phi :\{ (i,{\mu _1}):1 \le i \le M, M + 1 \le {\mu _1} \le M + N\} \to \{ 1, \ldots ,{\gamma _{\max }} = MN\} .\]

Recall that we relabel \[{X^V} = (({X_V}{)_{i{\mu _1}}},i \in {{\mathcal{I}}_1}, {\mu _1} \in {{\mathcal{I}}_2})\], and similarly for X ^G. For any 1 ≤ γ ≤ γ_max, we define the matrix \[{X_\gamma } = \left( {x_{i{\mu _1}}^\gamma } \right)\] such that \[x_{i{\mu _1}}^\gamma = X_{i{\mu _1}}^G\] if Φ (i, μ₁) > γ and \[x_{i{\mu _1}}^\gamma = X_{i{\mu _1}}^V\] otherwise. Note that X ₀ = X ^G and X _{γ ^max} = X ^V. With the above definitions, we have

\[[{{\mathbb{E}}^G} - {{\mathbb{E}}^V}]\theta (\int_I x(E)q(y(E)) {\rm{d}}E) = \sum\limits_{\gamma = 1}^{{\gamma _{\max }}} [{{\mathbb{E}}^{\gamma - 1}} - {{\mathbb{E}}^\gamma }]\theta (\int_I x(E)q(y(E)) {\rm{d}}E).\]

For simplicity, we rewrite the above equation as

\[{\mathbb{E}}[\theta (\int_I {x^G}q({y^G}) {\rm{d}}E) - \theta (\int_I {x^V}q({y^V}) {\rm{d}}E)] \\ \quad \quad = \sum\limits_{\gamma = 1}^{{\gamma _{\max }}} {\mathbb{E}}[\theta (\int_I {x_{\gamma - 1}}q({y_{\gamma - 1}}) {\rm{d}}E) - \theta (\int_I {x_\gamma }q({y_\gamma }) {\rm{d}}E)].\]

The key step of the Green function comparison argument is to use the Lindeberg replacement strategy. We focus on the indices \[s,t \in {\mathcal{I}}\]; the special case \[\mu ,\nu \in {{\mathcal{I}}_2}\] follows. Define Y _γ := Σ^1/2 X ^γ and

\[{H^\gamma } : = \left( {\matrix{ 0 & {{z^{1/2}}{Y_\gamma }} \cr {{z^{1/2}}Y_\gamma ^*} & 0 \cr } } \right),\quad \quad {G^\gamma } : = {\left( {\matrix{ { - zI} & {{z^{1/2}}{Y_\gamma }} \cr {{z^{1/2}}Y_\gamma ^*} & { - zI} \cr } } \right)^{ - 1}}.\](3.46)

As Σ is diagonal, for each fixed γ, H ^γ and H ^γ–1 differ only at the (i, μ ₁) and (μ ₁, i) elements, where Φ (i, μ₁) = γ. Then we define the \[(N + M) \times (N + M)\] matrices V and W by

\[{V_{ab}} = {z^{1/2}}\left( {{{\bf{1}}_{\{ (a,b) = (i,{\mu _1})\} }} + {{\bf{1}}_{\{ (a,b) = ({\mu _1},i)\} }}} \right)\sqrt {{\sigma _i}} X_{i{\mu _1}}^G, \\ {W_{ab}} = {z^{1/2}}\left( {{{\bf{1}}_{\{ (a,b) = (i,{\mu _1})\} }} + {{\bf{1}}_{\{ (a,b) = ({\mu _1},i)\} }}} \right)\sqrt {{\sigma _i}} X_{i{\mu _1}}^V,\]

so that H ^γ and H ^γ–1 can be written as

\[{H^{\gamma - 1}} = O + V,\quad \quad {H^\gamma } = O + W,\]

for some N + M) × (N + M) matrix O satisfying \[{O_{i{\mu _1}}} = {O_{{\mu _1}i}} = 0,\] with O independent of V and W. Define

\[S : = ({H^{\gamma - 1}} - z{)^{ - 1}},\quad \quad R : = (O - z{)^{ - 1}},\quad \quad T : = ({H^\gamma } - z{)^{ - 1}}.\](3.47)

With the above definitions, we can write

\[{\mathcal{E}}[\theta (\int_I {x^G}q({y^G}) {\rm{d}}E) - \theta (\int_I {x^V}q({y^V}) {\rm{d}}E)] \\ \quad \quad = \sum\limits_{\gamma = 1}^{{\gamma _{\max }}} {\mathcal{E}}[\theta (\int_I {x^S}q({y^S}) {\rm{d}}E) - \theta (\int_I {x^T}q({y^T}) {\rm{d}}E)].\](3.48)

The comparison argument is based on the resolvent expansion

\[S = R - RVR + {(RV)^2}R - {(RV)^3}R + {(RV)^4}S.\](3.49)

For any integer m > 0, by Equation (6.11) of [Reference Ding and Yang16], we have

\[{([RV{]^m}R)_{ab}} = \sum\limits_{({a_i},{b_i}) \in \{ (i,{\mu _1}),({\mu _1},i)\} , 1 \le i \le m} {(z)^{m/2}}{({\sigma _i})^{m/2}}{(X_{i{\mu _1}}^G)^m}{R_{a{a_1}}}{R_{{b_1}{a_2}}} \ldots {R_{{b_m}b}}, \](3.50)

\[{([RV{]^m}S)_{ab}} = \sum\limits_{({a_i},{b_i}) \in \{ (i,{\mu _1}),({\mu _1},i)\} , 1 \le i \le m} {(z)^{m/2}}{({\sigma _i})^{m/2}}{(X_{i{\mu _1}}^G)^m}{R_{a{a_1}}}{R_{{b_1}{a_2}}} \ldots {S_{{b_m}b}}.\](3.51)

Define

\[\Delta {X_{\mu \nu ,k}} : = {S_{\mu k}}{\overline S _{\nu k}} - {R_{\mu k}}{\overline R _{\nu k}}.\](3.52)

In [Reference Knowles and Yin24], the discussion relied on a crucial parameter (see [Reference Knowles and Yin24, Equation (3.32)]), which counts the maximum number of diagonal resolvent elements in Δ X _{μ ν, k}. We will follow this strategy using a different counting parameter, and, furthermore, use (3.50) and (3.51) as our key ingredients. Our discussion is slightly easier due to the loss of a free index (i.e. i ≠ μ ₁).

Inserting (3.49) into (3.52), by (3.50) and (3.51), we find that there exists a random variable A ₁, which depends on the randomness only through O and the first two moments of \[X_{i{\mu _1}}^G\]. Taking the partial expectation with respect to the (i, μ ₁)th entry of X ^G(recall they are i.i.d.), by (1.2), we have the following result.

Lemma 3.7. Recall (2.7), and let 𝔼_γ be the partial expectation with respect to \[X_{i{\mu _1}}^G\]. Then there exists some constant C > 0, and with probability \[1 - {N^{ - {D_1}}}\], we have

\[|{{\mathbb{E}}_\gamma }\Delta {X_{\mu \nu ,k}} - {A_1}| \le {N^{ - 3/2 + C{\varepsilon _0}}}\Psi {(z)^{3 - s}},\quad \quad M + 1 \le k \ne \mu ,\nu \le M + N,\]

where s counts the maximum number of resolvent elements in ΔX _{μ ν, k} involving the index μ ₁ and is defined as

\[s : = {\bf{1}}((\{ \mu ,\nu \} \cap \{ {\mu _1}\} \ne \emptyset ) \cup (\{ k = {\mu _1}\} )).\](3.53)

Proof. Inserting (3.49) into (3.52), the terms in the expansion containing \[X_{i{\mu _1}}^G\] and \[{(X_{i{\mu _1}}^G)^2}\] will be included in A ₁; we consider only the terms containing \[{(X_{i{\mu _1}}^G)^m}, m \ge 3\]. We consider m = 3 and discuss the terms

\[{R_{\mu k}}{\overline {[(RV{)^3}R]} _{\nu k}},\quad \quad {[RVR]_{\mu k}}{\overline {[(RV{)^2}R]} _{\nu k}}.\]

By (3.50), we have

\[{R_{\mu k}}{\overline {[(RV{)^3}R]} _{\nu k}} = {R_{\mu k}}(\sum {({\sigma _i})^{3/2}}{(X_{i{\mu _1}}^G)^3}\overline {{{(z)}^{3/2}}{R_{\nu {a_1}}}{R_{{b_1}{a_2}}}{R_{{b_2}{a_3}}}{R_{{b_3}k}}} ).\]

In the worst scenario, \[{R_{{b_1}{a_2}}}\] and \[{R_{{b_2}{a_3}}}\] are assumed to be the diagonal entries of R. Similarly, we have

\[{[RVR]_{\mu k}}{\overline {[(RV{)^2}R]} _{\nu k}} = (\sum {z^{1/2}}\sigma _i^{1/2}X_{i{\mu _1}}^G{R_{\mu {a_1}}}{R_{{b_1}k}})(\sum {\sigma _i}{(X_{i{\mu _1}}^G)^2}\overline {z{R_{\nu {a_1}}}{R_{{b_1}{a_2}}}{R_{{b_2}k}}} ),\]

and the worst scenario is the case when \[{R_{{b_1}{a_2}}}\] is a diagonal term. As μ, ν ≠ i always holds and there are only a finite number of terms in the summation, by (1.2) and (3.36), for some constant C, we have

\[{{\mathbb{E}}_\gamma }|{R_{\mu k}}{\overline {[(RV{)^3}R]} _{\nu k}}| \le {N^{ - 3/2 + C{\varepsilon _0}}}\Psi {(z)^{3 - s}}.\]

Similarly, we have

\[{{\mathcal{E}}_\gamma }|[RVR{]_{\mu k}}{\overline {[(RV{)^2}R]} _{\nu k}}| \le {N^{ - 3/2 + C{\varepsilon _0}}}\Psi {(z)^{3 - s}}.\]

The cases in which 4 ≤ m ≤ 8 can be handled similarly. This completes the proof. □

Lemma 3.5 follows from the following lemma. Recall (3.38), and define

\[\Delta x(E) : = {x^S}(E) - {x^R}(E),\quad \quad \Delta y(E) : = {y^S}(E) - {y^R}(E).\]

Lemma 3.8. For any fixed μ, ν, and μ, there exists a random variable A, which depends on the randomness only through O and the first two moments of X^G, such that

\[{\mathbb{E}}\theta (\int_I {x^S}q({y^S}) {\rm{d}}E) - {\mathbb{E}}\theta (\int_I {x^R}q({y^R}) {\rm{d}}E) = A + o({N^{ - 2 + t}}),\](3.54)

where t := |μ, ν ∩ μ ₁|.

The proof of Lemma 3.8 given in the supplementary material [Reference Ding14]. We now show how Lemma 3.8 implies Lemma 3.5.

Proof of Lemma 3.5. It is easy to check that Lemma 3.8 still holds when we replace S with T. Note that in (3.48) there are O(N) terms when t = 1 and O(N ²) terms when t = 0. By (3.54), we have

\[{\mathbb{E}}[\theta (\int_I {x^G}q({y^G}) {\rm{d}}E) - \theta (\int_I {x^V}q({y^V}){\kern 1pt} {\rm{d}}E)] = o(1),\]

where we have used the assumption that the first two moments of X ^V are the same as those of X ^G. Combine with (3.37) to complete the proof. □

It is clear that our proof can be extended to the left singular vectors. For the proof of Theorem 1.1, the only difference is that we use the mean value theorem in ℝ² whenever it is needed. Moreover, for the proof of Theorem 1.2, we need to use n intervals defined by

\[{I_i}{\kern 1pt} : = [{a_{2{k_i} - 1}} - {N^{ - 2/3 + \varepsilon }},{a_{2{k_i} - 1}} + {N^{ - 2/3 + \varepsilon }}],\quad \quad i = 1,2, \ldots ,n.\]

3.3. Extension to singular values

In this section we discuss how the arguments of Section 3.2 can be applied to the general function θ defined in (1.15) containing singular values. We mainly focus on discussing the proof of Corollary 1.1.

On the one hand, similarly to Lemma 3.3, we can write the singular values in terms of an integral of smooth functions of Green functions. Using the comparison argument with θ ∈ ℝ³ and the mean value theorem in ℝ³ completes our proof. Similar discussions and results have been derived in [Reference Erdös, Yau and Yin18, Corollary 6.2 and Theorem 6.3]. For completeness, we basically follow the strategy in [Reference Knowles and Yin24, Section 4] to prove Corollary 1.1. The basic idea is to write the function θ in terms of Green functions by using integration by parts. We mainly look at the right edge of the kth bulk component.

Proof of Corollary 1.1. Let F ^V be the law of λ _α′, and consider a smooth function θ: ℝ→ℝ For δ defined in Lemma 3.2, when \[l \le N_k^\delta \], by (1.14) and (2.18), it is easy to check that

\[{E^V}\theta ({{{N^{2/3}}} \over \varpi }({\lambda _{{\alpha ^{'}}}} - {a_{2k - 1}})) = \int_I \theta ({{{N^{2/3}}} \over \varpi }(E - {a_{2k - 1}})){\kern 1pt} {\rm{d}}{F^V}(E) + O({N^{ - {D_1}}}),\](3.55)

where ϖ := ϖ_{2k – 1} and I is defined in (3.6). Using integration by parts on (3.55), we have

\[[{{\mathcal{E}}^V} - {{\mathcal{E}}^G}]\theta ({{{N^{2/3}}} \over \varpi }({\lambda _{{\alpha ^{'}}}} - {a_{2k - 1}})) \\ \quad \quad = - [{{\mathcal{E}}^V} - {{\mathcal{E}}^G}]\int_I {{{N^{2/3}}} \over \varpi }{\theta ^{'}}({{{N^{2/3}}} \over \varpi }(E - {a_{2k - 1}})){\bf{1}}({\lambda _{{\alpha ^{'}}}} \le E)dE + O({N^{ - {D_1}}}),\]

where we have used (1.14) and (2.18). Similarly to (3.27), recalling (11), choose a smooth nonincreasing function f _l that vanishes on the interval \[[l + {\textstyle{2 \over 3}},\infty )\] and is equal to 1 on the interval \[( - \infty ,l + {\textstyle{1 \over 3}}]\]. Recall that E _U = a _{2k – 1} + 2N ^{– 2/3 + ε} and \[{\mathcal{N}}(E,{E_U})\] denotes the number of eigenvalues of Q ₁ located in the interval [E,E _U] By (3.56), we have

\[[{{\mathbb{E}}^V} - {{\mathbb{E}}^G}]\theta ({{{N^{2/3}}} \over \varpi }({\lambda _{{\alpha ^{'}}}} - {a_{2k - 1}})) \\ \quad \quad = - [{{\mathbb{E}}^V} - {{\mathcal{E}}^G}]\int_I {{{N^{2/3}}} \over \varpi }{\theta ^{'}}({{{N^{2/3}}} \over \varpi }(E - {a_{2k - 1}})){f_l}(N(E,{E_U}))dE + O({N^{ - {D_1}}}).\]

Recall that \[\tilde \eta = {N^{ - 2/3 - 9{\varepsilon _0}}}\]. Similarly to the discussion of (3.31), with probability \[1 - {N^{ - {D_1}}}\], we have

\[{N^{2/3}}\int_I |{\rm Tr}({{\bf{1}}_{[E,{E_U}]}}*{\vartheta _{\tilde \eta }}({Q_1}))) - {\rm Tr}({{\bf{1}}_{[E,{E_U}]}}({Q_1}))|{\kern 1pt} {\rm{d}}E \le {N^{ - {\varepsilon _0}}}.\]

This yields

\[[{{\mathbb{E}}^V} - {{\mathbb{E}}^G}]\theta ({{{N^{2/3}}} \over \varpi }({\lambda _{{\alpha ^{'}}}} - {a_{2k - 1}})) \\ \quad = - [{{\mathbb{E}}^V} - {{\mathbb{E}}^G}]\int_I {{{N^{2/3}}} \over \varpi }{\theta ^{'}}({{{N^{2/3}}} \over \varpi }(E - {a_{2k - 1}})){f_l}({\rm Tr}({{\bf{1}}_{[E,{E_U}]}}*{\vartheta _{\tilde \eta }}({Q_1}))){\kern 1pt} {\rm{d}}E + O({N^{ - {D_1}}}).\]

Integration by parts yields

\[[{{\mathbb{E}}^V} - {{\mathbb{E}}^G}]\theta ({{{N^{2/3}}} \over \varpi }({\lambda _{{\alpha ^{'}}}} - {a_{2k - 1}})) \\ \quad \quad = {N \over \pi }[{{\mathbb{E}}^V} - {{\mathcal{E}}^G}]\int_I \theta ({{{N^{2/3}}} \over \varpi }({\lambda _{{\alpha ^{'}}}} - {a_{2k - 1}})) \\ \times f_l^{'}({\rm Tr}({{\bf{1}}_{[E,{E_U}]}}*{\vartheta _{\tilde \eta }}({Q_1})))Im{m_2}(E + {\rm{i}}\tilde \eta ){\kern 1pt} {\rm{d}}E + o(1),\]

where we have used (3.42). Now we extend θ to the general case defined in (1.15). By Theorem 1.1, it is easy to check that

\[[{{\mathbb{E}}^V} - {{\mathbb{E}}^G}]\theta ({{{N^{2/3}}} \over \varpi }({\lambda _{{\alpha ^{'}}}} - {a_{2k - 1}}),N{\xi _{{\alpha ^{'}}}}(i){\xi _{{\alpha ^{'}}}}(j),N{\zeta _{{\alpha ^{'}}}}(\mu ){\zeta _{{\alpha ^{'}}}}(\nu )) \\ \quad \quad = {1 \over \pi }[{{\mathbb{E}}^V} - {{\mathbb{E}}^G}]\int_I \theta ({{{N^{2/3}}} \over \varpi }({\lambda _{{\alpha ^{'}}}} - {a_{2k - 1}}),{\phi _{{\alpha ^{'}}}},{\varphi _{{\alpha ^{'}}}}) \\ \times f_l^{'}({\rm Tr}({{\bf{1}}_{[E,{E_U}]}}*{\vartheta _{\tilde \eta }}({Q_1})))NIm{m_2}(E + {\rm{i}}\tilde \eta ){\kern 1pt} {\rm{d}}E + o(1),\](3.57)

where

\[{\phi _{{\alpha ^{'}}}} = {N \over \pi }\int_I {\tilde G_{ij}}(\tilde E + i\eta ){q_1}[{\rm Tr}({{\bf{1}}_{[{{\tilde E}^ - },{E_U}]}}*{\vartheta _{\tilde \eta }}({Q_1}))]{\kern 1pt} {\rm{d}}\tilde E, \\ {\varphi _{{\alpha ^{'}}}} = {N \over \pi }\int_I {\tilde G_{\mu \nu }}(\tilde E + i\eta ){q_2}[{\rm Tr}({{\bf{1}}_{[{{\tilde E}^ - },{E_U}]}}*{\vartheta _{\tilde \eta }}({Q_1}))]d\tilde E,\]

and q ₁ and q ₂ are the functions defined in (3.27). Therefore, the randomness on the right-hand side of (3.57) is expressed in terms of Green functions. Hence, we can apply the Green function comparison argument to (3.57) as in Section 3.2. The complications are notational and we will not reproduce the details here. □

Finally, the proof of Corollary 1.2 is very similar to that of Corollary 1.1 except that we use n different intervals and a multidimensional integral. We will not reproduce the details here.

4.1. Proof of Lemma 4.2

The proof strategy is very similar to that of Lemma 3.1. Our first step is an analogue of Lemma 3.2. The proof is quite similar (actually easier as the window size is much smaller). We omit further details.

Lemma 4.3. Under the assumptions of Lemma 4.2, there exists a 0 < δ < 1 such that

\[\mathop {\lim }\limits_{N \to \infty } \mathop {\max }\limits_{\delta {N_k} \le l \le (1 - \delta ){N_k}} \mathop {\max }\limits_{\mu ,\nu } |{{\mathbb{E}}^V}\theta (N{\zeta _{{\alpha ^{'}}}}(\mu ){\zeta _{{\alpha ^{'}}}}(\nu )) - {{\mathbb{E}}^V}\theta [{N \over \pi }\int_I {\tilde G_{\mu \nu }}(z){\mathcal X}(E){\kern 1pt} {\rm{d}}E]| = 0,\](4.5)

where \[{\mathcal X}(E)\] is defined in (3.7) and, for ε satisfying (3.5),

\[I{\kern 1pt} : = [{\gamma _{{\alpha ^{'}}}} - {N^{ - 1 + \varepsilon }},{\gamma _{{\alpha ^{'}}}} + {N^{ - 1 + \varepsilon }}].\](4.6)

Next we express the indicator function in (4.5) using Green functions. Recall (3.28), a key observation is that the size of E ^–, E _U is of order N ^{– 2/3} due to (3.4). As we now use (4.2) and (101) in the bulks, the size here is of order 1. So we cannot use the delta approximation function to estimate \[{\mathcal X}(E)\]. Instead, we use Helffer–Sjöstrand functional calculus. This has been used many times when the window size η takes the form of (4.2), for example, in the proofs of rigidity of eigenvalues in [Reference Ding and Yang16], [Reference Erdös, Yau and Yin18], and [Reference Pillai and Yin33].

For any 0 < E ₁, E ₂ ≤ τ^{– 1}, let \[f(\lambda ) \equiv {f_{{E_1},{E_2},{\eta _d}}}(\lambda )\] be the characteristic function of E ₁, E ₂ smoothed on the scale

\[{\eta _d}{\kern 1pt} : = {N^{ - 1 - d{\varepsilon _0}}},\quad \quad d \gt 2,\]

where f = 1 when λ ∈ E ₁, E ₂ and f = 0 when λ ∈ ℝ\ [E ₁ – η_d, E ₂ + η_d], and

\[|{f^{'}}| \le C\eta _d^{ - 1},\quad \quad |{f^{''}}| \le C\eta _d^{ - 2},\](4.7)

for some constant C > 0 By Equation (B.12) of [Reference Erdös, Ramirez, Schlein and Yau19], with \[{f_E} \equiv {f_{{E^ - },{E_U},{\eta _d}}},\] we have

\[{f_E}(\lambda ) = {1 \over {2\pi }}\int_{{{\mathbb{R}}^2}} {{{\rm{i}}\sigma f_E^{''}(e)\chi (\sigma ) + {\rm{i}}{f_E}(e){\chi ^{'}}(\sigma ) - \sigma f_E^{'}(e){\chi ^{'}}(\sigma )} \over {\lambda - e - {\rm{i}}\sigma }}{\kern 1pt} {\rm{d}}e{\kern 1pt} {\rm{d}}\sigma ,\](4.8)

where X(γ) is a smooth cutoff function with support [–1, 1] and χ(γ)=1 for \[|y| \le {1 \over 2}\] with bounded derivatives. Using a similar argument to that used for Lemma 3.3, we have the following result, whose proof is given in the supplementary material [Reference Ding14].

Lemma 4.4. Recall the smooth cutoff function q defined in (3.27). Under the assumptions of Lemma 4.3, there exists a 0 < δ < 1 such that

\[\mathop {\lim }\limits_{N \to \infty } \mathop {\max }\limits_{\delta {N_k} \le l \le (1 - \delta ){N_k}} \mathop {\max }\limits_{\mu ,\nu } |{{\mathbb{E}}^V}\theta ({N \over \pi }\int_I {\tilde G_{\mu \nu }}(z){\mathcal X}(E)){\kern 1pt} {\rm{d}}E \\ - {{\mathbb{E}}^V}\theta ({N \over \pi }\int_I {\tilde G_{\mu \nu }}(z)q({\rm Tr}{f_E}({Q_1}))){\kern 1pt} {\rm{d}}E| \\ \quad \quad = 0.\](4.9)

Finally, we apply the Green function comparison argument, where we will follow the basic approach of Section 3.2 and [Reference Knowles and Yin24, Section 5]. The key difference is that we will use (4.2) and (4.3).

Lemma 4.5. Under the assumptions of Lemma 4.4, there exists a 0 < δ < 1 such that

\[\mathop {\lim }\limits_{N \to \infty } \mathop {\max }\limits_{\delta {N_k} \le l \le (1 - \delta ){N_k}} \mathop {\max }\limits_{\mu ,\nu } [{{\mathbb{E}}^V} - {{\mathbb{E}}^G}]\theta [{N \over \pi }\int_I {\tilde G_{\mu \nu }}(E + {\rm{i}}\eta )q({\rm Tr}{f_E}({Q_1})){\kern 1pt} {\rm{d}}E] = 0.\](4.10)

Proof. Recall (4.8). By (2.5), we have

\[{\rm Tr}{f_E}({Q_1}) = {N \over {2\pi }}\int_{{{\mathbb{R}}^2}} ({\rm{i}}\sigma f_E^{''}(e)\chi (\sigma ) + {\rm{i}}{f_E}(e){\chi ^{'}}(\sigma ) - \sigma f_E^{'}(e){\chi ^{'}}(\sigma )){m_2}(e + {\rm{i}}\sigma ){\kern 1pt} {\rm{d}}e{\kern 1pt} {\rm{d}}\sigma .\](4.11)

Define \[{\tilde \eta _d}{\kern 1pt} : = {N^{ - 1 - (d + 1){\varepsilon _0}}}\]. We can decompose the right-hand side of (4.11) as

\[{\rm Tr}{f_E}({Q_1}) = {N \over {2\pi }}\int \int_{{{\mathbb{R}}^2}} ({\rm{i}}{f_E}(e){\chi ^{'}}(\sigma ) - \sigma f_E^{'}(e){\chi ^{'}}(\sigma )){m_2}(e + {\rm{i}}\sigma ){\kern 1pt} {\rm{d}}e{\kern 1pt} {\rm{d}}\sigma \\ \quad {\kern 1pt} + {{{\rm{i}}N} \over {2\pi }}\int_{|\sigma | \gt {{\tilde \eta }_d}} \sigma \chi (\sigma )\int f_E^{''}(e){m_2}(e + {\rm{i}}\sigma ){\kern 1pt} {\rm{d}}\sigma {\kern 1pt} {\rm{d}}e \\ \quad {\kern 1pt} + {{{\rm{i}}N} \over {2\pi }}\int_{ - {{\tilde \eta }_d}}^{{{\tilde \eta }_d}} \sigma \chi (\sigma )\int f_E^{''}(e){m_2}(e + {\rm{i}}\sigma ){\kern 1pt} {\rm{d}}\sigma {\kern 1pt} {\rm{d}}e.\]

By (4.3) and (4.7), for some constant C > 0 with probability \[1 - {N^{ - {D_1}}}\], we have

\[|{{{\rm{i}}N} \over {2\pi }}\int_{ - {{\tilde \eta }_d}}^{{{\tilde \eta }_d}} \sigma \chi (\sigma )\int f_E^{''}(e){m_2}(e + {\rm{i}}\sigma ){\kern 1pt} {\rm{d}}\sigma {\kern 1pt} {\rm{d}}e| \le {N^{ - C{\varepsilon _0}}}.\](4.12)

Recall (3.35) and (3.38). Similarly to Lemma 3.6, we first drop the diagonal terms. By (4.1), with probability \[1 - {N^{ - {D_1}}}\], we have (recall (3.41))

\[\int_I |{N \over \pi }{\tilde G_{\mu \nu }}(E + {\rm{i}}\eta ) - x(E)|{\kern 1pt} {\rm{d}}E \le {N^{ - 1 + C{\varepsilon _0}}}\]

for some constant C > 0 Hence, by the mean value theorem, we need only prove that

\[\mathop {\lim }\limits_{N \to \infty } \mathop {\max }\limits_{\delta {N_k} \le l \le (1 - \delta ){N_k}} \mathop {\max }\limits_{\mu ,\nu } [{{\mathbb{E}}^V} - {{\mathbb{E}}^G}]\theta (\int_I x(E)q({\rm Tr}{f_E}({Q_1})){\kern 1pt} {\rm{d}}E) = o(1).\]

Furthermore, by Taylor’s expansion (4.12), and the definition of χ, it suffices to prove that

\[\mathop {\lim }\limits_{N \to \infty } \mathop {\max }\limits_{\delta {N_k} \le l \le (1 - \delta ){N_k}} \mathop {\max }\limits_{\mu ,\nu } [{{\mathbb{E}}^V} - {{\mathbb{E}}^G}]\theta (\int_I x(E)q(y(E) + \tilde y(E)){\kern 1pt} {\rm{d}}E) = o(1),\](4.13)

where

\[y(E){\kern 1pt} : = {N \over {2\pi }}\int_{{{\mathbb{R}}^2}} {\rm{i}}\sigma f_E^{''}(e)\chi (\sigma ){m_2}(e + {\rm{i}}\sigma ){\bf{1}}(|\sigma | \ge {\tilde \eta _d}){\kern 1pt} {\rm{d}}e{\kern 1pt} {\rm{d}}\sigma ,\](4.14)

\[\tilde y(E){\kern 1pt} : = {N \over {2\pi }}\int_{{R^2}} ({\rm{i}}{f_E}(e){\chi ^{'}}(\sigma ) - \sigma f_E^{'}(e){\chi ^{'}}(\sigma )){m_2}(e + {\rm{i}}\sigma ){\kern 1pt} {\rm{d}}e{\kern 1pt} {\rm{d}}\sigma .\](4.15)

Next we will use the Green function comparison argument to prove (4.13). In the proof of Lemma 3.5, we used the resolvent expansion until an order of four. However, due to the larger bounds in (4.3), we will use the expansion

\[S = R - RVR + {(RV)^2}R - {(RV)^3}R + {(RV)^4}R - {(RV)^5}S.\](4.16)

Recall (3.47) and (3.48). We have

\[[{{\mathbb{E}}^V} - {{\mathbb{E}}^G}]\theta (\int_I x(E)q(y(E) + \tilde y(E)){\kern 1pt} {\rm{d}}E) \\ \quad \quad = \sum\limits_{\gamma = 1}^{{\gamma _{\max }}} {\mathbb{E}}\left( {\theta ((\int_I {x^S}q({y^S} + {{\tilde y}^S}))) - \theta ((\int_I {x^T}q({y^T} + {{\tilde y}^T})))} \right).\](4.17)

We still use the same notation Δ x(E) := x^S(E) - x^R(E). We basically follow the approach of Section 3.2, where the control (3.36) is replaced by (4.3). We first deal with x(E). Let Δ x ^(k)(E) denote the summations of the terms in Δ x(E) containing k numbers of \[X_{i{\mu _1}}^G\]. Similarly to the discussion of Lemma 3.7, recalling (3.52), by (1.2) and (4.3), with probability \[1 - {N^{ - {D_1}}}\], we have

\[|\Delta {x^{(5)}}(E)| \le {N^{ - 3/2 + C{\varepsilon _0}}},\quad \quad M + 1 \le k \ne \mu ,\nu \le M + N.\]

This yields

\[\Delta x(E) = \sum\limits_{p = 1}^4 \Delta {x^{(p)}}(E) + O({N^{ - 3/2 + C{\varepsilon _0}}}).\](4.18)

Let

\[\Delta \tilde y(E) = {\tilde y^S}(E) - {\tilde y^R}(E),\quad \quad \Delta {m_2}{\kern 1pt} : = m_2^S - m_2^R = {1 \over N}\sum\limits_{\mu = M + 1}^{M + N} ({S_{\mu \mu }} - {R_{\mu \mu }}).\]

We first deal with (4.15). By the definition of χ, we need to restrict \[{1 \over 2} \le |\sigma | \le 1\]; hence, by (2.17), with probability \[1 - {N^{ - {D_1}}}\], we have

\[\mathop {\max }\limits_\mu |{G_{\mu \mu }}| \le {N^{{\varepsilon _1}}},\quad \quad \mathop {\max }\limits_{\mu \ne \nu } |{G_{\mu \nu }}| \le {N^{ - 1/2 + {\varepsilon _1}}}.\](4.19)

By (3.50), (3.51), (4.16), and (4.19), with probability \[1 - {N^{ - {D_1}}}\], we have \[|\Delta m_2^{(5)}| \le {N^{ - 7/2 + 9{\varepsilon _1}}}\]. This yields the decomposition

\[\Delta \tilde y(E) = \sum\limits_{p = 1}^4 \Delta {\tilde y^{(p)}}(E) + O({N^{ - 5/2 + C{\varepsilon _0}}}).\](4.20)

Next we will control (4.14). Define \[\Delta y(E){\kern 1pt} : = {y^S}(E) - {y^R}(E)\]. By (3.50), (3.51) and (4.1), using a similar discussion to that used for Equation (5.22) of [Reference Knowles and Yin24], with probability \[1 - {N^{ - {D_1}}}\], for \[\sigma \ge {\tilde \eta _d},\] we have

\[|\Delta m_2^{(5)}| \le {N^{ - 5/2 + C{\varepsilon _0}}}({N^{ - 1}} + \Lambda _\sigma ^2),\](4.21)

where \[{\Lambda _\sigma }{\kern 1pt} : = \mathop {\sup }\nolimits_{|e| \le {\tau ^{ - 1}}} \mathop {\max }\nolimits_{\mu \ne \nu } |{G_{\mu \nu }}(e + {\rm{i}}\sigma )|\], recalling that \[\mu ,\nu \in {{\mathcal{I}}_2}\]. In order to estimate Δγ(E), we integrate (4.14) by parts, first in e then in σ. By Equation (5.24) of [Reference Knowles and Yin24], with probability \[1 - {N^{ - {D_1}}}\], we have

\[|{N \over {2\pi }}\int_{{{\mathbb{R}}^2}} {\rm{i}}\sigma f_E^{''}(e)\chi (\sigma ){\Delta ^{(5)}}{m_2}(e + {\rm{i}}\sigma ){\bf{1}}(|\sigma | \ge {\tilde \eta _d}){\kern 1pt} {\rm{d}}e{\kern 1pt} {\rm{d}}\sigma | \\ \quad \quad \le CN|\int f_E^{'}(e){\tilde \eta _d}\Delta m_2^{(5)}(e + {\rm{i}}{\tilde \eta _d}){\kern 1pt} {\rm{d}}e| + CN|\int f_E^{'}(e){\kern 1pt} {\rm{d}}e\int_{{{\tilde \eta }_d}}^\infty {\chi ^{'}}(\sigma )\sigma \Delta m_2^{(5)(e + {\rm{i}}\sigma )}{\kern 1pt} {\rm{d}}\sigma | \\ \quad \quad + CN|\int f_E^{'}(e){\kern 1pt} {\rm{d}}e\int_{{{\tilde \eta }_d}}^\infty \chi (\sigma )\Delta m_2^{(5)}(e + {\rm{i}}\sigma ){\kern 1pt} {\rm{d}}\sigma |.\](4.22)

By (4.21), with probability \[1 - {N^{ - {D_1}}}\], the first two items of (4.22) can be easily bounded by \[{N^{ - 5/2 + C{\varepsilon _0}}}\]. For the last item, by (4.21), (4.1), and a similar discussion to the equation below [Reference Knowles and Yin24, Equation (5.24)], it can be bounded by

\[CN\int_{{{\tilde \eta }_d}}^1 ({1 \over {N\sigma }} + {1 \over {{{(N\sigma )}^2}}} + {1 \over N}){N^{ - 5/2 + C{\varepsilon _0}}} \le {N^{ - 5/2 + C{\varepsilon _0}}}.\]

Hence, with probability \[1 - {N^{ - {D_1}}}\], we have the decomposition

\[\Delta y(E) = \sum\limits_{p = 1}^4 \Delta {y^{(p)}}(E) + O({N^{ - 5/2 + C{\varepsilon _0}}}).\](4.23)

Similarly to the discussion of (4.18), (4.20), and (4.23), it is easy to check that, with probability \[1 - {N^{ - {D_1}}}\], we have

\[\matrix{ {\int_I |\Delta {x^{(p)}}(E)|{\kern 1pt} {\rm{d}}E \le {N^{ - p/2 + C{\varepsilon _0}}},\quad \quad |\Delta {{\tilde y}^{(p)}}(E)| \le {N^{ - p/2 + C{\varepsilon _0}}},} \cr {|\Delta {y^{(p)}}(E)| \le {N^{ - p/2 + C{\varepsilon _0}}},} \cr } \](4.24)

where p = 1, 2, 3, 4 and C > 0 is some constant. Furthermore, by (4.1), with probability \[1 - {N^{{D_1}}}\], we have

\[\int_I |x(E)|{\kern 1pt} {\rm{d}}E \le {N^{C{\varepsilon _0}}}.\](4.25)

Due to the similarity of (4.20) and (4.23), letting \[\;\bar y = y + \tilde y,\] we have

\[\Delta \bar y = \sum\limits_{p = 1}^4 \Delta {\bar y^{(p)}}(E) + O({N^{ - 5/2 + C{\varepsilon _0}}}).\](4.26)

By (4.24), (4.26), and Taylor’s expansion, we have

\[q({\bar y^S}) = q({\bar y^R}) + {q^{'}}({\bar y^R})(\sum\limits_{p = 1}^4 \Delta {\bar y^{(p)}}(E)) + {1 \over 2}{q^{''}}({\bar y^R})(\sum\limits_{p = 1}^3 \Delta {\bar y^{(p)}}(E{))^2} \\ \quad {\kern 1pt} + {1 \over 6}{q^{(3)}}({\bar y^R})(\sum\limits_{p = 1}^2 \Delta {\bar y^{(p)}}(E{))^3} + {1 \over {24}}{q^{(4)}}({\bar y^R})(\Delta {\bar y^{(1)}}(E{))^4} + o({N^{ - 2}}).\](4.27)

By (4.4), we have

\[\theta (\int_I {x^S}q({\bar y^S}){\kern 1pt} {\rm{d}}E) - \theta (\int_I {x^R}q({\bar y^R}){\kern 1pt} {\rm{d}}E) \\ \quad \quad = \sum\limits_{s = 1}^4 {1 \over {s!}}{\theta ^{(s)}}(\int_I {x^R}q({\bar y^R}){\kern 1pt} {\rm{d}}E)[\int_I {x^S}q({\bar y^S}){\kern 1pt} {\rm{d}}E - \int_I {x^R}q({\bar y^R})dE{]^s} + o({N^{ - 2}}).\](4.28)

Inserting \[{x^S} = {x^R} + \sum\nolimits_{p = 1}^4 \Delta {x^{(p)}}\] and (4.27) into (4.28), using the partial expectation argument as in Section 3.2, by (4.4), (4.24), and (4.25), we find that there exists a random variable B that depends on the randomness only through O and the first four moments of \[X_{i{\mu _1}}^G\] such that

\[{\mathbb{E}}\theta (\int_I {x^S}q{(y + \tilde y)^S}{\kern 1pt} {\rm{d}}E) - {\mathbb{E}}\theta (\int_I {x^R}q{(y + \tilde y)^R}{\kern 1pt} {\rm{d}}E) = B + o({N^{ - 2}}).\]

Hence, together with (4.17), this proves (4.13), which implies (4.10). This completes our proof. □