\chapter{Complex Hermitian matrices}\label{s:hermitian}

In this section, we sketch the proof of \zcref{t:hermitian}. The argument uses the comparison estimate of \zcref{s:comparison} and the induction on the scale of \zcref{s:pairs}, but not the resampling argument of \zcref{s:linearization,s:llt}; we explain the reason at the beginning of \zcref{s:herm-strategy}. We indicate the changes to the arguments of the previous sections and give the details of the two estimates that are new. The first is a relative bound on the number of triples of nearby eigenvalues, \zcref{p:herm-triples}, and the second is a bound on the probability that a Gaussian perturbation of a $3 \times 3$ diagonal matrix has a small gap, \zcref{l:herm-3x3}. We prove both in full, given the inputs collected in \zcref{s:herm-inputs}. Where we only sketch an argument, we name the result of the earlier sections whose proof we follow and the properties that this proof uses, and we explain why these properties hold in the complex Hermitian case.

\section{Setup}\label{s:herm-setup}

Throughout this section, $H$ is a complex Hermitian Wigner matrix as in \zcref{d:hermitian}, and $\lambda_1 \le \dots \le \lambda_N$, $g_i$, and $c_2(E) = (\pi^2/3) \rhosc(E)^4$ are as in \zcref{t:hermitian}; we also let $E_i = (\lambda_i + \lambda_{i + 1})/2$. We use the notation of the previous sections with the following changes. In this section, $G$ denotes a complex Hermitian matrix from the Gaussian unitary ensemble (GUE), independent of $H$, whose entries $G_{ab}$ with $a < b$ are independent complex Gaussian variables with $\E G_{ab} = \E G_{ab}^2 = 0$ and $\E |G_{ab}|^2 = 1/N$, and whose diagonal entries are independent real Gaussian variables with variance $1/N$, independent of the entries above the diagonal. The Ornstein--Uhlenbeck interpolation $H_t$ is defined by \eqref{e:prelim-Ht} with this $G$. We set
\begin{equation}\label{e:herm-scales}
d_2 = N^{-4/3}, \qquad M^{\C}_\alpha = N^4 \ell_\alpha^3 = N^{1 - 3\alpha} \quad (\alpha \in (0, 1/3)).
\end{equation}
\zcref[S]{t:hermitian} states that the smallest gaps in the bulk are of order $d_2$, and we call $d_2$ the \emph{critical scale}. By \zcref{l:herm-gue} below, for $\alpha \in (0, 1/4)$, the number of gaps of the GUE of size at most $\ell_\alpha = N^{-1 - \alpha}$ with midpoints in a bulk interval is of order $M^{\C}_\alpha$.

Let $p \ge 1$ be an integer, let $\chi \in C_c^\infty((-2, 2))$ be nonnegative, and let $q > 0$. We define the triple and quadruple statistics of a Hermitian matrix $A$ by
\begin{equation}\label{e:herm-U}
\begin{aligned}
U^{(3)}_q(A) &= \sum_{i < j < k} \chi_N\Big( \frac{\lambda_i(A) + \lambda_j(A) + \lambda_k(A)}{3} \Big) \prod_{\{ i', i'' \} \subset \{ i, j, k \}} f_p\Big( \frac{\lambda_{i'}(A) - \lambda_{i''}(A)}{q} \Big), \\
U^{(4)}_q(A) &= \sum_{i < j < k < l} \chi_N\Big( \frac{\lambda_i(A) + \lambda_j(A) + \lambda_k(A) + \lambda_l(A)}{4} \Big) \\
&\qquad \times \prod_{\{ i', i'' \} \subset \{ i, j, k, l \}} f_p\Big( \frac{\lambda_{i'}(A) - \lambda_{i''}(A)}{q} \Big),
\end{aligned}
\end{equation}
where $\chi_N$ is the smoothing \eqref{e:comparison-phiN} of $\chi$, and the products are over the three, respectively six, two-element subsets. We suppress the dependence on $p$ and $\chi$ in the notation. Observe that if $\lambda_k(A) - \lambda_i(A) \le 4q$ and $\chi_N \ge 1$ at the mean of $\lambda_i(A)$, $\lambda_j(A)$, and $\lambda_k(A)$, then the term of $(i, j, k)$ in $U^{(3)}_q(A)$ is at least $17^{-3p}$, since each of its three rational factors is at least $f_p(4) = 17^{-p}$. Similarly, if $\lambda_l(A) - \lambda_i(A) \le 6q$ and $\chi_N \ge 1$ at the mean of the four eigenvalues, then the term of $(i, j, k, l)$ in $U^{(4)}_q(A)$ is at least $37^{-6p}$, since $f_p(6) = 37^{-p}$.

\section{Strategy}\label{s:herm-strategy}

The smallest gaps of $H$ are of order $d_2 = N^{-4/3}$. A single entry moves the eigenvalues by about $N^{-3/2} = o(d_2)$, so the strip estimate of \zcref{l:strip} holds at the scale $d_2$, which corresponds to the exponent $\alpha = 1/3 < 1/2$. Consequently, the comparison estimate of \zcref{p:comparison} applies to pair statistics at the scale $d_2$, with the error
\begin{equation}\label{e:herm-critcomp}
C t N^{1/2} (N d_2)^{-3} N^{C\eps} = C t N^{3/2 + C\eps}.
\end{equation}
This is small if $t = t_{\alpha_1} = N^{-1 - 2\alpha_1}$ with $\alpha_1 > 1/4$. For such $t$, the Gaussian part of $H_t$ is of order $\ell_{\alpha_1} = N^{-1 - \alpha_1}$, and the computation of \zcref{s:pairs} through independent $2 \times 2$ Gaussian blocks determines the gaps of $H_t$ at the scale $d_2$, provided that the gap counts of $H$ at the scale $\ell_{\alpha_1}$ are known and that eigenvalues of $H$ forming clusters of three or more at a threshold $q = N^{-1 - \kappa}$ do not produce gaps of order $d_2$. The gap counts are supplied by an induction on the scale as in \zcref{s:pairs}, which reaches the exponent $\alpha_1 = 8/25$ that we use.

We do not use the resampling argument, which would not be available in the same form. Heuristically, a resampling step moves the traceless part of the $2 \times 2$ compression of the matrix to the span of $u_i$ and $u_{i + 1}$ (described by the map $L_i$ of \eqref{e:reduction-Li}, with a third coordinate in the complex Hermitian case) by $O(N^{-3/2}) = o(d_2)$, so order $N^{1/3}$ steps would be needed to move it by $d_2$, which is beyond the range of the crude entrywise bound \eqref{e:ov-W}.

The main difference with \zcref{s:pairs} concerns clusters of three eigenvalues. In \zcref{s:pairs}, we used the bound of Nguyen, Tao, and Vu on three consecutive eigenvalues, \zcref{p:three}. In the complex Hermitian case, the expected number of triples of span $N^{-1 - \kappa}$ for the GUE is of order $N (N^{-\kappa})^8$, and triples become absent when $\kappa > 1/8$. However, transferring information on the scale $N^{-1 - \kappa}$ from the GUE to $H$ through the coupling of \zcref{p:coupling} and the comparison of \zcref{p:comparison} requires a time $T$ with
\begin{equation*}
N^{-1 + \kappa} \ll T \ll N^{-1/2 - 3\kappa}.
\end{equation*}
The lower bound makes the coupling error $N^{\eps}/(N^2 T)$ smaller than $N^{-1 - \kappa}$, and the upper bound makes the comparison error, which is of order $T N^{1/2 + 3\kappa}$, small. Such a time exists only for $\kappa < 1/8$, so triples cannot be excluded in this way.

We therefore use relative bounds. For $\kappa < 1/8$, the number of triples of span at most $4 N^{-1 - \kappa}$ in the bulk is at most $N^{1 - 8\kappa + \eps}$ with probability tending to one, by \zcref{p:herm-triples}. In the induction step, this is negligible compared to $M^{\C}_{\alpha'} = N^{1 - 3\alpha'}$ when $8\kappa > 3\alpha'$. In the final step, \zcref{l:herm-3x3} shows that each triple produces a gap of order $d_2$ with probability bounded by $N^{C \eps} (d_2/\ell_{\alpha_1})^3 = N^{-1 + 3\alpha_1 + C\eps}$, and the expected number of such gaps is at most $N^{3\alpha_1 - 8\kappa + C\eps}$, which tends to zero when $8\kappa > 3\alpha_1$. Since $\alpha_1 < 1/3$, this is compatible with $\kappa < 1/8$.

\section{Inputs}\label{s:herm-inputs}

\paragraph{Regularity and coupling.} In this section, $\mathcal G_\eps$ denotes the set of complex Hermitian matrices with the three properties of \zcref{d:regular}, where unit eigenvectors are taken in $\C^N$. The estimates of \zcref{p:regularity} and \zcref{l:prelim-rigidity} hold for $H_t$ and $G$ with the same proofs, since \cite[Definition~2.2]{benaychgeorges2016lectures} concerns complex Hermitian matrices and \cite[Theorem~2.2]{erdos2012rigidity} covers complex Hermitian generalized Wigner matrices. These proofs use the entries only through their independence, their centering, their sub-Gaussian tails, and their variances, which are now $\E |(H_t)_{ab}|^2 = 1/N$ for $a \neq b$ and $\E (H_t)_{aa}^2 = (e^{-t} \E Z^2 + 1 - e^{-t})/N$.

The coupling of \zcref{p:coupling} holds with GUE in place of GOE. In this case, $\mathsf B$ in \eqref{e:prelim-ou} is a complex Hermitian matrix of Brownian motions whose diagonal entries are standard real Brownian motions and whose entries above the diagonal are standard complex Brownian motions with $\E |\mathsf B_{ab}(t)|^2 = t$, all independent, and the Brownian motions in \eqref{e:prelim-dbm} satisfy $\E \beta_i(t)^2 = t$. This is the dynamics with symmetry parameter $2$ in \cite[(2.2)]{bourgade2022extreme}, whose equilibrium measure is the GUE, and \cite[Theorem~3.1 and Corollary~3.3]{bourgade2022extreme} are stated for this dynamics as well; by \cite[Remark~3.2]{bourgade2022extreme}, they require only optimal rigidity estimates for the initial conditions. The verification of these estimates in the proof of \zcref{p:coupling} applies without change, since it uses only \zcref{l:prelim-rigidity} and \zcref{p:regularity}.

One step requires a modification. In the proof of \zcref{p:coupling}, the dynamics is started from $H$, whose spectrum is simple with overwhelming probability by \cite[Corollary~2.2]{nguyen2017random}. The extension of this result to Hermitian matrices stated in \cite[Section~2]{nguyen2017random} assumes that the real and imaginary parts of the entries are independent and identically distributed, which we do not assume. We therefore start the dynamics from $H_{t_*}$ with $t_* = N^{-10}$. Conditionally on $H$, the matrix $H_{t_*}$ has a density with respect to Lebesgue measure on the space of Hermitian matrices, and the Hermitian matrices with a multiple eigenvalue form a Lebesgue null set (the zero set of the discriminant of the characteristic polynomial), so the spectrum of $H_{t_*}$ is simple almost surely.

Moreover, for $t' \ge 0$, we have
\begin{equation*}
H_{t_* + t'} \stackrel{d}{=} e^{-t'/2} H_{t_*} + (1 - e^{-t'})^{1/2} \widetilde G,
\end{equation*}
where $\widetilde G$ is a GUE matrix independent of $H_{t_*}$, and the rigidity estimates for these matrices follow from \zcref{l:prelim-rigidity}, whose proof applies for all times in $[0, 2]$. Applying \cite[Corollary~3.3]{bourgade2022extreme} with the initial condition $\lambda(H_{t_*})$ at time $t - t_*$, and using $t - t_* \ge t/2$, we obtain \eqref{e:prelim-coupling} with $2 N^{\eps}/(N^2 t)$ in place of $N^\eps/(N^2 t)$, for all $t$ with $N^{-1 + \eps} \le t < 1$, and the spectra of $H_t$ and of a GUE matrix can be coupled accordingly.

\paragraph{Comparison.} The definition of $\varphi_N$, the strip estimate of \zcref{l:strip}, the holomorphic extension of \zcref{d:complexS} and \zcref{l:holomorphic}, and the derivative bound of \zcref{l:globalderivative} carry over verbatim to complex Hermitian matrices, since their proofs use only the spectral decomposition of a Hermitian matrix, Sylvester's determinant identity, and the properties of $\mathcal G_\eps$. For $a < b$, we let $\widetilde V^{ab} = \iu (e_a e_b^T - e_b e_a^T)$, so that the entry $(H_t)_{ab}$ contributes $\Re (H_t)_{ab} V^{ab} + \Im (H_t)_{ab} \widetilde V^{ab}$ to $H_t$. For complex $\zeta_1$ and $\zeta_2$, we have
\begin{equation*}
\zeta_1 V^{ab} + \zeta_2 \widetilde V^{ab} = (\zeta_1 + \iu \zeta_2) e_a e_b^T + (\zeta_1 - \iu \zeta_2) e_b e_a^T,
\end{equation*}
and the proof of \zcref{l:strip}, which uses only the factorization of a perturbation supported on the rows and columns $a$ and $b$ through Sylvester's identity, applies verbatim to perturbations $\zeta e_a e_b^T + \zeta' e_b e_a^T$ with $|\zeta|, |\zeta'| \le 8 c_p N^{-\alpha - 4\eps}$.

In \zcref{l:comparison-generator}, the sum runs over the diagonal entries, with $v^G_{aa} = 1/N$, and over the real and imaginary parts of the entries above the diagonal, with the directions $V^{ab}$ and $\widetilde V^{ab}$ and with $v^G = 1/(2N)$ for each of them; its proof uses only Gaussian integration by parts in the independent real coordinates of $G$.

The real and imaginary parts of $X_{\C}$ need not be independent, so in Steps 2 and 3 of the proof of \zcref{p:comparison}, we treat the two terms of each entry above the diagonal together. For $t' \in (0, 1]$ and $a < b$, let $H_{t'}^{[ab]}$ denote the matrix obtained from $H_{t'}$ by setting its $(a, b)$ and $(b, a)$ entries to zero. This matrix is independent of $(\Re (H_{t'})_{ab}, \Im (H_{t'})_{ab})$. Let $\mathcal E_{ab}(t')$ be the event that $\Spec(H_{t'}^{[ab]} + \zeta_1 V^{ab} + \zeta_2 \widetilde V^{ab}) \subset \mathfrak R_{\alpha, p}$ for all complex $\zeta_1, \zeta_2$ with $|\zeta_1|, |\zeta_2| \le 2 r_N$. As in Step 2 of the proof of \zcref{p:comparison}, this event is determined by $H_{t'}^{[ab]}$. By the preceding paragraphs, it contains $\{ H_{t'} \in \mathcal G_\eps \} \cap \{ |(H_{t'})_{ab}| \le r_N \}$, since the corresponding perturbation of $H_{t'}$ has coefficients of modulus at most $6 r_N \le 8 c_p N^{-\alpha - 4\eps}$. (Indeed, $H_{t'}^{[ab]} + \zeta_1 V^{ab} + \zeta_2 \widetilde V^{ab}$ is obtained from $H_{t'}$ by adding $\zeta_1 - \Re (H_{t'})_{ab}$ times $V^{ab}$ and $\zeta_2 - \Im (H_{t'})_{ab}$ times $\widetilde V^{ab}$, and both coefficients have modulus at most $3 r_N$.) Hence the complement of $\mathcal E_{ab}(t')$ has probability at most $C_D N^{-D}$, as in \eqref{e:comparison-eventprob}.

On $\mathcal E_{ab}(t')$, the function $(\zeta_1, \zeta_2) \mapsto F(H_{t'}^{[ab]} + \zeta_1 V^{ab} + \zeta_2 \widetilde V^{ab})$ is holomorphic and bounded by one in modulus on the closed polydisk of radius $2 r_N$, by \zcref{l:holomorphic}. Cauchy's estimates on the polydisks of radius $r_N$ centered at the points of $[-r_N, r_N]^2$ bound its partial derivatives of order $n$ on $[-r_N, r_N]^2$ by $C_n r_N^{-n}$. We now Taylor expand jointly in the two real variables. We write $\mathbf x = (\Re (H_{t'})_{ab}, \Im (H_{t'})_{ab})$ and
\begin{equation*}
\Upsilon(x_1, x_2) = F\big( H_{t'}^{[ab]} + x_1 V^{ab} + x_2 \widetilde V^{ab} \big), \qquad (x_1, x_2) \in \R^2.
\end{equation*}
The contribution of the entry $(a, b)$ to the integrand in \eqref{e:comparison-integrated} is then $\E[(2N)^{-1} (\partial_1^2 + \partial_2^2) \Upsilon(\mathbf x) - \mathbf x \cdot \nabla \Upsilon(\mathbf x)]$, where $\nabla$ denotes the gradient in $(x_1, x_2)$, and $\partial_1$ and $\partial_2$ denote the partial derivatives in $x_1$ and $x_2$. The vector $(\Re X_{\C}, \Im X_{\C})$ is centered with covariance $2^{-1} \Id_2$, since $\E X_{\C}^2 = 0$ and $\E |X_{\C}|^2 = 1$, and this agrees with the covariance of $N^{1/2} (\Re G_{ab}, \Im G_{ab})$. Hence $\E \mathbf x = 0$ and $\E \mathbf x \mathbf x^T = (2N)^{-1} \Id_2$.

We expand $\nabla \Upsilon$ to first order and $(\partial_1^2 + \partial_2^2) \Upsilon$ to zeroth order around the origin, on the event $\{ |\mathbf x| \le r_N \}$, as in the proof of \zcref{l:taylor}. By the last two identities, the terms of order zero and one cancel, and we obtain, on $\mathcal E_{ab}(t')$,
\begin{equation*}
\Big| \E\Big[ \frac{(\partial_1^2 + \partial_2^2) \Upsilon(\mathbf x)}{2N} - \mathbf x \cdot \nabla \Upsilon(\mathbf x) \, \Big| \, H_{t'}^{[ab]} \Big] \Big| \le C r_N^{-3} \E |\mathbf x|^3 + C_D N^{-D} \le C N^{-3/2} r_N^{-3} + C_D N^{-D}.
\end{equation*}
Here the term $C_D N^{-D}$ bounds the contribution of the event $\{ |\mathbf x| > r_N \}$, which we control with the crude bound of \zcref{l:globalderivative} as in Step 3 of the proof of \zcref{p:comparison}; the complement of $\mathcal E_{ab}(t')$ is treated in the same way. The diagonal entries are treated as in the proof of \zcref{p:comparison}, and the variance mismatch $|\E (H_{t'})_{aa}^2 - 1/N| \le C N^{-1}$ gives the term $r_N^{-2}$. Summing over the coordinates as in Step 4 of that proof, we obtain the analogue of \eqref{e:comparison-bound} for complex Hermitian matrices,
\begin{equation}\label{e:herm-comp-pair}
| \E F(H_t) - \E F(H) | \le C t \big( N^{1/2} r_N^{-3} + r_N^{-2} \big) + C_D N^{-D} \le C t N^{1/2 + 3\alpha + 12\eps} + C_D N^{-D}
\end{equation}
for $F$ as in \zcref{d:complexS}, $\alpha + 5\eps < 1/2$, and $r_N = c_p N^{-\alpha - 4\eps}$.

\paragraph{Triple and quadruple statistics.} The comparison estimate also applies to the statistics \eqref{e:herm-U}. We first write them as traces. Let $\mathsf s_3(x_1, x_2, x_3)$ and $\mathsf s_4(x_1, x_2, x_3, x_4)$ denote the symmetric functions such that the summands of $U^{(3)}_q$ and $U^{(4)}_q$ equal $\mathsf s_3(\lambda_i, \lambda_j, \lambda_k)$ and $\mathsf s_4(\lambda_i, \lambda_j, \lambda_k, \lambda_l)$, where $\lambda_i = \lambda_i(A)$. Summing over all tuples of indices and removing those with coincident indices by inclusion--exclusion over the partitions of the set of positions, we find
\begin{equation}\label{e:herm-incl-excl}
U^{(3)}_q(A) = \frac{1}{6} \Big( \sum_{i, j, k} \mathsf s_3(\lambda_i, \lambda_j, \lambda_k) - 3 \sum_{i, j} \mathsf s_3(\lambda_i, \lambda_i, \lambda_j) + 2 \sum_i \mathsf s_3(\lambda_i, \lambda_i, \lambda_i) \Big)
\end{equation}
and
\begin{equation}\label{e:herm-incl-excl-four}
\begin{aligned}
24 \, U^{(4)}_q(A) = {} & \sum_{i, j, k, l} \mathsf s_4(\lambda_i, \lambda_j, \lambda_k, \lambda_l) - 6 \sum_{i, j, k} \mathsf s_4(\lambda_i, \lambda_i, \lambda_j, \lambda_k) + 3 \sum_{i, j} \mathsf s_4(\lambda_i, \lambda_i, \lambda_j, \lambda_j) \\
& + 8 \sum_{i, j} \mathsf s_4(\lambda_i, \lambda_i, \lambda_i, \lambda_j) - 6 \sum_i \mathsf s_4(\lambda_i, \lambda_i, \lambda_i, \lambda_i).
\end{aligned}
\end{equation}
The coefficients are the values $\prod_{B} (-1)^{|B| - 1} (|B| - 1)!$ of the M\"obius function of the lattice of set partitions, where $B$ runs over the blocks, multiplied by the number of partitions of each type.

Each sum on the right sides of \eqref{e:herm-incl-excl} and \eqref{e:herm-incl-excl-four} is the trace of a function of commuting matrices obtained from $A$ by tensoring with identity matrices, exactly as in \zcref{d:complexS}. For example, the first sum in \eqref{e:herm-incl-excl} is the trace over $(\C^N)^{\otimes 3}$ of
\begin{equation*}
\chi_N\Big( \frac{A_1 + A_2 + A_3}{3} \Big) \prod_{1 \le i < j \le 3} \big( \Id + q^{-2} (A_i - A_j)^2 \big)^{-p},
\end{equation*}
where $A_1 = A \otimes \Id \otimes \Id$, $A_2 = \Id \otimes A \otimes \Id$, and $A_3 = \Id \otimes \Id \otimes A$. As in the proof of \zcref{l:holomorphic}, these traces are holomorphic functions of the entries of $A$ on the set of matrices whose eigenvalues $w$ satisfy $|\Im w| < \omega_p q$, and the identities \eqref{e:herm-incl-excl} and \eqref{e:herm-incl-excl-four} hold there with $\lambda_1, \dots, \lambda_N$ replaced by the eigenvalues of $A$, listed with algebraic multiplicity.

Next, we bound the arguments of the summands. Suppose now that $\Spec A$ lies in the rectangle $\mathfrak R_{\kappa, p}$ of \eqref{e:comparison-rectangle}, where $q = \ell_\kappa = N^{-1 - \kappa}$. The proof of \zcref{l:holomorphic} shows that every rational factor in \eqref{e:herm-U} has argument of absolute value at most $1/375$, and that the weight $\chi_N$ at the mean of the eigenvalues is either zero or has argument of absolute value at most $1/200$; the mean of three or four eigenvalues satisfies the same bounds on its real and imaginary parts as the mean of two. Hence every summand of $U^{(3)}_q(A)$ and $U^{(4)}_q(A)$ is zero or has argument of absolute value at most $1/200 + 6/375 < 1/40$, so $\Re U^{(r)}_q(A) \ge 0$ and $|\exp(- \theta U^{(r)}_q(A)/M)| \le 1$ for all $\theta, M > 0$ and $r \in \{ 3, 4 \}$. The proof of \zcref{l:globalderivative} applies to these observables with $\ell_\alpha$ replaced by $q$. Thus the observables $\exp(-\theta U^{(r)}_q/M)$ have the two properties listed in \zcref{r:comparison-inputs}, with the polydisk above in place of the disk. Hence the proof of \zcref{p:comparison}, with the modifications described in the paragraph on comparison above, gives, for $q = N^{-1 - \kappa}$ with $\kappa \in (0, 1/2)$, all $M > 0$, $\theta > 0$, $D > 0$, $t \in [0, 1]$, and $\eps > 0$ with $\kappa + 5\eps < 1/2$,
\begin{equation}\label{e:herm-comp-triple}
\Big| \E e^{-\theta U^{(r)}_q(H_t)/M} - \E e^{-\theta U^{(r)}_q(H)/M} \Big| \le C t N^{1/2 + 3\kappa + 12\eps} + C_D N^{-D}, \qquad r \in \{ 3, 4 \}.
\end{equation}

\paragraph{The Gaussian unitary ensemble.} The following estimates replace \zcref{p:goepairs}.

\begin{lemma}\label{l:herm-gue}
The following statements hold.
\begin{enumerate}[label=(\alph*)]
\item Let $\alpha \in (0, 1/4)$ and $p > 3/2$. Let $f \colon [0, \infty) \times \R \to \R$ be a continuous function for which there exist a bulk interval $I$ and a constant $C_f > 0$ such that $f(r, E) = 0$ for $E \notin I$ and $|f(r, E)| \le C_f (1 + r^2)^{-p}$ for all $(r, E)$. Then, in $L^1$,
\begin{equation}\label{e:herm-gue}
\frac{1}{M^{\C}_\alpha} \sum_{1 \le i < j \le N} f\Big( \frac{\lambda_j(G) - \lambda_i(G)}{\ell_\alpha}, \frac{\lambda_i(G) + \lambda_j(G)}{2} \Big) \longrightarrow \int_{-2}^2 \int_0^\infty c_2(E) \, r^2 f(r, E) \dd r \dd E .
\end{equation}
\item Let $\kappa > 0$, set $q = N^{-1 - \kappa}$, and let $p \ge 4$ be an integer. Then
\begin{equation}\label{e:herm-gue-triples}
\E U^{(3)}_q(G) \le C N (N q)^8 + o(1), \qquad \E U^{(4)}_q(G) \le C N (N q)^{15} + o(1).
\end{equation}
\end{enumerate}
\end{lemma}

\begin{proof}[Sketch of proof]
We use the kernel $\mathsf K_N$ of \zcref{d:goe-kernel} and the estimates for it proved in \zcref{s:goe}. Recall that the eigenvalues of $G$ form a determinantal point process with kernel $\mathsf K_N$. Thus their $n$-point correlation functions, defined as in \eqref{e:goe-corr-def} with the joint density of the eigenvalues of the GUE, are given by
\begin{equation}\label{e:herm-det}
R_{n, N}(x_1, \dots, x_n) = \det[\mathsf K_N(x_i, x_j)]_{i, j = 1}^n, \qquad n \in [N],
\end{equation}
(see \cite[Section~3.2]{anderson2010introduction}), and \eqref{e:goe-corr-identity} holds.

The proof follows \zcref{s:goe-proof}. The determinantal structure makes the correlation estimates simpler than for the GOE, and we give them below; we omit the routine verification of the dominated convergence arguments.

Throughout the proof, $\widetilde I$ denotes a bulk interval, constants may depend on $\widetilde I$, and we use \zcref{l:goe-gue} and \zcref{l:goe-density} for $x, y \in \widetilde I$. We first record two estimates for the correlation functions. We set
\begin{equation*}
\mathbf h(x) = \big( N^{1/4} \mathsf h_j(\sqrt N x) \big)_{j = 0}^{N - 1} \in \R^N, \qquad \text{so that} \qquad \mathsf K_N(x, y) = \mathbf h(x) \cdot \mathbf h(y).
\end{equation*}

\emph{The two-point function.} We begin with the expansion of $R_{2, N}$ near the diagonal. Since $\mathsf K_N$ is symmetric, \eqref{e:herm-det} with $n = 2$ gives
\begin{equation*}
R_{2, N}(x, y) = \mathsf K_N(x, x) \mathsf K_N(y, y) - \mathsf K_N(x, y)^2.
\end{equation*}
Using the symmetry of $\mathsf K_N$ again, we find
\begin{equation*}
\begin{gathered}
R_{2, N}(x, x) = \partial_y R_{2, N}(x, y) \big|_{y = x} = 0, \\
\partial_y^2 R_{2, N}(x, y) \big|_{y = x} = \mathsf K_N(x, x) \frac{\dd^2}{\dd x^2} \mathsf K_N(x, x) - 2 \big( \partial_2 \mathsf K_N(x, x) \big)^2 - 2 \mathsf K_N(x, x) \, \partial_y^2 \mathsf K_N(x, y) \big|_{y = x}.
\end{gathered}
\end{equation*}

We now bound the first two terms. By the proof of \zcref{l:goe-gue}(c), we have
\begin{equation*}
\frac{\dd}{\dd x} \mathsf K_N(x, x) = 2 \partial_2 \mathsf K_N(x, x) = - N^{3/2} \mathsf h_{N - 1}(\sqrt N x) \mathsf h_N(\sqrt N x).
\end{equation*}
Differentiating once more and using \eqref{e:goe-hermite-N} and $\mathsf K_N(x, x) \le C N$, we obtain
\begin{equation*}
\mathsf K_N(x, x) \frac{\dd^2}{\dd x^2} \mathsf K_N(x, x) = O(N^3), \qquad \big( \partial_2 \mathsf K_N(x, x) \big)^2 = O(N^2),
\end{equation*}
so the first two terms on the right side of the expression for $\partial_y^2 R_{2, N}(x, y) |_{y = x}$ are $O(N^3)$.

For the last term, \eqref{e:goe-K-second}, \eqref{e:goe-hermite-N}, \zcref{l:goe-density}, and the identity $1 - x^2/4 = \pi^2 \rhosc(x)^2$ give
\begin{equation*}
\begin{aligned}
- 2 \mathsf K_N(x, x) \, \partial_y^2 \mathsf K_N(x, y) \big|_{y = x} &= \frac{2\pi^2}{3} N^2 \rhosc(x)^2 \mathsf K_N(x, x)^2 + O(N^3) \\
&= \frac{2\pi^2}{3} N^4 \rhosc(x)^4 + O(N^{31/8}).
\end{aligned}
\end{equation*}
Further, \zcref{l:goe-gue}(a) gives
\begin{equation*}
\big| \partial_y^3 R_{2, N}(x, y) \big| \le C N^5, \qquad x, y \in \widetilde I.
\end{equation*}
Taylor's theorem in $y$ at $y = x$ therefore gives
\begin{equation}\label{e:herm-two}
\Big| R_{2, N}(x, x + h) - \frac{\pi^2}{3} N^4 \rhosc(x)^4 h^2 \Big| \le C \big( N^{31/8} h^2 + N^5 h^3 \big), \qquad x, x + h \in \widetilde I, \quad 0 < h \le N^{-1}.
\end{equation}
This is the source of the function $c_2(E)$. For larger separations, we use the bounds
\begin{equation}\label{e:herm-far}
R_{2, N}(x, y) \le \mathsf K_N(x, x) \mathsf K_N(y, y) \le C N^2, \qquad |\mathsf K_N(x, y)| \le C \min\{ N, |x - y|^{-1} \},
\end{equation}
which follow from \eqref{e:herm-det} and \zcref{l:goe-gue}(a) and (b).

\emph{Correlation functions of nearby points.} We claim that for $n \in \{ 2, 3, 4 \}$ and $x_1, \dots, x_n \in \widetilde I$,
\begin{equation}\label{e:herm-Rn}
R_{n, N}(x_1, \dots, x_n) \le C N^n \min\Big\{ 1, \prod_{1 \le i < j \le n} \big( N |x_i - x_j| \big)^2 \Big\}.
\end{equation}
The matrix $[\mathsf K_N(x_i, x_j)]_{i, j = 1}^n$ is the Gram matrix of the vectors $\mathbf h(x_1), \dots, \mathbf h(x_n)$. Hadamard's inequality bounds its determinant by its diagonal entries, which gives
\begin{equation*}
R_{n, N}(x_1, \dots, x_n) \le \prod_{i = 1}^n \mathsf K_N(x_i, x_i) \le C N^n.
\end{equation*}

To prove the second bound in \eqref{e:herm-Rn}, we may assume that the points are distinct. We replace $\mathbf h(x_1), \dots, \mathbf h(x_n)$ by the divided differences $\mathbf h[x_1], \mathbf h[x_1, x_2], \dots, \mathbf h[x_1, \dots, x_n]$. This is a triangular change of basis with determinant $\prod_{i < j} (x_j - x_i)^{-1}$, so
\begin{equation*}
R_{n, N}(x_1, \dots, x_n) = \prod_{1 \le i < j \le n} (x_j - x_i)^2 \det\big[ \mathbf h[x_1, \dots, x_i] \cdot \mathbf h[x_1, \dots, x_j] \big]_{i, j = 1}^n.
\end{equation*}

We now bound the divided differences. By the Hermite--Genocchi formula, $\mathbf h[x_1, \dots, x_k]$ is the integral of $\mathbf h^{(k - 1)}$ over a simplex of volume $1/(k - 1)!$ mapped into the convex hull of $x_1, \dots, x_k$. Hence the Cauchy--Schwarz inequality and the bound \eqref{e:goe-hermite-sums} in the proof of \zcref{l:goe-gue}, with $k - 1$ in place of $k$, give
\begin{equation*}
\begin{aligned}
|\mathbf h[x_1, \dots, x_k]|^2 &\le \sup_{y \in \widetilde I} |\mathbf h^{(k - 1)}(y)|^2 \\
&= \sup_{y \in \widetilde I} N^{k - 1/2} \sum_{j = 0}^{N - 1} \mathsf h_j^{(k - 1)}(\sqrt N y)^2 \le C N^{2k - 1}, \qquad k \le 4.
\end{aligned}
\end{equation*}

Applying Hadamard's inequality to the Gram determinant of the divided differences, we obtain
\begin{equation*}
\begin{aligned}
R_{n, N}(x_1, \dots, x_n) &\le \prod_{i < j} (x_j - x_i)^2 \prod_{k = 1}^n |\mathbf h[x_1, \dots, x_k]|^2 \\
&\le C \prod_{i < j} (x_j - x_i)^2 \prod_{k = 1}^n N^{2k - 1} = C N^{n^2} \prod_{i < j} |x_i - x_j|^2,
\end{aligned}
\end{equation*}
which is the second bound in \eqref{e:herm-Rn}, since $n + n(n - 1) = n^2$.

\emph{Proof of (a).} We first compute the mean. By \eqref{e:goe-corr-identity}, we have
\begin{equation*}
\E \sum_{1 \le i < j \le N} f\Big( \frac{\lambda_j(G) - \lambda_i(G)}{\ell_\alpha}, \frac{\lambda_i(G) + \lambda_j(G)}{2} \Big) = \int_\R \int_0^\infty f\Big( \frac{h}{\ell_\alpha}, x + \frac{h}{2} \Big) R_{2, N}(x, x + h) \dd h \dd x .
\end{equation*}

Let $\delta > 0$ be small. Since $|f(r, E)| \le C_f r^{-2p}$ and the integral of $R_{2, N}$ over $\R^2$ is $N(N - 1)$, the pairs at distance larger than $\delta$ contribute at most
\begin{equation*}
C_f \Big( \frac{\ell_\alpha}{\delta} \Big)^{2p} \int_\R \int_0^\infty R_{2, N}(x, x + h) \dd h \dd x \le C N^2 \Big( \frac{\ell_\alpha}{\delta} \Big)^{2p} = o(1).
\end{equation*}
For the remaining pairs, both points lie in a bulk interval $\widetilde I$, since $f(r, E) = 0$ for $E \notin I$.

We insert \eqref{e:herm-two} for $h \le N^{-1}$ and change variables $h = \ell_\alpha r$. We will use the identities
\begin{equation*}
N^4 \ell_\alpha^3 = M^{\C}_\alpha, \qquad N^{31/8} \ell_\alpha^3 = N^{-1/8} M^{\C}_\alpha, \qquad N^5 \ell_\alpha^4 = N^{-\alpha} M^{\C}_\alpha.
\end{equation*}
By the first identity, the main term is
\begin{equation*}
M^{\C}_\alpha \int_\R \int_0^{N^\alpha} f\Big( r, x + \frac{\ell_\alpha r}{2} \Big) \frac{\pi^2}{3} \rhosc(x)^4 \, r^2 \dd r \dd x,
\end{equation*}
and since $p > 3/2$, it converges to $M^{\C}_\alpha$ times the right side of \eqref{e:herm-gue} by dominated convergence. Using the other two identities, we find that the error terms in \eqref{e:herm-two} contribute at most
\begin{equation*}
C M^{\C}_\alpha \Big( N^{-1/8} + N^{-\alpha} \int_0^{N^\alpha} \frac{r^3 \dd r}{(1 + r^2)^p} \Big) \le C M^{\C}_\alpha \big( N^{-1/8} + N^{-\alpha} + N^{-\alpha(2p - 3)} \log N \big).
\end{equation*}

Finally, by \eqref{e:herm-far}, the pairs with $N^{-1} < h \le \delta$ contribute at most
\begin{equation*}
C N^2 \int_{N^{-1}}^\delta \Big( \frac{\ell_\alpha}{h} \Big)^{2p} \dd h \le C N^2 \ell_\alpha \int_{N^{\alpha}}^\infty r^{-2p} \dd r = C M^{\C}_\alpha N^{-\alpha(2p - 3)}.
\end{equation*}
Hence the mean of the sum truncated to $|\lambda_j(G) - \lambda_i(G)| \le \delta$, divided by $M^{\C}_\alpha$, converges to the right side of \eqref{e:herm-gue}.

We now bound the variance of the truncated sum. We set
\begin{equation*}
f^\sharp(x, y) = f\Big( \frac{|y - x|}{\ell_\alpha}, \frac{x + y}{2} \Big) \1_{|y - x| \le \delta}, \qquad x, y \in \R.
\end{equation*}
As in \eqref{e:goe-var-expansion}, with $f^\sharp$ in place of $g^\sharp$, the variance is the sum of three integrals, against $R_{4, N} - R_{2, N} \otimes R_{2, N}$, against $R_{3, N}$, and against $R_{2, N}$. In the expansion of the determinant $R_{4, N}(x_1, x_2, x_3, x_4)$, the terms that do not appear in $R_{2, N}(x_1, x_2) R_{2, N}(x_3, x_4)$ come from permutations that do not preserve $\{ 1, 2 \}$, so they contain at least two factors $\mathsf K_N(x_i, x_j)$ connecting $\{ 1, 2 \}$ and $\{ 3, 4 \}$, in either direction. Hence, by the symmetry of $\mathsf K_N$ and \eqref{e:herm-far}, they are bounded by $C \min\{ N^4, N^2 \dist(\{ x_1, x_2 \}, \{ x_3, x_4 \})^{-2} \}$.

To integrate this bound, we write $x_2 = x_1 + h_1$ and $x_4 = x_3 + h_2$ and integrate first over $x_3$. Where $|x_3 - x_1| \ge 2(|h_1| + |h_2|)$, the distance between $\{ x_1, x_2 \}$ and $\{ x_3, x_4 \}$ is at least $|x_3 - x_1|/2$, and the remaining values of $x_3$ form a set of measure at most $4(|h_1| + |h_2|)$. Hence the integral of the last bound over $x_3 \in \R$ is at most
\begin{equation*}
C \int_\R \min\Big\{ N^4, \frac{4 N^2}{|x_3 - x_1|^2} \Big\} \dd x_3 + 4 C N^4 (|h_1| + |h_2|) \le C N^3 + C N^4 (|h_1| + |h_2|).
\end{equation*}

We now integrate over $h_1$ and $h_2$. Since $p > 3/2$, we have
\begin{equation*}
\int_\R |f^\sharp(x_1, x_1 + h)| \, (1 + N |h|) \dd h \le C \ell_\alpha (1 + N \ell_\alpha) \le C \ell_\alpha,
\end{equation*}
and we find that the first integral is bounded by
\begin{equation*}
C \ell_\alpha^2 \big( N^3 + N^4 \ell_\alpha \big) \le C N^3 \ell_\alpha^2 = C N^{1 - 2\alpha}.
\end{equation*}
By Fischer's inequality, we have
\begin{equation*}
R_{3, N}(x_1, x_2, x_3) \le \mathsf K_N(x_3, x_3) R_{2, N}(x_1, x_2) \le C N R_{2, N}(x_1, x_2),
\end{equation*}
so the second integral is bounded by $C N \ell_\alpha M^{\C}_\alpha$, and the third by $C M^{\C}_\alpha$, by the computation of the mean applied to $|f|$. Collecting the three bounds, we find that the variance divided by $(M^{\C}_\alpha)^2$ is at most
\begin{equation*}
C \, \frac{N^{1 - 2\alpha} + N \ell_\alpha M^{\C}_\alpha + M^{\C}_\alpha}{(M^{\C}_\alpha)^2} = C \big( N^{4\alpha - 1} + N^{2\alpha - 1} + N^{3\alpha - 1} \big),
\end{equation*}
which tends to zero since $\alpha < 1/4$. Hence the normalized sum converges in $L^2$ after truncation, and hence in $L^1$.

\emph{Proof of (b).} Let $\delta > 0$ be small. We first discard two kinds of tuples. The tuples containing two points at distance larger than $\delta$, and the tuples whose mean is at distance at least $\delta$ from $\supp \chi$, contribute at most
\begin{equation*}
\| \chi \|_\infty N^4 \Big( \frac{q}{\delta} \Big)^{2p} = o(1) \qquad \text{and} \qquad C N^4 \exp\Big( -\frac{\delta^2 N^{1/5}}{2} \Big) = o(1),
\end{equation*}
respectively. For the first bound, we used that the rational factor of the two distant points is at most $(q/\delta)^{2p}$, and for the second, part (4) of \zcref{l:smoothing}. For the remaining tuples, all points lie in a bulk interval $\widetilde I$.

We now bound the remaining tuples. Using \eqref{e:goe-corr-identity}, the second bound in \eqref{e:herm-Rn}, and the substitution $x_i = x_1 + q a_i$ for $i \ge 2$, we find
\begin{equation*}
\begin{aligned}
\E U^{(3)}_q(G) &\le C N^9 q^8 \int_{\R^2} (a_2 a_3 (a_3 - a_2))^2 f_p(a_2) f_p(a_3) f_p(a_3 - a_2) \dd a_2 \dd a_3 + o(1) \\
&\le C N (N q)^8 + o(1),
\end{aligned}
\end{equation*}
where the integral is finite since $p \ge 3$ and
\begin{equation*}
(a_2 a_3 (a_3 - a_2))^2 \le 2 (a_2^4 a_3^2 + a_2^2 a_3^4).
\end{equation*}
Similarly, with $a_1 = 0$, we find
\begin{equation*}
\E U^{(4)}_q(G) \le C N^{16} q^{15} \int_{\R^3} \prod_{1 \le i < j \le 4} (a_i - a_j)^2 f_p(a_i - a_j) \dd a_2 \dd a_3 \dd a_4 + o(1).
\end{equation*}
To bound the integral, we bound the three factors with $2 \le i < j \le 4$ by one, using $y^2 f_p(y) \le 1$ for $y \in \R$. The other three factors are $a_j^2 f_p(a_j)$ for $j \in \{ 2, 3, 4 \}$, so the integral is at most $\big( \int_\R a^2 f_p(a) \dd a \big)^3$, which is finite since $2p - 2 > 1$. As $N^{16} q^{15} = N (N q)^{15}$, this gives the bound for $U^{(4)}_q$.
\end{proof}

\section{Triples of nearby eigenvalues}\label{s:herm-triples}

\begin{proposition}\label{p:herm-triples}
Let $I$ be a bulk interval and $\kappa \in (0, 1/8)$, and set $q = N^{-1 - \kappa}$. Then for every $\eps > 0$,
\begin{equation}\label{e:herm-triples}
\lim_{N \to \infty} \Prob\Big( \# \{ i \in [N - 2] : \lambda_i \in I, \ \lambda_{i + 2} - \lambda_i \le 4q \} > N^{1 - 8\kappa + \eps} \Big) = 0 .
\end{equation}
If moreover $\kappa > 1/15$, then with probability tending to one there exists no $i \in [N - 3]$ such that $\lambda_i \in I$ and $\lambda_{i + 3} - \lambda_i \le 6q$.
\end{proposition}

\begin{proof}
We first reduce both claims to bounds on the statistics $U^{(3)}_q$ and $U^{(4)}_q$ of \eqref{e:herm-U}. Let $I_+$ be a bulk interval whose interior contains $I$. Fix a nonnegative $\chi \in C_c^\infty((-2, 2))$ with $\chi \ge 2$ on $I_+$, so that $\chi_N \ge 1$ on $I_+$ for $N \ge N_0$ by part (5) of \zcref{l:smoothing}, and fix an integer $p \ge 4$.

Let $i$ be a label with $\lambda_i \in I$ and $\lambda_{i + 2} - \lambda_i \le 4q$. Then the mean of $\lambda_i$, $\lambda_{i + 1}$, and $\lambda_{i + 2}$ belongs to $I_+$ for $N \ge N_0$, so the term of $(i, i + 1, i + 2)$ in $U^{(3)}_q(H)$ is at least $17^{-3p}$ by the observation after \eqref{e:herm-U}. Distinct labels $i$ give distinct triples, and all terms of $U^{(3)}_q(H)$ are nonnegative, so
\begin{equation*}
\# \{ i \in [N - 2] : \lambda_i \in I, \ \lambda_{i + 2} - \lambda_i \le 4q \} \le 17^{3p} U^{(3)}_q(H).
\end{equation*}
Similarly, a label $i$ with $\lambda_i \in I$ and $\lambda_{i + 3} - \lambda_i \le 6q$ gives $U^{(4)}_q(H) \ge 37^{-6p}$.

It therefore suffices to show that
\begin{equation}\label{e:herm-trip-goal}
\frac{U^{(3)}_q(H)}{N^{1 - 8\kappa + \eps}} \to 0, \qquad U^{(4)}_q(H) \to 0 \quad (\kappa > 1/15),
\end{equation}
in probability. We prove \eqref{e:herm-trip-goal} for $H_T$, at a time $T$ chosen below, by comparison with the GUE, and we then remove the Gaussian component with \eqref{e:herm-comp-triple}.

We now choose the time. Let $\eps' \in (0, \eps/10)$ be small, and set
\begin{equation}\label{e:herm-trip-time}
T = N^{-1 + \kappa + 4\eps'}, \qquad \frac{2 N^{\eps'}}{N^2 T} = 2 N^{-3\eps'} q, \qquad T N^{1/2 + 3\kappa + 12\eps'} = N^{-1/2 + 4\kappa + 16\eps'} \to 0.
\end{equation}
Then $T \ge N^{-1 + \eps'}$, and the limit holds if $\eps'$ is small enough, since $\kappa < 1/8$. The second quantity in \eqref{e:herm-trip-time} is the coupling error at time $T$, and the third is the comparison error in \eqref{e:herm-comp-triple}.

Next, we compare $H_T$ with the GUE. Let $\delta \in (0, 1)$ satisfy $\supp \chi \subset [-2 + 3\delta, 2 - 3\delta]$, and let $c_*$ and $\mathscr J$ be as in \zcref{l:localized}, defined with this $\delta$. By the coupling of \zcref{p:coupling} at time $T$, in the form of \zcref{s:herm-inputs} and applied with $\eps'$ in place of $\eps$, there is a GUE matrix $G'$ such that, with overwhelming probability,
\begin{equation*}
\max_{c_* N \le i \le (1 - c_*) N} \big| \big( \lambda_{i + 1}(H_T) - \lambda_i(H_T) \big) - \big( \lambda_{i + 1}(G') - \lambda_i(G') \big) \big| \le 2 N^{-3\eps'} q.
\end{equation*}
We intersect this event with the events of \zcref{p:regularity}, for $H_T$ and $G'$ and with $\eps'$ in place of $\eps$. On the resulting event, whose probability tends to one, we claim that
\begin{equation}\label{e:herm-trip-coupling}
U^{(r)}_q(H_T) \le (1 + o(1)) U^{(r)}_q(G') + o(1), \qquad r \in \{ 3, 4 \}.
\end{equation}

To prove \eqref{e:herm-trip-coupling}, we follow the proof of \zcref{l:localized}, applied to each of the three, respectively six, rational factors of the summands and to the weight $\chi_N$ at the mean. We write $x_i = \lambda_i(H_T)$ and $x'_i = \lambda_i(G')$, and we first consider the tuples whose labels all belong to $\mathscr J$. For labels $i < j$ in $\mathscr J$, there are at most $N^{\eps'} (1 + N (x_j - x_i))$ gaps between $x_i$ and $x_j$, by condition (iii) of \zcref{d:regular}. As in \eqref{e:pairs-localized-telescope}, it follows that
\begin{equation*}
\big| (x'_j - x'_i) - (x_j - x_i) \big| \le N^{\eps'} \big( 1 + N (x_j - x_i) \big) \cdot 2 N^{-3\eps'} q \le 2 N^{-2\eps'} (q + x_j - x_i),
\end{equation*}
where we used $N q \le 1$.

We now compare the terms. Let $y = (x_j - x_i)/q$ and $y' = (x'_j - x'_i)/q$. The last display gives $|y - y'| \le 2 N^{-2\eps'} (1 + y)$, so, as in \eqref{e:pairs-localized-gap},
\begin{equation*}
| \log f_p(y) - \log f_p(y') | \le \frac{8p \, |y - y'|}{1 + y} \le 16 p N^{-2\eps'}
\end{equation*}
for $N \ge N_0$.

We next compare the weights. By \zcref{p:regularity}, $|x_i - \gamma_i|$ and $|x'_i - \gamma_i|$ are at most $N^{-1 + \eps'}$ for $i \in \mathscr J$, so the means $\bar x$ and $\bar x'$ of a tuple for $x$ and for $x'$ satisfy $|\bar x - \bar x'| \le 2 N^{-1 + \eps'}$. Recall that the smoothing \eqref{e:comparison-phiN} has width $h_N = N^{-1/10}$. By \eqref{e:comparison-relative-real} and $h_N^{-2} = N^{1/5}$, we have
\begin{equation*}
e^{-16 N^{-4/5 + \eps'}} \chi_N(\bar x') \le \chi_N(\bar x) \le e^{16 N^{-4/5 + \eps'}} \chi_N(\bar x').
\end{equation*}
A summand has at most six rational factors, so the term of every such tuple for $x$ is at most $e^{96 p N^{-2\eps'} + 16 N^{-4/5 + \eps'}}$ times the corresponding term for $x'$. Hence the sum of the terms of these tuples for $H_T$ is at most $(1 + o(1)) U^{(r)}_q(G')$.

We now consider the tuples with a label outside $\mathscr J$, for $x$ and $x'$ alike. Suppose first that such a tuple has its mean within distance $\delta$ of $\supp \chi$, so that the mean lies in $[-2 + 2\delta, 2 - 2\delta]$. If its smallest label is smaller than $c_* N$, then \eqref{e:pairs-edge-points} shows that its smallest point is at most $-2 + \delta$, and hence at least $\delta$ below the mean and below its largest point; the case where its largest label exceeds $(1 - c_*) N$ is symmetric. In both cases, two of its points are at distance at least $\delta$, so its term is bounded by $\| \chi \|_\infty (q/\delta)^{2p}$.

If instead the mean of such a tuple is at distance more than $\delta$ from $\supp \chi$, then its term is at most $C \exp(-\delta^2 N^{1/5}/2)$ by part (4) of \zcref{l:smoothing}. There are at most $N^4$ tuples, so these tuples contribute at most
\begin{equation*}
N^4 \Big( \| \chi \|_\infty \Big( \frac{q}{\delta} \Big)^{2p} + C \exp\Big( -\frac{\delta^2 N^{1/5}}{2} \Big) \Big) = o(1),
\end{equation*}
since $p \ge 4$. This proves \eqref{e:herm-trip-coupling}.

We now use the estimates for the GUE. By \eqref{e:herm-gue-triples} and Markov's inequality, we have
\begin{equation*}
\frac{U^{(3)}_q(G')}{N^{1 - 8\kappa + \eps}} \to 0, \qquad U^{(4)}_q(G') \to 0 \quad (\kappa > 1/15),
\end{equation*}
in probability, where for the second limit we used $N (N q)^{15} = N^{1 - 15\kappa}$. Combining this with \eqref{e:herm-trip-coupling}, we obtain \eqref{e:herm-trip-goal} with $H_T$ in place of $H$. Since these random variables are nonnegative, the bounded convergence theorem gives, for every $\theta > 0$,
\begin{equation}\label{e:herm-trip-laplace}
\lim_{N \to \infty} \E e^{-\theta U^{(3)}_q(H_T)/N^{1 - 8\kappa + \eps}} = 1, \qquad \lim_{N \to \infty} \E e^{-\theta U^{(4)}_q(H_T)} = 1 \quad (\kappa > 1/15).
\end{equation}

Finally, we remove the Gaussian component. By \eqref{e:herm-comp-triple} with $t = T$ and $\eps'$ in place of $\eps$, and by the last relation in \eqref{e:herm-trip-time}, the same limits hold with $H$ in place of $H_T$. Since $U^{(r)}_q(H) \ge 0$, these limits imply \eqref{e:herm-trip-goal}. This completes the proof.
\end{proof}

\section{Gaussian perturbations of \texorpdfstring{$3 \times 3$}{3 x 3} matrices}\label{s:herm-3x3}

The following lemma controls the gaps produced by a cluster of three eigenvalues under a Gaussian perturbation. We state it for a general $3 \times 3$ diagonal matrix. Let $\sigma > 0$, and let $W$ be a $3 \times 3$ complex Hermitian random matrix whose density with respect to Lebesgue measure on the space of $3 \times 3$ Hermitian matrices $B$ is proportional to $\exp(-\Tr B^2/(2\sigma^2))$. Equivalently, the entries $W_{ab}$ with $a \le b$ are independent, the diagonal entries are real centered Gaussian variables with variance $\sigma^2$, and the entries above the diagonal are complex Gaussian variables with $\E W_{ab} = \E W_{ab}^2 = 0$ and $\E |W_{ab}|^2 = \sigma^2$. For a $3 \times 3$ Hermitian matrix $B$, we let
\begin{equation*}
\mathrm{gap}(B) = \min\{ \lambda_2(B) - \lambda_1(B), \lambda_3(B) - \lambda_2(B) \}
\end{equation*}
denote its smallest eigenvalue gap.

\begin{lemma}\label{l:herm-3x3}
There exists a constant $C > 0$ such that the following holds. Let $y_1 \le y_2 \le y_3$ be real numbers, let $D_y = \diag(y_1, y_2, y_3)$, and let $W$ be as above. Then for all $\delta \in (0, \sigma]$ and $R \ge 1$,
\begin{equation}\label{e:herm-3x3}
\Prob\big( \mathrm{gap}(D_y + W) \le \delta \big) \le C R^4 \Big( \frac{\delta}{\sigma} \Big)^3 + C e^{-R^2/C}.
\end{equation}
\end{lemma}

\begin{proof}
Replacing $D_y$, $W$, and $\delta$ by $D_y/\sigma$, $W/\sigma$, and $\delta/\sigma$, we may assume that $\sigma = 1$, so that $\delta \le 1 \le R$. The Gaussian tail bound for the entries of $W$ gives
\begin{equation}\label{e:herm-normW}
\Prob(\|W\| > R) \le C e^{-R^2/C}.
\end{equation}
We distinguish three cases according to the configuration of $y_1, y_2, y_3$. Every configuration belongs to at least one of them.

\emph{Case 1: $y_2 - y_1 \ge 10R$ and $y_3 - y_2 \ge 10R$.} On the event $\{ \|W\| \le R \}$, Weyl's inequality shows that each eigenvalue of $D_y + W$ differs from the corresponding $y_j$ by at most $R$, so
\begin{equation*}
\lambda_{j + 1}(D_y + W) - \lambda_j(D_y + W) \ge y_{j + 1} - y_j - 2R \ge 8R > \delta, \qquad j \in \{ 1, 2 \}.
\end{equation*}
Hence the probability in \eqref{e:herm-3x3} is bounded by $C e^{-R^2/C}$, by \eqref{e:herm-normW}.

\emph{Case 2: $y_3 - y_1 \le 30R$.} We use the Weyl integration formula (see \cite[Section~4.1]{anderson2010introduction}). The density of $D_y + W$ with respect to Lebesgue measure on Hermitian matrices $B$ is proportional to
\begin{equation*}
\exp\Big( - \frac{\|B - D_y\|_F^2}{2} \Big),
\end{equation*}
where $\| \cdot \|_F$ denotes the Frobenius norm. Writing $B = O \diag(\mu) O^*$ with $\mu_1 \le \mu_2 \le \mu_3$ and $O$ unitary, the Weyl integration formula shows that the ordered eigenvalues of $D_y + W$ have the density
\begin{equation}\label{e:herm-weyl}
\mu \mapsto c \prod_{1 \le i < j \le 3} (\mu_j - \mu_i)^2 \int_{\mathrm U(3)} \exp\Big( - \frac{\| O \diag(\mu) O^* - D_y \|_F^2}{2} \Big) \dd O
\end{equation}
on $\{ \mu_1 \le \mu_2 \le \mu_3 \}$, where $\dd O$ is the Haar probability measure on the unitary group $\mathrm U(3)$ and $c > 0$ is an absolute constant.

By the Hoffman--Wielandt inequality for Hermitian matrices \cite{horn2013matrix}, we have
\begin{equation*}
\| O \diag(\mu) O^* - D_y \|_F \ge |\mu - y|, \qquad y = (y_1, y_2, y_3).
\end{equation*}
Hence the density \eqref{e:herm-weyl} at $\mu$ is bounded by
\begin{equation}\label{e:herm-weyl-bound}
c \prod_{1 \le i < j \le 3} (\mu_j - \mu_i)^2 \, e^{-|\mu - y|^2/2}.
\end{equation}

We now bound the probability that $\mu_2 - \mu_1 \le \delta$. On this event, we have
\begin{equation*}
\begin{gathered}
\prod_{1 \le i < j \le 3} (\mu_j - \mu_i)^2 \le \delta^2 (\mu_3 - \mu_1)^4, \qquad \mu_3 - \mu_1 \le |\mu_1 - y_1| + |\mu_3 - y_3| + 30R, \\
e^{-|\mu - y|^2/2} \le e^{-((\mu_1 - y_1)^2 + (\mu_3 - y_3)^2)/2},
\end{gathered}
\end{equation*}
where the second bound uses $y_3 - y_1 \le 30R$. Integrating over $\mu_2 \in [\mu_1, \mu_1 + \delta]$, and then over $\mu_1$ and $\mu_3$, we find
\begin{equation}\label{e:herm-case2}
\begin{aligned}
\Prob(\mu_2 - \mu_1 \le \delta) &\le c \, \delta^3 \int_{\R^2} \big( |\mu_1 - y_1| + |\mu_3 - y_3| + 30R \big)^4 e^{-((\mu_1 - y_1)^2 + (\mu_3 - y_3)^2)/2} \dd \mu_1 \dd \mu_3 \\
&\le C R^4 \delta^3 .
\end{aligned}
\end{equation}
The same bound holds for $\Prob(\mu_3 - \mu_2 \le \delta)$, so the probability in \eqref{e:herm-3x3} is bounded by $C R^4 \delta^3$ in this case.

\emph{Case 3: neither Case 1 nor Case 2 holds.} Then $y_3 - y_1 > 30R$, and one of the gaps $y_2 - y_1$ and $y_3 - y_2$ is smaller than $10R$; the other is then larger than $20R$. Let $P$ denote the permutation matrix that reverses the order of the coordinates. The matrix
\begin{equation*}
- P (D_y + W) P^T = \diag(-y_3, -y_2, -y_1) - P W P^T
\end{equation*}
has the gaps of $D_y + W$ in reverse order, and $- P W P^T$ has the law of $W$. Replacing $(y_1, y_2, y_3)$ by $(-y_3, -y_2, -y_1)$ if necessary, we may therefore assume that $y_2 - y_1 < 10R$ and $y_3 - y_2 > 20R$.

We work on the event $\|W\| \le R$, whose complement is controlled by \eqref{e:herm-normW}, and we set $\mu_j = \lambda_j(D_y + W)$. By Weyl's inequality, we have
\begin{equation*}
\begin{gathered}
\mu_3 \ge y_3 - R \ge y_1 + 19R, \qquad \mu_1, \mu_2 \in [y_1 - R, y_2 + R] \subset [y_1 - R, y_1 + 11R], \\
\mu_3 - \mu_2 \ge 19R - 11R = 8R > \delta.
\end{gathered}
\end{equation*}
Thus the only possible small gap is $\mu_2 - \mu_1$, and $\mu_1$ and $\mu_2$ are the eigenvalues of $D_y + W$ in the interval
\begin{equation*}
I_R = [y_1 - 12R, y_1 + 13R].
\end{equation*}

We reduce to a $2 \times 2$ problem by a Schur complement. Let $W_{[2]}$ denote the upper left $2 \times 2$ block of $W$, let $b = (W_{13}, W_{23})^T$, and for real $w$ with $y_3 + W_{33} - w \neq 0$, let
\begin{equation}\label{e:herm-schur}
A(w) = \diag(y_1, y_2) + W_{[2]} - \frac{b b^*}{y_3 + W_{33} - w}.
\end{equation}
Let $w \in I_R$, and suppose that $|W_{33}| \le R$ and $|b| \le R$, which hold if $\|W\| \le R$. Since $y_3 - y_1 > 20R$, we have
\begin{equation*}
\begin{gathered}
y_3 + W_{33} - w \ge 20R - R - 13R = 6R, \\
\Big\| \frac{b b^*}{y_3 + W_{33} - w} \Big\| \le \frac{R}{6}, \qquad \Big\| \frac{\dd}{\dd w} \frac{b b^*}{y_3 + W_{33} - w} \Big\| \le \frac{R^2}{(6R)^2} = \frac{1}{36},
\end{gathered}
\end{equation*}
and the last term in \eqref{e:herm-schur} is negative semidefinite.

We now locate the eigenvalues of $D_y + W$ in $I_R$. By Weyl's inequality, the eigenvalues $\lambda_1(A(w)) \le \lambda_2(A(w))$ of the Hermitian matrix $A(w)$ are Lipschitz functions of $w \in I_R$ with constant $1/36$, so the functions $w \mapsto \lambda_k(A(w)) - w$ are strictly decreasing on $I_R$. Since $\|W_{[2]}\| \le R$ and $y_2 < y_1 + 10R$, the bounds above give
\begin{equation*}
\lambda_k(A(w)) \in \Big[ y_1 - \frac{7R}{6}, y_2 + R \Big] \subset \Big[ y_1 - \frac{7R}{6}, y_1 + 11R \Big], \qquad w \in I_R,
\end{equation*}
so the functions $w \mapsto \lambda_k(A(w)) - w$ are positive at $w = y_1 - 12R$ and negative at $w = y_1 + 12R$. Hence each of them has exactly one zero $w_k$ in $I_R$, and $w_1 \le w_2$ since $\lambda_1(A(w)) \le \lambda_2(A(w))$.

We now identify $\mu_1$ and $\mu_2$. By the Schur complement formula, a point $w \in I_R$ is an eigenvalue of $D_y + W$ if and only if $\det(A(w) - w) = 0$, which holds exactly when $w \in \{ w_1, w_2 \}$. Moreover, the multiplicity of such an eigenvalue is the dimension of the kernel of $A(w) - w$. When $w_1 < w_2$, the two eigenvalues of $D_y + W$ in $I_R$ are therefore $w_1$ and $w_2$. When $w_1 = w_2$, we have $A(w_1) = w_1 \Id_2$, so $w_1$ is an eigenvalue of multiplicity two. In both cases, $\mu_k = w_k$ for $k \in \{ 1, 2 \}$.

We now transfer a small gap of $D_y + W$ to $A(w)$. Since $\lambda_k(A(\mu_k)) = \mu_k$ and $w \mapsto \lambda_2(A(w))$ is Lipschitz with constant $1/36$, if $\mu_2 - \mu_1 \le \delta$, then
\begin{equation}\label{e:herm-nu-gap}
\lambda_2(A(\mu_1)) - \lambda_1(A(\mu_1)) = \lambda_2(A(\mu_1)) - \mu_1 \le \lambda_2(A(\mu_2)) + \frac{\mu_2 - \mu_1}{36} - \mu_1 \le 2\delta .
\end{equation}

We now discretize. We set
\begin{equation*}
w_j' = y_1 - 12R + j\delta, \qquad j \in \Z, \quad 0 \le j \le \frac{24R}{\delta} + 1,
\end{equation*}
and we note that these points lie in $I_R$, since $\delta \le R$. If $\mu_2 - \mu_1 \le \delta$, we choose $j$ with $|w_j' - \mu_1| \le \delta$. Then the Lipschitz bound on $A$ and \eqref{e:herm-nu-gap}, together with Weyl's inequality, give
\begin{equation*}
\begin{gathered}
\|A(w_j') - A(\mu_1)\| \le \frac{\delta}{36}, \\
\lambda_2(A(w_j')) - \lambda_1(A(w_j')) \le 2\delta + \frac{\delta}{18} \le 3\delta, \\
|\lambda_1(A(w_j')) - w_j'| \le |\lambda_1(A(w_j')) - \lambda_1(A(\mu_1))| + |\mu_1 - w_j'| \le \frac{\delta}{36} + \delta \le 2\delta.
\end{gathered}
\end{equation*}
Thus the matrix $A(w_j')$ has eigenvalue gap at most $3\delta$ and an eigenvalue within distance $2\delta$ of $w_j'$.

We bound the probability of this event for fixed $j$. Condition on $W_{33}$ and $b$, and restrict to the event $\{ |W_{33}| \le R, |b| \le R \}$, on which $A(w_j') = M_j + W_{[2]}$ with a deterministic Hermitian matrix $M_j$. The block $W_{[2]}$ is independent of $(W_{33}, b)$. We set
\begin{equation*}
\begin{gathered}
v = \Big( \frac{(M_j)_{11} - (M_j)_{22}}{2}, \Re (M_j)_{12}, \Im (M_j)_{12} \Big), \\
\xi = \Big( \frac{W_{11} - W_{22}}{2}, \Re W_{12}, \Im W_{12} \Big).
\end{gathered}
\end{equation*}
Then the eigenvalues of the Hermitian $2 \times 2$ matrix $M_j + W_{[2]}$ are $(\Tr M_j + W_{11} + W_{22})/2 \pm |v + \xi|$.

The four real variables $(W_{11} + W_{22})/2$, $(W_{11} - W_{22})/2$, $\Re W_{12}$, and $\Im W_{12}$ are independent centered Gaussian variables with variance $1/2$. The event above requires
\begin{equation*}
|v + \xi| \le \frac{3\delta}{2}, \qquad \Big| \frac{\Tr M_j + W_{11} + W_{22}}{2} - w_j' \Big| \le 4\delta.
\end{equation*}
The densities of $\xi$ and of $(W_{11} + W_{22})/2$ are bounded, so this event has conditional probability at most $C\delta^3 \cdot C\delta$. Summing over the at most $24R/\delta + 2$ values of $j$, and using $\delta \le 1 \le R$, we conclude that in Case 3,
\begin{equation}\label{e:herm-case3}
\Prob\big( \mathrm{gap}(D_y + W) \le \delta \big) \le C \Big( \frac{R}{\delta} + 1 \Big) \delta^4 + C e^{-R^2/C} \le C R \delta^3 + C e^{-R^2/C}.
\end{equation}
Combining the three cases proves \eqref{e:herm-3x3}.
\end{proof}

\section{Gap counts at the scales \texorpdfstring{$N^{-1 - \alpha}$}{N\textasciicircum(-1-alpha)}}\label{s:herm-pairs}

For $\alpha \in (0, 1/3)$, we say that the statement $\mathcal P^{\C}(\alpha)$ holds if for every nonnegative $\varphi \in C_c^\infty((-2, 2))$ and every integer $p \ge p_*$, we have
\begin{equation}\label{e:herm-pairlaw}
\frac{1}{M^{\C}_\alpha} \mathcal X_\alpha(H; \varphi_N, f_p) \to \int_{-2}^{2} \varphi(E) c_2(E) \dd E \int_0^\infty y^2 f_p(y) \dd y
\end{equation}
in probability, where $\mathcal X_\alpha$ is defined by \eqref{e:comparison-pair} for Hermitian matrices. This is the analogue of the statement $\mathcal P(\alpha)$ of \zcref{d:pairlaw}.

We first record the results of \zcref{s:pairs} that carry over. The proofs of \zcref{l:pairs-unsmooth}, \zcref{l:pairs-transfer}, and \zcref{l:extension} apply with $M^{\C}_\alpha$, $c_2(E)$, and $\int_0^\infty y^2 f(y) \dd y$ in place of $M_\alpha$, $c(E)$, and $\int_0^\infty y f(y) \dd y$, and with \eqref{e:herm-comp-pair} in place of \zcref{p:comparison}; the integrals $\int_0^\infty y^2 f_p(y) \dd y$ are finite for $p \ge p_*$. Indeed, the proof of \zcref{l:pairs-unsmooth} is deterministic and uses only the bound $\| A \| \le 3$ and \zcref{l:smoothing}. The proof of \zcref{l:pairs-transfer} uses the comparison estimate only through the bound \eqref{e:comparison-bound} and the nonnegativity of the pair statistic, which holds for Hermitian matrices as well. The proof of \zcref{l:extension} uses only $\mathcal P(\alpha)$, the Weierstrass approximation theorem, and \zcref{l:pairs-unsmooth}, with \eqref{e:pairs-radial} replaced by the finiteness of the integrals above.

\paragraph{Base case.} The proof of \zcref{p:base} applies with the following changes, and gives $\mathcal P^{\C}(\alpha)$ for every $\alpha \in (0, 1/8)$. The GUE input is \zcref{l:herm-gue}(a) in place of \zcref{p:goepairs}. The proof of \zcref{l:localized} is deterministic, and in it the pairs with a label outside $\mathscr J$ contribute at most
\begin{equation*}
C N^{-2\alpha} + C N^2 \exp(-\delta^2 N^{1/5}/4) = o(M^{\C}_\alpha)
\end{equation*}
by \eqref{e:pairs-localized-outside}, so its proof gives \eqref{e:pairs-localized} with $M^{\C}_\alpha$ in place of $M_\alpha$. With $T = N^{-3/4 - \alpha}$ as in \zcref{p:base}, the coupling error of \zcref{s:herm-inputs} and the comparison error of \eqref{e:herm-comp-pair} satisfy
\begin{equation*}
\frac{2 N^{\eps}}{N^2 T} = 2 N^{-5/4 + \alpha + \eps} \le N^{-2\eps} \ell_\alpha, \qquad T N^{1/2 + 3\alpha + 12\eps} = N^{-1/4 + 2\alpha + 12\eps} \to 0
\end{equation*}
for $\alpha < 1/8$ and $\eps$ small. These exponents are the same as in the real symmetric case, and they do not use any condition on the third moments.

\paragraph{Induction step.} Suppose that $\mathcal P^{\C}(\alpha)$ holds, and let $\alpha' \in (\alpha, 1/3)$ and $\kappa > 0$ satisfy
\begin{equation}\label{e:herm-induction}
\frac{3\alpha'}{8} < \kappa < \min\Big\{ \alpha, \ 2\alpha - \alpha', \ \frac{1}{8} \Big\}, \qquad \alpha' - \kappa \ge \frac{1}{30}, \qquad \alpha' < \frac16 + \frac{2\alpha}{3}.
\end{equation}
We follow \zcref{s:pairs-induction}, with $t_\alpha = N \ell_\alpha^2$, $q = N^{-1 - \kappa}$, and $\eps > 0$ sufficiently small depending on the margins in \eqref{e:herm-induction}. The conditions \eqref{e:herm-induction} replace the choice of $\kappa$ in \eqref{e:pairs-induction-parameters} and the conditions \eqref{e:pairs-margins}, and we indicate below where each of them is used.

We begin with the Gaussian blocks. Conditionally on $H$, the matrix $H_{t_\alpha}$ has the eigenvalues of $D_y + W$, where $y_i = e^{-t_\alpha/2} \lambda_i$ and $W = (1 - e^{-t_\alpha})^{1/2} (u_i^* G u_j)_{i, j = 1}^N$ for orthonormal eigenvectors $u_1, \dots, u_N$ of $H$; by the unitary invariance of the GUE, $W$ is independent of $H$ and has the law of $(1 - e^{-t_\alpha})^{1/2} G$. As in the proof of \zcref{l:pairs-reduction}, \zcref{c:threshold} applies with $w_0 = N^{\eps} \ell_\alpha$, and the error in \eqref{e:prelim-threshold-error} is $o(\ell_{\alpha'})$, since
\begin{equation*}
\begin{gathered}
N^{\eps} w_0 (q^{-1} + N \log N) \le 2 N^{2\eps - \alpha + \kappa} \to 0, \\
24 N^{2\eps} w_0^2 (q^{-1} + N \log N) \le C N^{4\eps - 1 - 2\alpha + \kappa} = o(\ell_{\alpha'}),
\end{gathered}
\end{equation*}
where we used $\kappa < \alpha$ in the first bound and $\kappa < 2\alpha - \alpha'$ in the last relation. For a two-element cluster $\mathcal K = \{ i, i + 1 \}$, since the eigenvalues of a $2 \times 2$ Hermitian matrix differ by the Euclidean norm of the vector formed by the difference of its diagonal entries and twice the real and imaginary parts of its off-diagonal entry, the gap $\hat g_{\mathcal K}$ of the block $(D_y + W)_{\mathcal K \mathcal K}$ is
\begin{equation}\label{e:herm-block}
\hat g_{\mathcal K} = \big| \big( e^{-t_\alpha/2} g_i + W_{i + 1, i + 1} - W_{ii}, \ 2 \Re W_{i, i + 1}, \ 2 \Im W_{i, i + 1} \big) \big|.
\end{equation}

Next, we compute the conditional mean of the block statistic, following the proof of \zcref{l:pairs-mean}. The first coordinate of the vector in \eqref{e:herm-block} minus $e^{-t_\alpha/2} g_i$ and the other two coordinates are independent centered Gaussian variables with variance $v^{\C}_N \ell_\alpha^2$, where
\begin{equation*}
v^{\C}_N = \frac{2 (1 - e^{-t_\alpha})}{N \ell_\alpha^2} = \frac{2 (1 - e^{-t_\alpha})}{t_\alpha} \in [2 - t_\alpha, 2]
\end{equation*}
(compare \eqref{e:pairs-vN}). The vector in \eqref{e:herm-block} is three-dimensional because degeneracies of Hermitian matrices have codimension three. Thus $\hat g_{\mathcal K}/\ell_\alpha$ has the law of $|\mathsf G_x|$ in \zcref{l:gaussian} with $n = 3$, $x = g_i/\ell_\alpha$, $\varrho = e^{-t_\alpha/2}$, and $v = v^{\C}_N$, and we apply \zcref{l:gaussian} with $v_0 = 2$ at the scale $N^{\alpha - \alpha'} = \ell_{\alpha'}/\ell_\alpha$; its hypothesis $p > 2$ holds since $p \ge p_*$. In the notation \eqref{e:prelim-gaussian-density} with $n = 3$, we have $\mathfrak g_2(x, 0, 0) = (4\pi)^{-3/2} e^{-x^2/4}$, and $\int_{\R^3} f(|y|) \dd y = 4\pi \int_0^\infty r^2 f(r) \dd r$ for radial integrands. Since $4\pi (4\pi)^{-3/2} = (2\sqrt\pi)^{-1}$, we obtain
\begin{equation}\label{e:herm-condmean}
\E\Big[ f_p\Big( \frac{\hat g_{\mathcal K}}{\ell_{\alpha'}} \Big) \, \Big| \, H \Big] = N^{3(\alpha - \alpha')} \Big[ \frac{e^{-g_i^2/(4 \ell_\alpha^2)}}{2\sqrt{\pi}} \int_0^\infty y^2 f_p(y) \dd y + o(1) f_p\Big( \frac{g_i}{\ell_\alpha} \Big) \Big],
\end{equation}
where the $o(1)$ term is deterministic and uniform in $H$ and in $\mathcal K$.

Recall the block statistic $\widehat{\mathcal X}_{\alpha'}(\varphi, p)$ of \eqref{e:pairs-block-statistic}, and the set $\mathfrak P_2$ of two-element clusters and the quantities $g_{\mathcal K}$ and $E_{\mathcal K}$ defined before it. We sum \eqref{e:herm-condmean}, multiplied by $\varphi_N(E_{\mathcal K})$, over $\mathcal K \in \mathfrak P_2$, and we bound the error terms by $\mathcal P^{\C}(\alpha)$. Using also $N^{3(\alpha - \alpha')} M^{\C}_\alpha = M^{\C}_{\alpha'}$, we find
\begin{equation*}
\begin{aligned}
\frac{1}{M^{\C}_{\alpha'}} \E\big[ \widehat{\mathcal X}_{\alpha'}(\varphi, p) \, \big| \, H \big] &= \frac{\int_0^\infty y^2 f_p(y) \dd y}{2 \sqrt\pi \, M^{\C}_\alpha} \sum_{\mathcal K \in \mathfrak P_2} \varphi_N(E_{\mathcal K}) e^{-g_{\mathcal K}^2/(4 \ell_\alpha^2)} + o_{\Prob}(1) \\
&\longrightarrow \int_{-2}^{2} \varphi(E) c_2(E) \dd E \int_0^\infty y^2 f_p(y) \dd y
\end{aligned}
\end{equation*}
in probability, which is the limit in \eqref{e:herm-pairlaw} with $\alpha'$ in place of $\alpha$. For the limit, we used $\mathcal P^{\C}(\alpha)$ and \zcref{l:extension} applied to the Gaussian test function $y \mapsto e^{-y^2/4}$, together with $\int_0^\infty r^2 e^{-r^2/4} \dd r = 2 \sqrt\pi$, and we removed the pairs that do not form a two-element cluster as explained below. The conditional variance bound is as in the proof of \zcref{l:pairs-concentration}, since the blocks of distinct clusters are conditionally independent given $H$.

We now treat the pairs of eigenvalues in a common cluster with at least three elements, which appear in \eqref{e:pairs-reduction} and in the evaluation of the conditional mean. Here we use \zcref{p:herm-triples}, which applies since $\kappa < 1/8$, in place of \zcref{p:three}. Let $\delta > 0$, and let $I$ be a bulk interval containing the closed $2\delta$-neighborhood of $\supp \varphi$. Recall from \zcref{l:pairs-clusters} that every cluster has at most $N^{\eps}$ elements, and observe that the smallest label $i$ of a cluster with at least three elements satisfies $\lambda_{i + 2} - \lambda_i < 2 e^{t_\alpha/2} q \le 4q$, as in the proof of that lemma. Hence the number of such pairs in clusters whose smallest eigenvalue lies in $I$ is at most $N^{2\eps}$ times the number of labels $i$ with $\lambda_i \in I$ and $\lambda_{i + 2} - \lambda_i \le 4q$. By \zcref{p:herm-triples}, with probability tending to one, this is at most $N^{1 - 8\kappa + 3\eps} = o(M^{\C}_{\alpha'})$, since $8\kappa > 3\alpha'$; this replaces the use of \zcref{p:three} in \eqref{e:pairs-N3-bound}. A cluster has diameter at most $2 N^{\eps} q$, so the pairs in clusters whose smallest eigenvalue lies outside $I$ have midpoints at distance at least $\delta$ from $\supp \varphi$, and by \zcref{l:smoothing} they contribute at most $C N^2 \exp(-\delta^2 N^{1/5}/2) = o(1)$.

The pairs in different clusters are at distance at least $q/2$, by \zcref{c:threshold}, so they contribute at most
\begin{equation*}
4^p \| \varphi \|_\infty N^2 \Big( \frac{\ell_{\alpha'}}{q} \Big)^{2p} = 4^p \| \varphi \|_\infty M^{\C}_{\alpha'} N^{1 + 3\alpha' - 2p(\alpha' - \kappa)} \le 4^p \| \varphi \|_\infty M^{\C}_{\alpha'} N^{2 - 61/15},
\end{equation*}
where we used $p \ge p_* = 61$, $\alpha' - \kappa \ge 1/30$, and $1 + 3\alpha' < 2$. In the evaluation of the conditional mean, the pairs of eigenvalues of $H$ in different clusters are at distance at least $q$, so, as in the proof of \zcref{l:pairs-mean}, they contribute at most $\| \varphi \|_\infty N^2 \exp(-q^2/(4 \ell_\alpha^2)) = o(1)$, since $\kappa < \alpha$. Here $4 \ell_\alpha^2$ replaces the denominator $8 \ell_\alpha^2$ of the real symmetric case, because the Gaussian factor in \eqref{e:herm-condmean} is $e^{-g_i^2/(4 \ell_\alpha^2)}$ instead of $e^{-g_i^2/(8 \ell_\alpha^2)}$. Finally, the comparison \eqref{e:herm-comp-pair} at the scale $\ell_{\alpha'}$ and time $t_\alpha$ has the error $t_\alpha N^{1/2 + 3\alpha' + 12\eps} = N^{-1/2 - 2\alpha + 3\alpha' + 12\eps}$, which tends to zero by the last condition in \eqref{e:herm-induction}. This completes the proof of $\mathcal P^{\C}(\alpha')$.

\paragraph{Iteration.} The conditions \eqref{e:herm-induction} hold for the triples $(\alpha, \alpha', \kappa)$ equal to
\begin{equation}\label{e:herm-chain}
\Big( \frac{3}{25}, \frac{17}{100}, \frac{67}{1000} \Big), \qquad \Big( \frac{17}{100}, \frac{6}{25}, \frac{19}{200} \Big), \qquad \Big( \frac{6}{25}, \frac{8}{25}, \frac{49}{400} \Big).
\end{equation}
Indeed, for these three triples, in this order, we have
\begin{equation*}
\begin{aligned}
\Big( \frac{3\alpha'}{8}, \kappa, \min\Big\{ \alpha, 2\alpha - \alpha', \frac18 \Big\} \Big) &= \Big( \frac{255}{4000}, \frac{268}{4000}, \frac{280}{4000} \Big), \ \Big( \frac{18}{200}, \frac{19}{200}, \frac{20}{200} \Big), \ \Big( \frac{48}{400}, \frac{49}{400}, \frac{50}{400} \Big), \\
\alpha' - \kappa &= \frac{103}{1000}, \ \frac{29}{200}, \ \frac{79}{400}, \\
\Big( \alpha', \ \frac16 + \frac{2\alpha}{3} \Big) &= \Big( \frac{51}{300}, \frac{74}{300} \Big), \ \Big( \frac{36}{150}, \frac{42}{150} \Big), \ \Big( \frac{48}{150}, \frac{49}{150} \Big).
\end{aligned}
\end{equation*}
The minima in the first line are $2\alpha - \alpha'$, $2\alpha - \alpha'$, and $1/8$, respectively. In each triple of the first line, the numbers increase, the differences in the second line exceed $1/30$, and in each pair of the third line the first number is smaller than the second. Starting from the base case at $\alpha = 3/25 < 1/8$, three induction steps give $\mathcal P^{\C}(\alpha_1)$ with $\alpha_1 = 8/25$.

\section{The critical scale}\label{s:herm-critical}

We fix $\alpha_1 = 8/25$ and $\kappa_1 = 49/400$, so that
\begin{equation}\label{e:herm-final-param}
\frac{1}{4} < \alpha_1 < \frac13, \qquad \frac{3\alpha_1}{8} < \kappa_1 < \frac{1}{8}, \qquad \frac{1}{15} < \kappa_1 < 2\alpha_1 - \frac13,
\end{equation}
and we set $q = N^{-1 - \kappa_1}$. Recall that $t_{\alpha_1} = N \ell_{\alpha_1}^2 = N^{-1 - 2\alpha_1}$. Note that $\kappa_1 < \alpha_1$. We fix $\eps > 0$ sufficiently small depending on the margins in \eqref{e:herm-final-param}; for example, we need $5\eps < 8\kappa_1 - 3\alpha_1 = 1/50$. Fix a bulk interval $I$, a bulk interval $I'$ whose interior contains $I$, and a bulk interval $I''$ whose interior contains $I'$. We also fix $x_0 \ge 1$.

\paragraph{The matrix $H_{t_{\alpha_1}}$.} As in the induction step, conditionally on $H$, the matrix $H_{t_{\alpha_1}}$ has the eigenvalues of $D_y + W$, where $y_i = e^{-t_{\alpha_1}/2} \lambda_i$ and $W$ is independent of $H$ with the law of $(1 - e^{-t_{\alpha_1}})^{1/2} G$. We form clusters at the threshold $q$ as in \zcref{s:pairs-induction}, and we work on the event $\{ H \in \mathcal G_{\eps/2} \} \cap \{ \max_{i \le j} |W_{ij}| \le N^{\eps} \ell_{\alpha_1} \}$, whose probability tends to one. On this event, \zcref{c:threshold} applies with $w_0 = N^\eps \ell_{\alpha_1}$, since $N^{2\eps} \ell_{\alpha_1} (q^{-1} + N \log N) \le C N^{2\eps - \alpha_1 + \kappa_1} \to 0$. It shows that the eigenvalues of $H_{t_{\alpha_1}}$ attached to a cluster differ from those of the corresponding diagonal block of $D_y + W$ by at most
\begin{equation*}
C N^{4\eps} \ell_{\alpha_1}^2 q^{-1} = C N^{-1 - 2\alpha_1 + \kappa_1 + 4\eps} = o(d_2),
\end{equation*}
since $\kappa_1 < 2\alpha_1 - 1/3$, and that eigenvalues attached to different clusters are at distance at least $q/2$, which is much larger than $d_2$.

We next use \zcref{p:herm-triples} to control the sizes of the clusters. The smallest label $i$ of a cluster with at least three elements satisfies $\lambda_{i + 2} - \lambda_i < 2 e^{t_{\alpha_1}/2} q \le 4q$, and if the cluster has at least four elements, then also $\lambda_{i + 3} - \lambda_i < 3 e^{t_{\alpha_1}/2} q \le 6q$. By \zcref{p:herm-triples}, applied with $\kappa = \kappa_1 \in (1/15, 1/8)$ and with $I''$ in place of $I$, with probability tending to one no cluster whose smallest eigenvalue lies in $I''$ has four or more elements, and at most $N^{1 - 8\kappa_1 + \eps}$ such clusters have three elements. A cluster that contains an eigenvalue in $I'$ has its smallest eigenvalue in $I''$ for $N \ge N_0$, since its diameter is at most $2 N^{\eps} q$.

\paragraph{Two-element clusters.} Let $Q(y, E) = \sum_{j = 1}^{n} \theta_j \varphi_j(E) f_{p_j}(y)$ with $\theta_j \ge 0$, nonnegative $\varphi_j \in C_c^\infty((-2, 2))$ with support in the interior of $I'$, and integers $p_j \ge p_*$, and set $p_0 = \min_j p_j$. Recall that $E_i = (\lambda_i + \lambda_{i + 1})/2$. For a two-element cluster $\mathcal K = \{ i, i + 1 \}$, the block gap $\hat g_i = \hat g_{\mathcal K}$ is given by \eqref{e:herm-block} with $\alpha_1$ in place of $\alpha$. For $E \in \R$, the function $\psi_E(y) = 1 - e^{-Q(y, E)}$ satisfies $0 \le \psi_E(y) \le Q(y, E) \le C_Q f_{p_0}(y)$. We set
\begin{equation}\label{e:herm-psibar}
\bar\psi_i = \E\big[ \psi_{E_i}(\hat g_i/d_2) \, \big| \, H \big], \qquad m_Q(E) = \int_0^\infty y^2 \psi_E(y) \dd y .
\end{equation}
By \zcref{l:gaussian} as in \eqref{e:herm-condmean}, now applied to $\psi_{E_i}/C_Q$ at the scale $d_2/\ell_{\alpha_1}$, we have
\begin{equation*}
\bar\psi_i = \Big( \frac{d_2}{\ell_{\alpha_1}} \Big)^3 \Big[ \frac{e^{-g_i^2/(4 \ell_{\alpha_1}^2)}}{2\sqrt\pi} m_Q(E_i) + o(1) f_{p_0}\Big( \frac{g_i}{\ell_{\alpha_1}} \Big) \Big],
\end{equation*}
uniformly in $i$. In particular, $\max_i \bar\psi_i \le C (d_2/\ell_{\alpha_1})^3 \to 0$.

We now sum the $\bar\psi_i$. The function $m_Q$ is nonnegative and smooth, with support in the interior of $I'$. By $\mathcal P^{\C}(\alpha_1)$, \zcref{l:extension} applied with the unsmoothed weight $m_Q$ and the test function $y \mapsto e^{-y^2/4}$, and the identities $(d_2/\ell_{\alpha_1})^3 M^{\C}_{\alpha_1} = 1$ and $\int_0^\infty r^2 e^{-r^2/4} \dd r = 2\sqrt\pi$, we find
\begin{equation}\label{e:herm-laplace-exponent}
\sum_i \bar\psi_i \longrightarrow \int_{-2}^{2} \int_0^\infty c_2(E) \, y^2 \big( 1 - e^{-Q(y, E)} \big) \dd y \dd E
\end{equation}
in probability, where the sum is over the first labels of the two-element clusters. Here the pairs that do not form a two-element cluster were removed from the statistic of $\mathcal P^{\C}(\alpha_1)$ as in the induction step, using $8\kappa_1 > 3\alpha_1$ for the clusters with three elements. The blocks of distinct clusters are conditionally independent given $H$, so the conditional expectation of $\exp(-\sum_i Q(\hat g_i/d_2, E_i))$ given $H$ is $\prod_i (1 - \bar\psi_i)$. Since $\max_i \bar\psi_i \to 0$ and $\sum_i \bar\psi_i = O_{\Prob}(1)$, this product converges in probability to the exponential of minus the right side of \eqref{e:herm-laplace-exponent}.

\paragraph{Three-element clusters.} Let $\mathcal K = \{ i, i + 1, i + 2 \}$ be a cluster whose smallest eigenvalue lies in $I''$, and let $B_{\mathcal K} = (D_y + W)_{\mathcal K \mathcal K}$. This block is a deterministic diagonal matrix plus a random matrix $W_{\mathcal K \mathcal K}$ independent of $H$ as in \zcref{l:herm-3x3}, with $\sigma^2 = (1 - e^{-t_{\alpha_1}})/N \in [\ell_{\alpha_1}^2/2, \ell_{\alpha_1}^2]$. By \zcref{l:herm-3x3} with $R = N^{\eps}$, for all $\delta \in (0, \sigma]$,
\begin{equation*}
\Prob\big( \mathrm{gap}(B_{\mathcal K}) \le \delta \, \big| \, H \big) \le C N^{4\eps} \Big( \frac{\delta}{\ell_{\alpha_1}} \Big)^3 + C e^{-N^{2\eps}/C}.
\end{equation*}

We now bound the contribution of $B_{\mathcal K}$. Every pair of eigenvalues of $B_{\mathcal K}$ is at distance at least $\mathrm{gap}(B_{\mathcal K})$, and $f_{p_0}$ is decreasing on $[0, \infty)$. Let $k \ge 0$ be the smallest integer such that $\mathrm{gap}(B_{\mathcal K}) \le 2^k d_2$. Then $f_{p_0}(\mathrm{gap}(B_{\mathcal K})/d_2) \le 4^{p_0} 2^{-2p_0 k}$, and we find
\begin{equation*}
\begin{aligned}
\E\Big[ f_{p_0}\Big( \frac{\mathrm{gap}(B_{\mathcal K})}{d_2} \Big) \, \Big| \, H \Big] &\le C N^{4\eps} \Big( \frac{d_2}{\ell_{\alpha_1}} \Big)^3 \sum_{k \ge 0} 2^{(3 - 2p_0) k} + C \Big( \frac{d_2}{\sigma} \Big)^{2p_0} + C e^{-N^{2\eps}/C} \log N \\
&\le C N^{4\eps} \Big( \frac{d_2}{\ell_{\alpha_1}} \Big)^3,
\end{aligned}
\end{equation*}
where the second term bounds the contribution of the $k$ with $2^k d_2 > \sigma$, and the third the error terms of the $O(\log N)$ values of $k$ with $2^k d_2 \le \sigma$. On our event, the eigenvalues of $H_{t_{\alpha_1}}$ attached to $\mathcal K$ differ from those of $B_{\mathcal K}$ by $o(d_2)$, so the terms in $\sum_{i < i'} Q((\lambda_{i'} - \lambda_i)/d_2, (\lambda_i + \lambda_{i'})/2)$ of the pairs of these eigenvalues are at most $C f_{p_0}(\mathrm{gap}(B_{\mathcal K})/d_2)$. Summing over the at most $N^{1 - 8\kappa_1 + \eps}$ three-element clusters, we find that their total contribution, and also the conditional probability that some three-element cluster produces a gap at most $x_0 d_2$, is bounded in conditional expectation by
\begin{equation}\label{e:herm-triple-contribution}
C N^{1 - 8\kappa_1 + 5\eps} \Big( \frac{d_2}{\ell_{\alpha_1}} \Big)^3 = C N^{1 - 8\kappa_1 + 5\eps} N^{-1 + 3\alpha_1} = C N^{3\alpha_1 - 8\kappa_1 + 5\eps} \to 0,
\end{equation}
since $5\eps < 8\kappa_1 - 3\alpha_1$.

\paragraph{The remaining pairs.} With probability tending to one, every cluster containing an eigenvalue in $I'$ has at most three elements. The pairs of eigenvalues of $H_{t_{\alpha_1}}$ attached to different clusters are at distance at least $q/2$, so, since $p_0 \ge p_*$, they contribute at most
\begin{equation*}
C N^2 \Big( \frac{2 d_2}{q} \Big)^{2p_0} = C N^{2 - 2p_0 (1/3 - \kappa_1)} = o(1).
\end{equation*}
Pairs whose midpoint lies outside $I'$ do not contribute. For a two-element cluster $\{ i, i + 1 \}$, the gap of $H_{t_{\alpha_1}}$ differs from $\hat g_i$ by $o(d_2)$, so the rational factors change by a relative amount $o(1)$, and the midpoint differs from $E_i$ by at most $C N^\eps \ell_{\alpha_1}$, as in the proof of \zcref{l:pairs-reduction}, so the weights $\varphi_j$ change by at most $C N^\eps \ell_{\alpha_1}$. The conditional expectation of $\sum_i f_{p_0}(\hat g_i/d_2)$ over the two-element clusters meeting $I'$ is $O_{\Prob}(1)$, by \zcref{l:gaussian} and the upper bound \eqref{e:pairs-upper} in \zcref{l:extension}, so these changes contribute $o_{\Prob}(1)$.

Combining these facts with the two preceding paragraphs and using the bounded convergence theorem, we conclude that, for every $Q$ as above,
\begin{equation}\label{e:herm-Ht-limit}
\begin{aligned}
&\E \exp\Big( - \sum_{i < i'} Q\Big( \frac{\lambda_{i'}(H_{t_{\alpha_1}}) - \lambda_i(H_{t_{\alpha_1}})}{d_2}, \frac{\lambda_i(H_{t_{\alpha_1}}) + \lambda_{i'}(H_{t_{\alpha_1}})}{2} \Big) \Big) \\
&\qquad \longrightarrow \exp\Big( - \int_{-2}^{2} \int_0^\infty c_2(E) \, y^2 \big( 1 - e^{-Q(y, E)} \big) \dd y \dd E \Big).
\end{aligned}
\end{equation}
The same argument shows that the statistic
\begin{equation*}
\sum_{i < i'} \chi\Big( \frac{\lambda_i + \lambda_{i'}}{2} \Big) f_{p_*}\Big( \frac{\lambda_{i'} - \lambda_i}{d_2} \Big), \qquad \lambda_i = \lambda_i(H_{t_{\alpha_1}}),
\end{equation*}
is tight for every nonnegative $\chi \in C_c^\infty((-2, 2))$. Therefore, as in the proof of \zcref{l:extension}, using \zcref{l:pairs-unsmooth} at the scale $d_2 = \ell_{1/3}$, \eqref{e:herm-Ht-limit} also holds after replacing the weights $\varphi_j$ on the left side by their smoothings $\varphi_{j, N}$, since $|e^{-a} - e^{-b}| \le |a - b|$ for $a, b \ge 0$.

\paragraph{Return to $H$.} The observable on the left side of \eqref{e:herm-Ht-limit}, with smoothed weights, is of the form considered in \zcref{d:complexS} at the scale $d_2 = \ell_{1/3}$. By \eqref{e:herm-comp-pair} with $\alpha = 1/3$ and $t = t_{\alpha_1}$, which is \eqref{e:herm-critcomp}, the comparison error is $C N^{1/2 - 2\alpha_1 + 12\eps} = C N^{-7/50 + 12\eps} = o(1)$, since $\alpha_1 > 1/4$. Hence \eqref{e:herm-Ht-limit}, with smoothed weights, holds with $H$ in place of $H_{t_{\alpha_1}}$.

Next, we show that close triples are absent for $H$ at the scale $d_2$. Let $U^{(3)}_{d_2}$ be defined by \eqref{e:herm-U} with $d_2$ in place of $q$, $p = p_*$, and a nonnegative $\chi \in C_c^\infty((-2, 2))$ with support in the interior of $I'$ and with $\chi \ge 2$ on a neighborhood of $I$, and let $\eta_\chi > 0$ be such that the $2\eta_\chi$-neighborhood of $\supp \chi$ lies in $I'$. We first show that $U^{(3)}_{d_2}(H_{t_{\alpha_1}}) \to 0$ in probability. A triple of eigenvalues of $H_{t_{\alpha_1}}$ attached to a single three-element cluster has its term bounded by $C f_{p_*}(\mathrm{gap}(B_{\mathcal K})/d_2)$, so the total contribution of these triples tends to zero in probability by \eqref{e:herm-triple-contribution}. A triple attached to at least two clusters contains two eigenvalues at distance at least $q/2$, so these triples contribute at most $C N^3 (2 d_2/q)^{2p_*} = o(1)$. With probability tending to one, every cluster containing an eigenvalue in $I'$ has at most three elements, and the triples whose mean lies at distance at least $\eta_\chi$ from $\supp \chi$ contribute at most $C N^3 \exp(-\eta_\chi^2 N^{1/5}/2) = o(1)$. Hence $U^{(3)}_{d_2}(H_{t_{\alpha_1}}) \to 0$ in probability, and $\E e^{-U^{(3)}_{d_2}(H_{t_{\alpha_1}})} \to 1$.

We now transfer this to $H$. Applying \eqref{e:herm-comp-triple} with $r = 3$, $d_2$ in place of $q$ (that is, $\kappa = 1/3$), $\theta = M = 1$, and $t = t_{\alpha_1}$, we find that $\E e^{-U^{(3)}_{d_2}(H)} \to 1$, so $U^{(3)}_{d_2}(H) \to 0$ in probability. If $\lambda_i \in I$ and $\lambda_{i + 2} - \lambda_i \le x_0 d_2$, then the term of $(i, i + 1, i + 2)$ in $U^{(3)}_{d_2}(H)$ is at least $(1 + x_0^2)^{-3p_*}$ for $N \ge N_0$. Consequently, with probability tending to one, there exists no $i$ with $\lambda_i \in I$ and $\lambda_{i + 2} - \lambda_i \le x_0 d_2$. In particular, every pair $i < i'$ with $\lambda_i \in I$ and $\lambda_{i'} - \lambda_i \le x_0 d_2$ is then a pair of consecutive eigenvalues.

\paragraph{Conclusion.} We have shown that \eqref{e:herm-Ht-limit} holds for $H$, with smoothed weights. Taking $Q = \theta \chi f_{p_*}$ and letting $\theta \to 0$, we see that the statistics
\begin{equation*}
\sum_{i < i'} \chi_N\Big( \frac{\lambda_i + \lambda_{i'}}{2} \Big) f_{p_*}\Big( \frac{\lambda_{i'} - \lambda_i}{d_2} \Big)
\end{equation*}
are tight, and \zcref{l:smoothing} then allows us to remove the smoothing of the weights as in the proof of \zcref{l:extension}. Since the Laplace transform determines the law of a random vector with nonnegative coordinates, it follows that for all nonnegative $\varphi_1, \dots, \varphi_n \in C_c^\infty((-2, 2))$ and integers $p_1, \dots, p_n \ge p_*$, the vector of statistics
\begin{equation*}
\sum_{i < i'} \varphi_j\Big( \frac{\lambda_i + \lambda_{i'}}{2} \Big) f_{p_j}\Big( \frac{\lambda_{i'} - \lambda_i}{d_2} \Big), \qquad j \in [n],
\end{equation*}
converges jointly in law to the vector of integrals of the functions $(y, E) \mapsto \varphi_j(E) f_{p_j}(y)$ against a Poisson process $\Pi$ on $[0, \infty) \times (-2, 2)$ with intensity $c_2(E) y^2 \dd y \dd E$.

We now pass to general test functions. Every continuous function $f$ with compact support in $[0, \infty) \times (-2, 2)$ can be approximated by such combinations, with signed coefficients, up to an error $\varpi f_{p_*}(y) \1_{I'}(E)$, by the weighted polynomial approximation in the variable $(1 + y^2)^{-1}$ in the proof of \zcref{l:extension} and the partition of unity in the proof of \zcref{p:pairsl}. Together with the tightness of the statistics $\sum_{i < i'} \chi((\lambda_i + \lambda_{i'})/2) f_{p_*}((\lambda_{i'} - \lambda_i)/d_2)$ shown above, this shows that $\sum_{i < i'} f((\lambda_{i'} - \lambda_i)/d_2, (\lambda_i + \lambda_{i'})/2)$ converges in law to $\int f \dd \Pi$, and hence that the point process $\sum_{i < i'} \delta_{((\lambda_{i'} - \lambda_i)/d_2, (\lambda_i + \lambda_{i'})/2)}$ converges in distribution to $\Pi$ with respect to the vague topology.

Finally, we pass to consecutive eigenvalues. By the absence of close triples for $H$ at the scale $d_2$, shown above and applied to bulk intervals $I$ containing the $E$-supports of the test functions, the pairs with $\lambda_{i'} - \lambda_i \le x_0 d_2$ are pairs of consecutive eigenvalues with probability tending to one, and midpoints may be replaced by left endpoints since they differ by $O(d_2)$. This gives the convergence of the point processes \eqref{e:XiNherm}. The limit \eqref{e:minimumherm} follows from \zcref{l:reduction-void}, whose proof applies with $\Pi$ and its intensity $c_2(E) y^2 \dd y \dd E$ in place of $\Xi$ and $\nu$, since $\int_0^x c_2(E) y^2 \dd y = \pi^2 \rhosc(E)^4 x^3/9$.