2.1 ETH and its fluctuations for deformed Wigner matrices

In this section, we consider deformed Wigner matrices, $H = W + D$ , with increasingly ordered eigenvalues $\lambda _1\le \lambda _2\le \dots \le \lambda _N$ and corresponding orthonormal eigenvectors $\mathbf {u}_1,\dots , \mathbf {u}_N$ . Here, $D =D^* \in \mathbf {C}^{N \times N}$ is a self-adjoint matrix with uniformly bounded norm (i.e., $\lVert D\rVert \le C_D$ for some N-independent constant $C_D>0$ ). Although the Eigenstate Thermalisation Hypothesis (ETH) will be shown to hold for general deformations D, we shall require slightly stronger assumptions for proving the Gaussian fluctuations (see Assumption 2.8 below).

In order to state our results on the ETH and its fluctuations (Theorems 2.7 and 2.9, respectively), we need to introduce the concept of regular observables – first in a simple form in Definition 2.11 (later along the proofs we will need a more general version in Definition 4.2). For this purpose, we introduce $M(z)$ being the unique solution of the Matrix Dyson Equation (MDE):Footnote ⁵

(2.8) $$ \begin{align} -\frac{1}{M(z)}=z-D+\langle M(z)\rangle, \qquad\quad \Im M(z)\Im z>0. \end{align} $$

The self consistent density of states (scDos) is then defined as

(2.9) $$ \begin{align} \rho(e):=\frac{1}{\pi}\lim_{\eta\downarrow 0}\langle \Im M(e+\mathrm{i}\eta)\rangle\,. \end{align} $$

We point out that not only $\langle \Im M(e+\mathrm {i}\eta )\rangle $ has an extension to the real axis, but the whole matrix $M(e) := \lim _{\eta \downarrow 0} M(e + \mathrm {i} \eta )$ is well-defined (see Lemma B.1 (b) of the arXiv: 2301.03549 version of [Reference Cipolloni, Erdős, Henheik and Schröder21]). The scDos $\rho $ is a compactly supported Hölder- $1/3$ continuous function on $\mathbf {R}$ which is real-analytic on the set $\{ \rho> 0\}$ Footnote ⁶ . Moreover, for any small $\kappa> 0$ (independent of N), we define the $\kappa $ -bulk of the scDos as

(2.10) $$ \begin{align} \mathbf{B}_\kappa = \left\{ x \in \mathbf{R} \; : \; \rho(x) \ge \kappa^{1/3} \right\}\,, \end{align} $$

which is a finite union of disjoint compact intervals (see Lemma B.2 in the arXiv: 2301.03549 version of [Reference Cipolloni, Erdős, Henheik and Schröder21]). For $\Re z \in \mathbf {B}_\kappa $ , it holds that $\Vert M(z) \Vert \lesssim 1$ , as easily follows by taking the imaginary part of (2.8).

Definition 2.6 (Regular observables – One-point regularisation).

Fix $\kappa> 0$ and an energy $e\in \mathbf {B}_\kappa $ in the bulk. Given a matrix $A\in \mathbf {C}^{N\times N}$ , we define its one-point regularisation w.r.t. the energy e, denoted by $\mathring {A}^e$ , as

(2.11) $$ \begin{align} \mathring{A}=\mathring{A}^e:=A-\frac{\langle \Im M(e) A\rangle}{\langle \Im M(e)\rangle}\,. \end{align} $$

Moreover, we call A regular w.r.t. the energy e if and only if $A = \mathring {A}^e$ .

Notice that in the analysis of Wigner matrices without deformation, $D=0$ , in [Reference Cipolloni, Erdős and Schröder16, Reference Cipolloni, Erdős and Schröder17, Reference Cipolloni, Erdős and Schröder19, Reference Cipolloni, Erdős and Schröder20], M was a constant matrix and the regular observables were simply given by traceless matrices (i.e., $\mathring {A}= A-\langle A\rangle $ ). For deformed Wigner matrices, the concept of regular observables depends on the energy.

Next, we define the quantiles $\gamma _i$ of the density $\rho $ implicitly by

(2.12) $$ \begin{align} \int_{-\infty}^{\gamma_i} \rho(x)\,\mathrm{d} x=\frac{i}{N}, \qquad i\in[N]. \end{align} $$

We can now formulate the ETH in the bulk for deformed Wigner matrices which generalises the same result for Wigner matrices, $D=0$ , from [Reference Cipolloni, Erdős and Schröder16].

Theorem 2.7 (Eigenstate Thermalisation Hypothesis).

Let $\kappa> 0$ be an N-independent constant and fix a bounded deterministic $D =D^* \in \mathbf {C}^{N \times N}$ . Let $H = W + D$ be a deformed Wigner matrix, where W satisfies Assumption 2.1, and denote the orthonormal eigenvectors of H by $\{ \textbf {u}_i\}_{i \in [N]}$ . Then, for any deterministic $A\in \mathbf {C}^{N\times N}$ with $\Vert A \Vert \lesssim 1$ , it holds that

(2.13) $$ \begin{align} \max_{i, j}\left|\langle \textbf{u}_i,\mathring{A}^{\gamma_i}\textbf{u}_j\rangle \right| = \max_{i, j}\left|\langle \textbf{u}_i,A\textbf{u}_j\rangle-\delta_{ij}\frac{\langle A\Im M(\gamma_i)\rangle}{\langle \Im M(\gamma_i)\rangle} \right| \prec \frac{1}{\sqrt{N}}\,, \end{align} $$

where the maximum is taken over all $i,j \in [N]$ such that the quantiles $\gamma _i, \gamma _j \in \mathbf {B}_\kappa $ defined in (2.12) are in the $\kappa $ -bulk of the scDos $\rho $ .

This ‘Law of Large Numbers’-type result (2.13) is complemented by the corresponding Central Limit Theorem (2.14), which requires slightly strengthened assumptions on the deformation D.

The requirements on D in Assumption 2.8 are natural and they hold for typical applications; see Remark 2.12 later for more details. We can now formulate our result on the Gaussian fluctuations in the ETH which generalises the analogous result for Wigner matrices, $D=0$ from [Reference Cipolloni, Erdős and Schröder18].

Theorem 2.9 (Fluctuations in ETH).

Fix $\kappa ,\sigma> 0 \ N$ -independent constants and let $H = W + D$ be a deformed Wigner matrix, where W satisfies Assumption 2.1 and D satisfies Assumption 2.8. Denote the orthonormal eigenvectors of H by $\{ \textbf {u}_i\}_{i \in [N]}$ and fix an index $i \in [N]$ , such that the quantile $\gamma _i \in \mathbf {B}_\kappa $ defined in (2.12) is in the bulk. Then, for any deterministic Hermitian matrix $A\in \mathbf {C}^{N\times N}$ with $\Vert A \Vert \lesssim 1$ (which we assume to be real in the case of a real Wigner matrix) satisfying $\langle (A-\langle {A}\rangle )^2\rangle \ge \sigma $ , it holds that

(2.14) $$ \begin{align} \sqrt{\frac{\beta N }{2 \, \mathrm{Var}_{\gamma_i}(A)}}\left[\langle \textbf{u}_i,A\textbf{u}_i\rangle-\frac{\langle A\Im M(\gamma_i)\rangle}{\langle \Im M(\gamma_i)\rangle} \right]\Longrightarrow \mathcal{N}(0,1) \end{align} $$

in the sense of moments (see Footnote 4), whereFootnote ⁷

(2.15) $$ \begin{align} \mathrm{Var}_{\gamma_i}(A) := \frac{1}{\langle \Im M (\gamma_i)\rangle^2} \left( \big\langle \big(\mathring{A}^{\gamma_i} \Im M(\gamma_i)\big)^2\big\rangle - \frac{1}{2} \Re\left[ \frac{\big\langle \big(M(\gamma_i)\big)^2 \mathring{A}^{\gamma_i} \big\rangle^2}{1 - \big\langle \big(M(\gamma_i)\big)^2 \big\rangle} \right] \right) \,. \end{align} $$

This variance is strictly positive with an effective lower bound

(2.16) $$ \begin{align} \mathrm{Var}_{\gamma_i}(A)\ge c \, \langle (A-\langle{A}\rangle)^2\rangle \end{align} $$

for some constant $c = c(\kappa , \Vert D \Vert )> 0.$

In the following two remarks we comment on the condition $\langle (A-\langle {A}\rangle )^2\rangle \ge \sigma $ and on Assumption 2.8.

In the following Section 3, we will prove our main result, Theorem 2.2, assuming Theorems 2.7 and 2.9 on deformed Wigner matrices as inputs. These will be proven in Sections 4 and 5, respectively. Both proofs crucially rely on an averaged local law for two resolvents and two regular observables, Proposition 4.4, which we prove in Section 6. Several additional technical and auxiliary results are deferred to the Appendix.

3.1 Computation of the expectation (3.1)

As in the proof of Theorem 2.2 above, we condition on $W_2$ and work in the probability space of $W_1$ (i.e., we consider $p_2 W_2$ as a deterministic deformation of $p_1W_1$ ). This allows us to use Theorem 2.9 asFootnote ⁹

(3.5) $$ \begin{align} \mathbf{E}_{W_1}\langle \mathbf{u}_i,p_2W_2 \mathbf{u}_i\rangle = \frac{\langle p_2W_2\Im M_2(\gamma_{i,2}))\rangle}{\langle \Im M_2(\gamma_{i,2})\rangle}+\mathcal{O}_\prec\left(N^{-1/2-\epsilon}\right) \end{align} $$

for some constant $\epsilon> 0$ . Here, $M_2(z)$ , depending on $W_2$ , is the unique solution of the MDE

(3.6) $$ \begin{align} -\frac{1}{M_2(z)}=z-p_2W_2+p_1^2\langle M_2(z)\rangle\,, \end{align} $$

corresponding to the matrix $p_1W_1+p_2W_2$ , where $p_2W_2$ is considered a deformation, and $\gamma _{i,2}$ is the $i^{\mathrm {th}}$ quantile of the scDos $\rho _2$ corresponding to (3.6). The subscript ‘ $2$ ’ for $M_2$ , $\rho _2$ and $\gamma _{i,2}$ in (3.5) and (3.6) indicates that these objects are dependent on $W_2$ and hence random.

The Stieltjes transform $m_2(z)$ of $\rho _2$ is given by the implicit equation

$$ \begin{align*}m_2(z):=\langle M_2(z)\rangle = \frac{1}{p_2}\cdot \Big\langle \frac{1}{W_2-\frac{1}{p_2}(z+p_1^2m_2(z))} \Big\rangle \end{align*} $$

with the usual side condition $\Im z\cdot \Im m_2(z)>0$ . Applying the standard local law for the resolvent of $W_2$ on the right-hand side shows that

(3.7) $$ \begin{align} \Big| m_2(z) - \frac{1}{p_2} m_{\mathrm{sc}}(w_2)\Big|\prec \frac{1}{N |\Im w_2|}, \qquad w_2:=\frac{1}{p_2}(z+p_1^2m_2(z)), \end{align} $$

where $m_{\mathrm {sc}}$ is the Stieltjes transform of the standard semicircle law (i.e., it satisfies the quadratic equation

(3.8) $$ \begin{align} m_{\mathrm{sc}}(w)^2+w m_{\mathrm{sc}}(w)+1=0\, \end{align} $$

with the side condition $\Im w\cdot \Im m_{\mathrm {sc}}(w)>0$ ). Note that in (3.7), $w_2$ is random, it depends on $W_2$ , but the local law for $\langle (W_2-w)^{-1}\rangle $ holds uniformly in the spectral parameter $|\Im w|\ge N^{-1}$ . Hence, a standard grid argument and the Lipschitz continuity of the resolvents shows that it holds for any (random) w with $|\Im w|\ge N^{-1+\xi }$ with any fixed $\xi>0$ .

Applying (3.8) at $w=w_2$ together with (3.7) implies that

(3.9) $$ \begin{align} - \frac{1}{\Vert \textbf{p} \Vert \, m_2(z)} =\frac{z}{\Vert \textbf{p}\Vert} + \Vert \textbf{p}\Vert\, m_2(z) + \mathcal{O}_\prec \Big( \frac{1}{N |\Im w_2|}\Big). \end{align} $$

We view this relation as a small additive perturbation of the exact equation

(3.10) $$ \begin{align} - \frac{1}{m_{\mathrm{sc}}\big(\frac{z}{\Vert \textbf{p}\Vert}\big)} =\frac{z}{\Vert \textbf{p}\Vert} + m_{\mathrm{sc}}\big(\tfrac{z}{\Vert \textbf{p}\Vert}\big) \end{align} $$

to conclude

(3.11) $$ \begin{align} \left| \Vert \textbf{p} \Vert \, m_2(z) - m_{\mathrm{sc}}\big(\tfrac{z}{\Vert \textbf{p}\Vert}\big)\right|\prec \frac{1}{N|\Im z|}\,, \qquad |\Im z|\ge N^{-1+\xi}\,, \end{align} $$

using that $|\Im w_2|\gtrsim |\Im z|$ from the definition of $w_2$ in (3.7) and that $\Im z\cdot \Im m_2(z)>0$ . The conclusion (3.11) requires a standard continuity argument, starting from a z with a large imaginary part and continuously reducing the imaginary part by keeping the real part fixed (the same argument is routinely used in the proof of the local law for Wigner matrices, see, for example, [Reference Erdős and Yau26]).

The estimate (3.11) implies that the quantiles of $\rho _2$ satisfy the usual rigidity estimate (i.e.,

(3.12) $$ \begin{align} |\gamma_{2,i} -\gamma_i|\prec \frac{1}{N} \end{align} $$

for bulk indices $i\in [\kappa N, (1-\kappa )N]$ with any N-independent $\kappa>0$ ). Moreover, (3.11) also implies that for any z in the bulk of the semicircle (i.e., $|\Im m_{sc}(z)|\ge c>0$ for some $c>0$ ) independent of N, we have $|\Im m_2(z)|\ge c/2$ as long as $|\Im z|\ge N^{-1+\xi }$ . Using the definition of $w_2$ in (3.7) again, this shows $|\Im w_2|\sim |\Im w|$ for the deterministic $w:=\frac {1}{p_2}(z+p_1^2 m_{\mathrm {sc}}(z))$ for any z with $|\Im z|\ge N^{-1+\xi }$ . Feeding this information into (3.9) and viewing it again as a perturbation of (3.10) but with the improved deterministic bound $\mathcal {O}_\prec (1/(N|\Im w|))$ , we obtain

(3.13) $$ \begin{align} \left| \Vert \textbf{p} \Vert \, m_2(z) - m_{\mathrm{sc}}\big(\tfrac{z}{\Vert \textbf{p}\Vert}\big)\right|\prec \frac{1}{N|\Im w|}\,, \qquad \mbox{with}\quad w=\frac{1}{p_2}(z+p_1^2 m_{\mathrm{sc}}(z))\,, \end{align} $$

uniformly in $|\Im z|\ge N^{-1+\xi }$ . In particular, when z is in the bulk of the semicircle, then we have that

$$ \begin{align*}\left| \Vert \textbf{p} \Vert \, m_2(z) - m_{\mathrm{sc}}\big(\tfrac{z}{\Vert \textbf{p}\Vert}\big)\right|\prec \frac{1}{N},\end{align*} $$

and this relation holds even down to the real axis by the Lipschitz continuity (in fact, real analyticity) of the Stieltjes transform $m_2(z)$ in the bulk.

In the following, we will use the shorthand notation $A\approx B$ for two (families of) random variables A and B if and only if $\vert A - B \vert \prec N^{-1}$ . Evaluating (3.6) at $z=\gamma _{i,2}$ , we have

(3.14) $$ \begin{align} M_2(\gamma_{i,2})=\frac{1}{p_2}\cdot \frac{1}{W_2-w_{i,2}}\,, \qquad w_{i,2}:=\frac{1}{p_2}(\gamma_{i,2}+p_1^2\,m_2(\gamma_{i,2}))\,, \end{align} $$

and note that $w_{i,2} \approx w_i:=\frac {1}{p_2}\left (\gamma _{i}+\frac {p_1^2}{\lVert \boldsymbol{p}\rVert }\,m_{\mathrm {sc}}\left (\frac {\gamma _{i}}{\lVert \boldsymbol{p}\rVert }\right )\right )$ by (3.13) and since $\gamma _{i,2}\approx \gamma _i$ in the bulk by rigidity (3.12).

Now we are ready to evaluate the rhs. of (3.5). By elementary manipulations using (3.14), we can now write the rhs. of (3.5) as

(3.15) $$ \begin{align} \frac{\langle p_2W_2\Im M_2(\gamma_{i,2})\rangle}{\langle \Im M_2(\gamma_{i,2})\rangle} = \gamma_{i,2} + \frac{p_1^2}{p_2} \frac{\Im \big[ \langle (W_2 - w_{i,2})^{-1} \rangle^2\big] }{\Im \, \langle (W_2 - w_{i,2})^{-1} \rangle}\,. \end{align} $$

Using (3.13), we obtain

(3.16) $$ \begin{align} \langle (W_2-w_{i,2})^{-1}\rangle \approx \langle (W_2-w_i)^{-1}\rangle \approx m_{\mathrm{sc}}(w_i) \end{align} $$

with very high $W_2$ -probability. Continuing with (3.15) and using $\gamma _{i,2} \approx \gamma _i$ , we thus find

(3.17) $$ \begin{align} \frac{\langle p_2W_2\Im M_2(\gamma_{i,2}))\rangle}{\langle \Im M_2(\gamma_{i,2})\rangle} \approx \gamma_{i} + \frac{p_1^2}{p_2} \frac{\Im \big[ m_{\mathrm{sc}}(w_i)^2\big]}{\Im m_{\mathrm{sc}}(w_i)}\,. \end{align} $$

Next, we combine (3.8) with $p_2 m_2(\gamma _{i}) \approx m_{\mathrm {sc}}(w_i)$ from (3.7), (3.14) and (3.16) and find that

(3.18) $$ \begin{align} m_{\mathrm{sc}}(w_i)^2 \approx -\frac{p_2^2}{p_1^2+p_2^2}\left(1+\frac{1}{p_2}\gamma_{i} \, m_{\mathrm{sc}}(w_i)\right)\,. \end{align} $$

Hence, plugging (3.18) into (3.17), we deduce

$$ \begin{align*} \frac{\langle p_2W_2\Im M_2(\gamma_{i,2}))\rangle}{\langle \Im M_2(\gamma_{i,2})\rangle} \approx \left(1 - \frac{p_1^2}{p_1^2+p_2^2}\right)\gamma_{i} = \frac{p_2^2}{\|\boldsymbol{p}\|^2}\gamma_i\,. \end{align*} $$

This completes the proof of (3.1).

3.2 Computation of the variance (3.2)

As in the calculation of the expectation in Section 3.1, we first condition on $W_2$ and work in the probability space of $W_1$ . So, we apply Theorem 2.9 to the matrix $p_1W_1+p_2W_2$ , where the second term is considered a fixed deterministic deformation. Indeed, using the same notations as in Section 3.1, this gives that the lhs. of (3.2) equals

(3.19) $$ \begin{align} p_2^2\, \mathrm{Var}_{\gamma_{i,2}}\left(W_2\right)= p_2^2\frac{1}{\langle \Im M_2 (\gamma_{i,2})\rangle^2} \left( \big\langle \big(\mathring{W}_2^{\gamma_{i,2}} \Im M_2(\gamma_{i,2})\big)^2\big\rangle - \frac{p_1^2}{2} \Re\left[ \frac{\big\langle \big(M_2(\gamma_{i,2})\big)^2 \mathring{W}_2^{\gamma_{i,2}} \big\rangle^2}{1 - p_1^2\big\langle \big(M_2(\gamma_{i,2})\big)^2 \big\rangle} \right] \right) \end{align} $$

up to an additive error of order $\mathcal {O}_\prec \big (N^{-\epsilon }\big )$ , which will appear on the rhs. of (3.2). The factor $p_1^2$ in the second term of (3.19) is a natural rescaling caused by applying Theorem 2.9 to a deformation of $p_1W_1$ instead of a Wigner matrix $W_1$ . Further, we express $M_2$ in terms of a Wigner resolvent $G:=(W_2-w)^{-1}$ and use local laws not only for a single resolvent $\langle G\rangle $ but also their multi-resolvent versions for $\langle G^2\rangle $ and $\langle GG^*\rangle $ (see [Reference Cipolloni, Erdős and Schröder19]). With a slight abuse of notation we shall henceforth drop the subscript ‘ $2$ ’ in $\gamma _{i,2}$ and $w_{i,2}$ and replace them by their deterministic values $\gamma _i$ and $w_i$ , respectively, at a negligible error of order $N^{-1}$ exactly as in Section 3.1. Note that $\Im w_i\gtrsim 1$ for bulk indices i, so all resolvents below are stable and all denominators are well separated away from zero; this is needed to justify the $\approx $ relations below.

The first term in (3.19) can be rewritten as (here $G=G(w_i)$ and $m_{\mathrm {sc}}:=m_{\mathrm {sc}}(w_i)$ for brevity)

(3.20) $$ \begin{align} \left\langle\big(\mathring{W}_2^{\gamma_i} \Im M_2(\gamma_i)\big)^2\right\rangle &\approx \frac{1}{2} \Re\left({\left\vert \frac{w_i}{p_2}-\frac{\gamma_i}{p_1^2+p_2^2}\right\vert}^2\langle GG^*\rangle-\left(\frac{w_i}{p_2}-\frac{\gamma_i}{p_1^2+p_2^2}\right)^2\langle G^2\rangle\right)\nonumber\\ &\approx \frac{1}{2}\Re\left({\left\vert \frac{w_i}{p_2}-\frac{\gamma_i}{p_1^2+p_2^2}\right\vert}^2 \frac{\vert m_{\mathrm{sc}}\vert^2}{1-\vert m_{\mathrm{sc}}\vert^2}-\left(\frac{w_i}{p_2} -\frac{\gamma_i}{p_1^2+p_2^2}\right)^2\frac{m_{\mathrm{sc}}^2}{1-m_{sc}^2}\right) \nonumber\\ &\approx\frac{p_1^4}{2p_2^2(p_1^2+p_2^2)^2}\Re\left(\frac{1}{1-\vert m_{\mathrm{sc}}\vert^2} - \frac{1}{1-m_{\mathrm{sc}}^2} \right), \end{align} $$

where in the last step we used (3.18). Similarly, for the second term in (3.19), we have

(3.21) $$ \begin{align} \frac{\left\langle\big(M_2(\gamma_i)\big)^2 \mathring{W}_2^{\gamma_i}\right\rangle^2}{1 - p_1^2\left\langle \big(M(\gamma_i)\big)^2 \right\rangle} &\approx \frac{ \Big[ \frac{1}{p_2}\left\langle \frac{1}{p_2}G +\left(\frac{w_i}{p_2}-\frac{\gamma_i}{p_1^2+p_2^2}\right)G^2\right\rangle\Big]^2}{1-\frac{p_1^2}{p_2^2}\langle G^2\rangle} \approx \frac{m_{\mathrm{sc}}^2(p_2^2-(p_1^2+p_2^2)m_{\mathrm{sc}}^2)}{p_2^2(p_1^2+p_2^2)^2(1-m_{\mathrm{sc}}^2)}\,. \end{align} $$

Plugging (3.20) and (3.21) into (3.19) we obtain

(3.22) $$ \begin{align} p_2^2 \, \mathrm{Var}_{\gamma_i}(W_2) \approx \frac{2p_2^2+p_2\gamma_i\Re m_{\mathrm{sc}}}{\left( \Im m_{\mathrm{sc}}\right)^2}\cdot \frac{p_1^2p_2^2}{2(p_1^2+p_2^2)^2}\,. \end{align} $$

Taking the imaginary part of (3.18), we find that $|m_{\mathrm {sc}}|^2 \approx \frac {p_2^2}{p_1^2 + p_2^2}$ , and hence, using (3.8) again, we infer

(3.23) $$ \begin{align} \frac{1}{p_1^2 + p_2^2} \frac{2p_2^2+p_2\gamma_i\Re m_{sc}}{\left( \Im m_{\mathrm{sc}}\right)^2} \approx \frac{\Re [|m_{\mathrm{sc}}|^2 - m_{\mathrm{sc}}^2]}{(\Im m_{\mathrm{sc}})^2} = 2\,. \end{align} $$

Combining (3.22) and (3.23) with (3.19), this completes the proof of (3.2). This proves Lemma 3.1.

5.1 Perfect matching observable and proof of Theorem 2.9

We introduce the notation

(5.6) $$ \begin{align} p_{ij}=p_{ij}(t)=\langle \mathbf{u}_i,A \mathbf{u}_j\rangle-\delta_{ij}C_0\, \end{align} $$

with A being a fixed real symmetric deterministic matrix A and $C_0$ being a fixed constant independent of i. Note that compared to [Reference Cipolloni, Erdős and Schröder18, Reference Cipolloni, Erdős and Schröder20] in (5.6), we define the diagonal $p_{ii}$ without subtracting their expectation (see (2.13) above), but rather a generic constant $C_0$ which we will choose later (see (5.48) below). The reason behind this choice is that in the current setting, unlike in the Wigner case [Reference Cipolloni, Erdős and Schröder18, Reference Cipolloni, Erdős and Schröder20], the expectation of $p_{ii}$ is now i–dependent. Hence, the flow (5.10) below would not be satisfied if we had defined (5.7) with the centred $p_{ii}$ ’s.

To study moments of the $p_{ij}$ ’s, we use the particle representation introduced in [Reference Bourgade and Yau14] and further developed in [Reference Bourgade, Yau and Yin15, Reference Marcinek and Yau35]. A particle configuration, corresponding to a certain monomials of $p_{ij}$ ’s, can be encoded by a function $\boldsymbol {\eta }:[N]\to \mathbf {N}_0$ . The image $\eta _j=\boldsymbol {\eta }(j)$ denotes the number of particle at the site j, and $\sum _j\eta _j=n$ denotes the total number of particles. Additionally, given a particle configuration $\boldsymbol {\eta }$ , by $\boldsymbol {\eta }^{ij}$ , with $i\ne j$ , we denote a new particle configuration in which a particle at the site i moved to a new site j; if there is no particle in i, then $\boldsymbol {\eta }^{ij}=\boldsymbol {\eta }$ . We denote the set of such configuration by $\Omega ^n$ .

Fix a configuration $\boldsymbol {\eta }$ , and then we define the perfect matching observable (see [Reference Bourgade, Yau and Yin15, Section 2.3]):

(5.7) $$ \begin{align} f_{\boldsymbol{\lambda},t,C_0,C_1}(\boldsymbol{\eta}):= \frac{N^{n/2}}{ [2 C_1]^{n/2}} \frac{1}{(n-1)!!}\frac{1}{\mathcal{M}(\boldsymbol{\eta}) }\mathbf{E}\left[\sum_{G\in\mathcal{G}_{\boldsymbol{\eta}}} P(G)\Bigg| \boldsymbol{\lambda}\right] \,, \quad \mathcal{M}(\boldsymbol{\eta}):=\prod_{i=1}^N (2\eta_i-1)!!\, \end{align} $$

with n being the total number of particles in the configuration η. The sum in (5.7) is taken over $\mathcal {G}_{\boldsymbol {\eta }}$ , which denotes the set of perfect matchings on the complete graph with vertex set

$$\begin{align*}\mathcal{V}_{\boldsymbol{\eta}}:=\{(i,a): 1\le i\le n, 1\le a\le 2\eta_i\}\,. \end{align*}$$

We also introduced the short–hand notation

(5.8) $$ \begin{align} P(G):=\prod_{e\in\mathcal{E}(G)}p(e), \qquad p(e):=p_{i_1i_2}\,, \end{align} $$

where $e=\{(i_1,a_1),(i_2,a_2)\}\in \mathcal {V}_{\boldsymbol {\eta }}^2$ , and $\mathcal {E}(G)$ denotes the edges of G. Note that in (5.7) we took the conditional expectation with respect to the entire trajectories of the eigenvalues, λ = {λ(t)} _{t∈[0, T]} for some fixed $0<T\ll 1$ . We also remark that the definition (5.7) differs slightly from [Reference Cipolloni, Erdős and Schröder18, Eq. (3.9)] and [Reference Cipolloni, Erdős and Schröder20, Eq. (4.6)], since we now do not normalise by $\langle (A-\langle A\rangle )^2\rangle $ but using a different constant $C_1$ which we will choose later in the proof (see (5.48) below); this is a consequence of the fact that the diagonal overlaps $p_{ii}$ are not correctly centred and normalised. Note that we did not incorporate the factor $2$ in (5.7) into the constant $C_1$ , since $C_1$ will be chosen as a normalisation constant to compensate the size of the matrix A, while the factor $2$ represents the fact that diagonal overlaps, after the proper centering and normalisation depending on A and i, would be centred Gaussian random variable of variance two. Furthermore, we consider eigenvalues paths $\{\boldsymbol {\lambda }({t})\}_{{t}\in [{0},{T}]}$ which lie in the event

(5.9) $$ \begin{align} \widetilde{\Omega}=\widetilde{\Omega}_\xi,:= \Big\{ \sup_{0\le t \le T} \max_{i\in [N]} \eta_{\mathrm{f}}(\gamma_i(t))^{-1}| \lambda_i(t)-\gamma_i(t)| \le N^\xi\Big\} \end{align} $$

for any $\xi>0$ , where $\eta _{\mathrm {f}}(\gamma _i(t))$ is the local fluctuation scale defined as in [Reference Erdős, Krüger and Schröder28, Definition 2.4]. In most instances, we will use this rigidity estimate in the bulk regime when $\eta _{\mathrm {f}}(\gamma _i(t))\sim N^{-1}$ – at the edges $\eta _{\mathrm {f}}(\gamma _i(t))\sim N^{-2/3}$ . We recall that here $\gamma _i(t)$ denote the quantiles of $\rho _t$ defined as in (2.12). The fact that the event $ \widetilde {\Omega }$ holds with very high probability follows by [Reference Alt, Erdős, Krüger and Schröder4, Corollary 2.9].

By [Reference Bourgade, Yau and Yin15, Theorem 2.6], it follows that $f_{\boldsymbol{\lambda},t}$ is a solution of the parabolic discrete partial differential equation (PDE):

(5.10) $$ \begin{align} \partial_t f_{\boldsymbol{\lambda},t}&=\mathcal{B}(t)f_{\boldsymbol{\lambda},t}\,,\qquad\qquad\qquad\qquad\quad\qquad\qquad \end{align} $$

(5.11) $$ \begin{align}\!\mathcal{B}(t)f_{\boldsymbol{\lambda},t}&=\sum_{i\ne j} c_{ij}(t) 2\eta_i(1+2\eta_j)\big(f_{\boldsymbol{\lambda},t}(\boldsymbol{\eta}^{ij}) -f_{\boldsymbol{\lambda},t}(\boldsymbol{\eta})\big)\,, \end{align} $$

where

(5.12) $$ \begin{align} c_{ij}(t):= \frac{1}{N(\lambda_i(t) - \lambda_j(t))^2}\,. \end{align} $$

In the remainder of this section, we may often omit $\boldsymbol {\lambda }$ from the notation since the paths of the eigenvalues are fixed within this proof.

The main result of this section is the following Proposition 5.1, which will readily prove Theorem 2.9. For this purpose, we define a version of $f_t(\boldsymbol {\eta })$ with centred and rescaled $p_{ii}$ :

(5.13) $$ \begin{align} q_{\boldsymbol{\lambda},t}(\boldsymbol{\eta}):= \left(\prod_{i=1}^N\frac{1}{ \mathrm{Var}_{\gamma_i}(A)^{\eta_i/2}}\right) \frac{N^{n/2}}{2^{n/2} (n-1)!!}\frac{1}{\mathcal{M}(\boldsymbol{\eta}) }\mathbf{E}\left[\sum_{G\in\mathcal{G}_{\boldsymbol{\eta}}} Q(G)\Bigg| \boldsymbol{\lambda}\right] \end{align} $$

with $\mathring {A}^{\gamma _i}$ denoting the regular component of A defined as in (2.11):

$$\begin{align*}\mathring{A}^{\gamma_i}:=A-\frac{\langle A\Im M(\gamma_i)\rangle }{\langle \Im M(\gamma_i)\rangle}, \end{align*}$$

and

(5.14) $$ \begin{align} Q(G):=\prod_{e\in\mathcal{E}(G)}q(e), \qquad q(e):=\langle \mathbf{u}_{i_1},\mathring{A}^{\gamma_{i_1}} \mathbf{u}_{i_2}\rangle. \end{align} $$

Note that the definition in (5.14) is not asymmetric for $i_1\ne i_2$ , since in this case, $\langle \mathbf {u}_{i_1},\mathring {A}^{\gamma _{i_1}} \mathbf {u}_{i_2}\rangle =\langle \mathbf {u}_{i_1},\mathring {A}^{\gamma _{i_2}} \mathbf {u}_{i_2}\rangle $ .

We now comment on the main difference between $q_t$ and $f_t$ from (5.13) and (5.7), respectively. First of all, we notice the $q(e)$ ’s in (5.14) are slightly different compared with the $p(e)$ ’s from (5.6). In particular, we choose the $q(e)$ ’s in such a way that the diagonal overlaps have very small expectation (i.e., much smaller than their fluctuations size). The price to pay for this choice is that the centering is i–dependent; hence, $q_t$ is not a solution of an equation of the form (5.10)-(5.11). We also remark that later within the proof, $C_0$ from (5.6) will be chosen as

$$\begin{align*}C_0 = \frac{\langle A\Im M(\gamma_{i_0})\rangle}{\langle \Im M(\gamma_{i_0})\rangle} \end{align*}$$

for some fixed $i_0$ such that $\gamma _{i_0}\in \mathbf {B}_\kappa $ is in the bulk (recall (2.10)). The idea behind this choice is that the analysis of the flow (5.10)–(5.11) will be completely local. We can thus fix a base point $i_0$ and ensure that the corresponding overlap is exactly centred; then the nearby overlaps for indices $|i-i_0|\le K$ , for some N–dependent $K>0$ , will not be exactly centred, but their expectation will be very small compared to the size of their fluctuations:

$$\begin{align*}\frac{\langle A\Im M(\gamma_{i_0})\rangle}{\langle \Im M(\gamma_{i_0})\rangle}-\frac{\langle A\Im M(\gamma_i)\rangle}{\langle \Im M(\gamma_i)\rangle}=\mathcal{O}\left(\frac{K}{N}\right)\,. \end{align*}$$

A consequence of this choice is also that the normalisation for $q_t$ and $f_t$ is different: for $q_t$ , we chose a normalisation that is i–dependent, whereas for $f_t$ , the normalisation $C_1$ is i–independent and later, consistently with the choice of $C_0$ , it will be chosen as

$$\begin{align*}C_1=\mathrm{Var}_{\gamma_{i_0}}(A)^{n/2}, \end{align*}$$

which is exactly the normalisation that makes $f_t(\boldsymbol {\eta })=1$ when $\boldsymbol {\eta }$ is such that $\eta _{i_0}=n$ and zero otherwise.

Proposition 5.1. For any $n\in \mathbf {N}$ , there exists $c(n)>0$ such that for any $\epsilon>0$ , and for any $T\ge N^{-1+\epsilon }$ , it holds

(5.15) $$ \begin{align} \sup_{\boldsymbol{\eta}}\ \left|q_T(\boldsymbol{\eta})- \mathbf{1}(\mathrm{n}\,\, \mathrm{even})\right|\lesssim N^{-c(n)}\,, \end{align} $$

with very high probability. The supremum is taken over configurations $\boldsymbol {\eta }$ supported on bulk indices, and the implicit constant in (5.15) depends on n and $\epsilon $ .

Proof of Theorem 2.9.

Fix $n\in \mathbf {N}$ , an index i such that $\gamma _i \in \mathbf {B}_\kappa $ is in the bulk, and choose a configuration $\boldsymbol {\eta }$ such that $\eta _i=n$ and $\eta _j=0$ for any $j\ne i$ . Then by Proposition 5.1, we conclude that

$$\begin{align*}\mathbf{E}\left[\sqrt{\frac{N}{2\mathrm{Var}_{\gamma_i}(A)}}\langle \mathbf{u}_i(T),\mathring{A}^{\gamma_i}\mathbf{u}_{i}(T)\rangle\right]^n=\mathbf{1}({n}\,\, \mathrm{even})({n}-1)!!+\mathcal{O}\left( {N}^{-{c}({n})}\right), \end{align*}$$

with $T=N^{-1+\epsilon }$ , for some very small fixed $\epsilon>0$ , and $c(n)>0$ . Here $\mathring {A}^{\gamma _i}$ is defined in (2.11) and $\mathrm {Var}_{\gamma _i}(A)$ is defined in (2.15). Then, by a standard GFT argument (see, for example, [Reference Cipolloni, Erdős and Schröder20, Appendix B]), we see that

$$\begin{align*}\mathbf{E}\left[\sqrt{\frac{N}{2\mathrm{Var}_{\gamma_i}(A)}}\langle \mathbf{u}_i(T),\mathring{A}^{\gamma_i}\mathbf{u}_{i}(T)\rangle\right]^n=\mathbf{E}\left[\sqrt{\frac{N}{2\mathrm{Var}_{\gamma_i}(A)}}\langle \mathbf{ u}_i(0),\mathring{A}^{\gamma_i}\mathbf{u}_{i}(0)\rangle\right]^n+\mathcal{O}\left( N^{-c(n)}\right). \end{align*}$$

This shows that the Gaussian component added by the dynamics (5.3) can be removed at the price of a negligible error implying (2.14).

The lower bound on the variance (2.16) is an explicit calculation relying on the definition of M from (2.8). In particular, we use that

1. (i) A, and hence $\mathring {A}^{\gamma _i}$ , are self-adjoint;

2. (ii) $\Im M(\gamma _i) \ge g$ for some $g = g(\kappa , \Vert D \Vert )> 0$ since we are in the bulk;

3. (iii) $\langle \mathring {A}^{\gamma _i} \Im M(\gamma _i)\rangle = 0$ by definition of the regularisation;

4. (iv) $[\Re M(\gamma _i), \Im M(\gamma _i)] = 0$ from (2.8).

Then, after writing $\mathrm {Var}_{\gamma _i}(A)$ as a sum of squares and abbreviating $\Im M = \Im M(\gamma _i)$ , we find

$$ \begin{align*} \mathrm{Var}_{\gamma_i}(A) \ge \frac{\big\langle \big(\sqrt{\Im M} \big[ A - \frac{\langle A (\Im M)^2\rangle}{ \langle (\Im M)^2\rangle}\big]\sqrt{\Im M} \big)^2 \big\rangle}{\langle (\Im M)^2\rangle} \ge g^2 \frac{\big\langle \big[ A - \frac{\langle A (\Im M)^2\rangle}{ \langle (\Im M)^2\rangle}\big]^2 \big\rangle}{\langle (\Im M)^2\rangle} \ge \frac{g^2}{\langle (\Im M)^2\rangle} \langle (A - \langle A \rangle)^2 \rangle\,, \end{align*} $$

where in the last step we used the trivial variational principle $\langle (A - \langle A \rangle )^2 \rangle = \inf _{t \in \mathbf {R}} \langle ( A - t )^2 \rangle $ . This completes the proof of Theorem 2.9.

5.2 DBM analysis

Similar to [Reference Cipolloni, Erdős and Schröder18, Section 4.1] and [Reference Cipolloni, Erdős and Schröder20, Section 4.2], we introduce an equivalent particle representation to encode moments of the $p_{ij}$ ’s. In particular, here, and previously in [Reference Cipolloni, Erdős and Schröder18, Reference Cipolloni, Erdős and Schröder20], we relied on the particle representation (5.16)–(5.18) below since our arguments heavily build on [Reference Marcinek and Yau35], which use this latter representation.

Consider a particle configuration $\boldsymbol {\eta }\in \Omega ^n$ for some fixed $n\in \mathbf {N}$ (i.e., $\boldsymbol {\eta }$ is such that $\sum _j\eta _j=n$ ). We now define the new configuration space

(5.16) $$ \begin{align} \Lambda^n:= \{ \mathbf{x}\in [N]^{2n} \, : \, n_i(\mathbf{x}) \mbox{ is even for every } i\in [N]\big\}, \end{align} $$

where

(5.17) $$ \begin{align} n_i(\mathbf{x}):=|\{a\in [2n]:x_a=i\}| \end{align} $$

for all $i\in \mathbf {N}$ .

By the correspondence

(5.18) $$ \begin{align} \boldsymbol{\eta} \leftrightarrow \mathbf{x}\qquad \eta_i=\frac{n_i( \mathbf{x})}{2}\,, \end{align} $$

it is easy to see that these two representations are basically equivalent. The only difference is that x uniquely determines η, but η determines only the coordinates of x as a multi-set and not its ordering.

From now on, given a function f defined on $\Omega ^n$ , we will always consider functions g on $\Lambda ^n\subset [N]^{2n}$ defined by

$$\begin{align*}f(\boldsymbol{\eta})= f(\phi(\mathbf{x}))= g(\mathbf{x}), \end{align*}$$

with ϕ:Λ ⁿ →Ω ⁿ , ϕ(x) = η being the projection from the x-configuration space to the η-configuration space using (5.18). We thus defined the observable

(5.19) $$ \begin{align} g_t(\mathbf{x})=g_{\boldsymbol{\lambda},t}(\mathbf{x}):= f_{\boldsymbol{\lambda},t}( \phi(\mathbf{x}))\, \end{align} $$

with f _{λ, t} from (5.7). Note that $g_t(\mathbf {x})$ is equivariant under permutation of the arguments (i.e., it depends on x only as a multi–set). Similarly, we define

(5.20) $$ \begin{align} r_t(\mathbf{x})=r_{\boldsymbol{\lambda},t}(\mathbf{x}):= q_{\boldsymbol{\lambda},t}( \phi(\mathbf{x}))\,. \end{align} $$

We remark that $g_t$ and $r_t$ are the counterpart of $f_t$ and $q_t$ , respectively, in the $\textbf {x}$ -configuration space.

We can thus now write the flow (5.10)–(5.11) in the $\textbf {x}$ –configuration space:

(5.21) $$ \begin{align} \!\!\!\!\partial_t g_t(\mathbf{x})&=\mathcal{L}(t)g_t(\mathbf{x})\qquad\qquad \end{align} $$

(5.22) $$ \begin{align} \mathcal{L}(t):=\sum_{j\ne i}\mathcal{L}_{ij}(t), \quad \mathcal{L}_{ij}(t)g(\mathbf{x}):&= c_{ij}(t) \frac{n_j(\mathbf{x})+1}{n_i(\mathbf{x})-1}\sum_{a\ne b\in[2 n]}\big(g(\mathbf{x}_{ab}^{ij})-g(\mathbf{x})\big), \end{align} $$

where

(5.23) $$ \begin{align} \mathbf{x}_{ab}^{ij}:=\mathbf{x}+\delta_{x_a i}\delta_{x_b i} (j-i) (\mathbf{e}_a+\mathbf{e}_b) \end{align} $$

with $\mathbf {e}_a\in \mathbf {R}^{2n}$ denoting the standard unit vector (i.e., $\mathbf {e}_a(b)=\delta _{ab}$ ). We remark that this flow is map on functions defined on Λ ⁿ ⊂ [N]²ⁿ which preserves equivariance.

For the following analysis, it is convenient to define the scalar product and the natural measure on $\Lambda ^n$ :

(5.24) $$ \begin{align} \langle f, g\rangle_{\Lambda^n}=\langle f, g\rangle _{\Lambda^n, \pi}:=\sum_{\mathbf{x}\in \Lambda^n}\pi(\mathbf{x})\bar f(\mathbf{x})g(\mathbf{x}), \qquad \pi(\mathbf{x}):=\prod_{i=1}^N ((n_i(\mathbf{x})-1)!!)^2, \end{align} $$

as well as the norm on $L^p(\Lambda ^n)$ :

(5.25) $$ \begin{align} \lVert f\rVert_p=\lVert f\rVert_{L^p(\Lambda^n,\pi)}:=\left(\sum_{\mathbf{x}\in \Lambda^n}\pi(\mathbf{x})|f(\mathbf{x})|^p\right)^{1/p}. \end{align} $$

The operator $\mathcal {L}=\mathcal {L}(t)$ is symmetric with respect to the measure $\pi $ and it is a negative in $L^2(\Lambda ^n)$ , with associated Dirichlet form (see [Reference Marcinek34, Appendix A.2]):

$$\begin{align*}D(g)=\langle g, (-\mathcal{L}) g\rangle_{\Lambda^n} = \frac{1}{2} \sum_{\mathbf{x}\in \Lambda^n}\pi(\mathbf{x}) \sum_{i\ne j} c_{ij}(t) \frac{n_j(\mathbf{x})+1}{n_i(\mathbf{x})-1} \sum_{a\ne b\in[2 n]}\big|g(\mathbf{x}_{ab}^{ij})-g(\mathbf{x})\big|^2. \end{align*}$$

Finally, by $\mathcal {U}(s,t)$ , we denote the semigroup associated to $\mathcal {L}$ ; that is, for any $0\le s\le t$ , it holds

(5.26) $$ \begin{align} \partial_t\mathcal{U}(s,t)=\mathcal{L}(t)\mathcal{U}(s,t), \quad \mathcal{U}(s,s)=I. \end{align} $$

5.3 Short-range approximation

As a consequence of the singularity of the coefficients $c_{ij}(t)$ in (5.22), the main contribution to the flow (5.21) comes from nearby eigenvalues; hence, its analysis will be completely local. For this purpose, we define the sets

(5.27) $$ \begin{align} \mathcal{J}=\mathcal{J}_\kappa:=\{ i\in [N]:\, \gamma_i(0)\in \mathbf{B}_{\kappa}\}, \end{align} $$

which correspond to indices with quantiles $\gamma _i(0)$ (recall (2.12)) in the bulk.

Fix a point $\mathbf {y}\in \mathcal {J}^{2n}$ and an N-dependent parameter K such that $1\ll K\ll \sqrt {N}$ . We remark that $\mathbf {y}\in \mathcal {J}^{2n}$ will be fixed for the rest of the analysis. Next, we define the averaging operator as a simple multiplication operator by a ‘smooth’ cut-off function:

(5.28) $$ \begin{align} \mathrm{Av}(K,\mathbf{y})h(\mathbf{x}):=\mathrm{Av}(\mathbf{x};K,\mathbf{y})h(\mathbf{x}), \qquad \mathrm{Av}(\mathbf{x}; K, \mathbf{y}):=\frac{1}{K}\sum_{j=K}^{2K-1} \mathbf{1}(\lVert\mathbf{x}-\mathbf{y}\rVert_1<\mathrm{j}), \end{align} $$

with $\lVert\mathbf{x}-\mathbf{y}\rVert_1 {:=} \sum^{2n}_{a=1} |x_a-y_a|$ . For notational simplicity, we may often omit $K,\textbf {y}$ from the notation since they are fixed throughout the proof:

(5.29) $$ \begin{align} \mathrm{Av}(\textbf{x})=\mathrm{Av}(\mathbf{x};K,\mathbf{y})h(\mathbf{x}), \qquad \mathrm{Av} h(\textbf{x})=\mathrm{Av}((\textbf{x}))h((\textbf{x})). \end{align} $$

Additionally, fix an integer ℓ with 1 ≪ ℓ ≪ K and define the short-range coefficients

(5.30) $$ \begin{align} c_{ij}^{\mathcal{S}}(t):=\begin{cases} c_{ij}(t) &\mathrm{if}\,\, i,j\in \mathcal{J} \,\, \mathrm{and}\,\, |i-j|\le \ell \\ 0 & \mathrm{otherwise}, \end{cases} \end{align} $$

where c _ij (t) is defined in (5.12). The parameter $\ell $ is the length of the short-range interaction.

We now define a short–range approximation of $r_t$ , with $r_t$ defined in (5.20). Note that in the definition of the short–range flow (5.31) below, there is a slight notational difference compared to [Reference Cipolloni, Erdős and Schröder18, Section 4.2] and [Reference Cipolloni, Erdős and Schröder20, Section 4.2.1]: we now choose an initial condition $h_0$ which depends on $r_0$ rather than $g_0$ . This minor difference is caused by the fact that in [Reference Cipolloni, Erdős and Schröder18, Reference Cipolloni, Erdős and Schröder20], the observable $g_t$ was already centred and rescaled, whereas in the current case, the centred and rescaled version of $g_t$ is given by $r_t$ . Hence, the definition in (5.31) is still conceptually the same as the one in [Reference Cipolloni, Erdős and Schröder18, Reference Cipolloni, Erdős and Schröder20] (see also the paragraph above (5.47) for a more detailed explanation). We point out that we make this choice to ensure that the infinite norm of the short-range approximation is always bounded by $N^\xi $ (see below (5.32)). The short-range approximation $h_t=h_t(\mathbf {x})$ is defined as the unique solution of the parabolic equation

(5.31) $$ \begin{align} \begin{aligned} \partial_t h_t(\mathbf{x}; \ell, K,\mathbf{y})&=\mathcal{S}(t) h_t(\mathbf{x}; \ell, K,\mathbf{y})\\ h_0(\mathbf{x};\ell, K,\mathbf{y})=h_0(\mathbf{x};K,\mathbf{y}):&=\mathrm{Av}(\mathbf{x}; K,\mathbf{y})(r_0(\mathbf{x})-\mathbf{1}(\mathrm{n} \,\, \mathrm{even})), \end{aligned} \end{align} $$

where

(5.32) $$ \begin{align} \mathcal{S}(t):=\sum_{j\ne i}\mathcal{S}_{ij}(t), \quad \mathcal{S}_{ij}(t)h(\mathbf{x}):=c_{ij}^{\mathcal{S}}(t)\frac{n_j(\mathbf{x})+1}{n_i(\mathbf{x})-1}\sum_{a\ne b\in [2n]}\big(h(\mathbf{x}_{ab}^{ij})-h(\mathbf{ x})\big). \end{align} $$

In the remainder of this section, we may often omit K, $\mathbf {y}$ and $\ell $ from the notation, since they are fixed for the rest of the proof. We conclude this section defining the transition semigroup $\mathcal {U}_{\mathcal {S}}(s,t)=\mathcal {U}_{\mathcal {S}}(s,t;\ell )$ associated to the short-range generator $\mathcal {S}(t)$ . Note that $\lVert h_t\rVert _\infty \le N^\xi $ , for any $t\ge 0$ and any small $\xi>0$ , since $\mathcal {U}_{\mathcal {S}}(s,t)$ is a contraction and $\lVert h_0\rVert _\infty \le N^\xi $ by (2.13), as a consequence of $h_t(\textbf {x})$ being supported on $\textbf {x}\in \mathcal {J}^{2n}$ .

5.4 $L^2$ –estimates

To prove the $L^\infty $ –bound in Proposition 5.1, we first prove an $L^2$ –bound in Proposition 5.3 below and then use an ultracontractivity argument for the parabolic PDE (5.21) (see [Reference Cipolloni, Erdős and Schröder18, Section 4.4]) to get an $L^\infty $ –bound. To get an $L^2$ –bound, we will analyse $h_t$ , the short–range version of the observable $g_t$ from (5.19), and then we will show that $h_t$ and $g_t$ are actually close to each other using the following finite speed of propagation (see [Reference Cipolloni, Erdős and Schröder18, Proposition 4.2, Lemmas 4.3–4.4]):

Lemma 5.2. Let 0 ≤ s ₁ ≤ s ₂ ≤ s ₁ + ℓN ⁻¹ and f be a function on Λ ⁿ . Then for any x ∈Λ ⁿ supported on J, it holds

(5.33) $$ \begin{align} \Big| (\mathcal{U}(s_1,s_2)-\mathcal{U}_{\mathcal{S}}(s_1,s_2;\ell) ) f(\mathbf{x}) \Big|\lesssim N^{1+n\xi}\frac{s_2-s_1}{\ell} \| f\|_\infty\,, \end{align} $$

for any small ξ > 0. The implicit constant in (5.15) depends on n, ϵ, δ.

To estimate several terms in the analysis of (5.31), we will rely on the multi–resolvent local laws from Proposition 4.4 (in combination with the extensions in Lemma A.1 in Lemma A.2). For this purpose, for a small ω > 2ξ > 0, we define the very high probability event (see Lemmas A.1–A.2)

(5.34) $$ \begin{align} \small & \widehat{\Omega}=\widehat{\Omega}_{\omega, \xi}:= \nonumber\\ &\bigcap_{\substack{e_i\in\mathbf{B}_\kappa, \atop |\Im z_i|\ge N^{-1+\omega}}}\Bigg[\bigcap_{k=2}^n \left\{\sup_{0\le t \le T}\Big|\langle G_t(z_1)\mathring{A}_1\dots G_t(z_k)\mathring{A}_k\rangle-\mathbf{1}({k}=2)\langle {M}(\mathrm{z}_1, \mathring{{A}}_1, {z}_2) \mathring{{A}}_2\rangle\Big|\le \frac{N^{\xi+k/2-1}}{\sqrt{N\eta}}\right\} \nonumber\\ &\qquad\quad\cap \left\{\sup_{0\le t \le T}\big|\langle G_t(z_1)\mathring{A}_1\rangle\big|\le \frac{N^\xi}{N\sqrt{|\Im z_1|}}\right\}\Bigg] \nonumber\\ &\bigcap_{z_1,z_2: e_1\in\mathbf{B}_\kappa,\atop |e_1-e_2|\ge c_1, |\Im z_i|\ge N^{-1+\omega}} \left\{\sup_{0\le t \le T}\big|\langle (G_t(z_1)B_1G_t(z_2)B_2\rangle\big|\le N^\xi\right\}\,, \end{align} $$

where $\mathring {A}_1,\dots ,\mathring {A}_k$ are regular matrices defined as in Definition 4.2 (here we used the short–hand notation $\mathring {A}_i=\mathring {A}_i^{z_i,z_{i+1}}$ ),

(5.35) $$ \begin{align} \langle M(z_1, \mathring{A}_1, z_2) \mathring{A}_2\rangle=\langle M(z_1)\mathring{A}_1 M(z_2)\mathring{A}_2\rangle +\frac{\langle M(z_1)\mathring{A}_1M(z_2)\rangle\langle M(z_2)\mathring{A}_2M(z_1)\rangle}{1-\langle M(z_1)M(z_2)\rangle}\,, \end{align} $$

η:= min{|ℑz _i |:i ∈ [k]}, $c_1>0$ is a fixed small constant, and $B_1,B_2$ are norm bounded deterministic matrices. We remark that for $|e_1-e_2|\ge c_1$ , we have the norm bound $\lVert M(z_1,B_1, z_2)\rVert \lesssim 1$ , with $M(z_1,B_1, z_2)$ being defined in (4.4). Then, by standard arguments (see, for example, [Reference Cipolloni, Erdős and Schröder20, Eq. (4.30)]), we conclude the bound (recall that $\widetilde {\Omega }_\xi $ from (5.9) denotes the rigidity event)

(5.36) $$ \begin{align} \max_{i,j\in \mathcal{J}}|\langle\mathbf{u}_i(t), A \mathbf{u}_j(t)\rangle |\le \frac{N^{\omega}}{\sqrt{N}} \qquad \,\, \mbox{on } \;\widehat{\Omega}_{\omega,\xi} \cap\widetilde{\Omega}_\xi\,, \end{align} $$

simultaneously for all $i,j\in \mathcal {J}$ and $0\le t\le T$ . Additionally, using the notation $\rho _{i,t}:=|\Im \langle M_t(z_i)\rangle |$ on $\widehat {\Omega }_{\omega ,\xi }\cap \widetilde {\Omega }_\xi $ , it also holds that

(5.37) $$ \begin{align} |\langle\mathbf{u}_i(t), A \mathbf{u}_j(t)\rangle|\le N^\omega \sqrt{\frac{\langle \Im G(\gamma_i(t)+\mathrm{i}\eta)A\Im G(\gamma_j(t)+\mathrm{i}\eta)A \rangle}{N\rho_{i,t}\rho_{j,t}}}\lesssim \frac{N^\omega}{N^{1/4}}\,, \end{align} $$

when one among i and j is in the bulk and $|i-j|\ge c N$ , for some small constant c depending on $c_1$ from (5.34). Here we used that for the index in the bulk – say i – we have $\rho _{i,t}\sim 1$ and for the other index $\rho _{j,t}\gtrsim N^{-1/2}$ as a consequence of $\eta \gg N^{-1}$ . We point out that this nonoptimal bound $N^{-1/4}$ , instead of the optimal $N^{-1/2}$ , follows from the fact that the bound from Lemma A.2 is not optimal when one of the two spectral parameters is close to an edge; this is exactly the same situation as in [Reference Cipolloni, Erdős and Schröder20, Eq. (4.31)], where we get an analogous nonoptimal bound for overlaps of eigenvectors that are not in the bulk.

We are now ready to prove the main technical proposition of this section. Note the additional term $KN^{-1/2}$ in the error $\mathcal {E}$ in (5.39) compared to [Reference Cipolloni, Erdős and Schröder18, Proposition 4.2] and [Reference Cipolloni, Erdős and Schröder20, Proposition 4.4]; this is a consequence of the fact the $p_{ii}$ ’s in (5.6) are not correctly centred. We stress that the base point $\textbf {y}$ in Proposition 5.3 is fixed throughout the remainder of this section.

Proposition 5.3. For any parameters satisfying N ⁻¹ ≪ η ≪ T ₁ ≪ ℓN ⁻¹ ≪ KN ⁻¹ ≪ N ^−1/2, and any small ϵ, ξ > 0, it holds

(5.38) $$ \begin{align} \lVert h_{T_1}(\cdot; \ell, K, \mathbf{y})\rVert_2\lesssim K^{n/2}\mathcal{E}, \end{align} $$

with

(5.39) $$ \begin{align} \mathcal{E}:= N^{n\xi}\left(\frac{N^\epsilon\ell}{K}+\frac{NT_1}{\ell}+\frac{N\eta}{\ell}+\frac{N^\epsilon}{\sqrt{N\eta}}+\frac{1}{\sqrt{K}}+\frac{K}{\sqrt{N}}\right), \end{align} $$

uniformly in particle configurations $\textbf {y}$ , such that $y_a=i_0$ for any $a\in [2n]$ and $i_0\in \mathcal {J}$ , and eigenvalue trajectory λ in the high probability event $\widetilde{\Omega}_\xi \cap \widehat{\Omega}_{\omega,\xi} $ .

Proof. The proof of this proposition is very similar to the one of [Reference Cipolloni, Erdős and Schröder18, Proposition 4.2] and [Reference Cipolloni, Erdős and Schröder20, Proposition 4.4]. We thus only explain the main differences here. In the following, the star over $\sum $ denotes that the summation runs over two n-tuples of fully distinct indices. The key idea in this proof is that in order to rely on the multi–resolvent local laws (5.34), we replace the operator $\mathcal {S}(t)$ in (5.31) with the new operator

(5.40) $$ \begin{align} \mathcal{A}(t):=\sum_{\mathbf{i}, \mathbf{j}\in [N]^n}^*\mathcal{A}_{ \mathbf{i}\mathbf{j}}(t), \quad \mathcal{A}_{\mathbf{i}\mathbf{j}}(t)h(\mathbf{x}):=\frac{1}{\eta}\left(\prod_{r=1}^n a_{i_r,j_r}^{\mathcal{S}}(t)\right)\sum_{\mathbf{a}, \mathbf{b}\in [2n]^n}^*(h(\mathbf{x}_{\mathbf{a}\mathbf{b}}^{\mathbf{i}\mathbf{j}})-h(\mathbf{x})), \end{align} $$

where

(5.41) $$ \begin{align} a_{ij}=a_{ij}(t):=\frac{\eta}{N((\lambda_i(t)-\lambda_j(t))^2+\eta^2)}, \end{align} $$

and a _ij ^S are their short-range version defined as in (5.30) with $c_{ij}(t)$ replaced with $a_{ij}(t)$ , and

(5.42) $$ \begin{align} \mathbf{x}_{\mathbf{a}\mathbf{b}}^{\mathbf{i}\mathbf{j}}:=\mathbf{x}+\left(\prod_{r=1}^n \delta_{x_{a_r}i_r}\delta_{x_{b_r}i_r}\right)\sum_{r=1}^n (j_r-i_r) (\mathbf{e}_{a_r}+\mathbf{e}_{b_r}). \end{align} $$

The main idea behind this replacement is that infinitesimally $\mathcal {S}(t)$ averages only in one direction at a time, whereas $\mathcal {A}(t)$ averages in all directions simultaneously. This is expressed by the fact that $\mathbf {x}^{ij}_{ab}$ from (5.23) changes two entries of $\mathbf {x}$ per time; instead, $\mathbf {x}_{\mathbf {a}\mathbf {b}}^{\mathbf {i}\mathbf {j}}$ changes all the coordinates of $\mathbf {x}$ at the same time (i.e., let $\mathbf {i}:=(i_1,\dots , i_n), \mathbf {j}:=(j_1,\dots , j_n)\in [N]^n$ , with $\{i_1,\dots ,i_n\}\cap \{j_1,\dots , j_n\}=\emptyset $ ; then $\mathbf {x}_{\mathbf {a}\mathbf {b}}^{\mathbf {i}\mathbf {j}}\ne \mathbf {x}$ if and only if for all $r\in [n]$ , it holds that $x_{a_r}=x_{b_r}=i_r$ ). Technically, the replacement of $\mathcal {S}(t)$ by $\mathcal {A}(t)$ can be performed at the level of Dirichlet forms.

Lemma 5.4 (Lemma 4.6 of [Reference Cipolloni, Erdős and Schröder18]).

Let S(t), $\mathcal{A}$ (t) be defined in (5.32) and (5.40), respectively, and let $\mu $ denote the uniform measure on $\Lambda ^n$ for which $\mathcal{A}$ (t) is reversible. Then there exists a constant C(n) > 0 such that

(5.43) $$ \begin{align} \langle h, \mathcal{S}(t) h\rangle_{\Lambda^n, \pi}\le C(n) \langle h, \mathcal{A}(t) h\rangle_{\Lambda^n,\mu}\le 0 \end{align} $$

for any h ∈ L ²(Λ ⁿ ), on the very high probability event $\widetilde{\Omega}_\xi \cap \widehat{\Omega}_{\omega,\xi}$ .

We start noticing the fact that by (5.31), it follows

(5.44) $$ \begin{align} \partial_t \lVert h_t\rVert_2^2=2\langle h_t, \mathcal{S}(t) h_t\rangle_{\Lambda^n}. \end{align} $$

Then, combining this with (5.43), and using that ${\mathbf x}_{{\mathbf a}{\mathbf b}}^{{\mathbf i}{\mathbf j}}={\mathbf x}$ unless ${\mathbf x}_{a_r}={\mathbf x}_{b_r}=i_r$ for all r ∈ [n], we conclude that

(5.45) $$ \begin{align} \begin{aligned} \partial_t \lVert h_t\rVert_2^2&\le C(n) \langle h_t,\mathcal{A}(t) h_t\rangle_{\Lambda^n,\mu} \\[4pt] &=\frac{C(n) }{2\eta}\sum_{\mathbf{x}\in \Lambda^n}\sum_{\mathbf{i},\mathbf{j}\in [N]^n}^* \left(\prod_{r=1}^n a_{i_r j_r}^{\mathcal{S}}(t)\right)\sum_{\mathbf{a}, \mathbf{b}\in [2n]^n}^*\overline{h_t}(\mathbf{ x})\big(h_t(\mathbf{x}_{\mathbf{a}\mathbf{b}}^{\mathbf{i} \mathbf{j}})-h_t(\mathbf{x})\big)\left(\prod_{r=1}^n \delta_{x_{a_r}i_r}\delta_{x_{b_r}i_r}\right). \end{aligned} \end{align} $$

Then, proceeding as in the proof of [Reference Cipolloni, Erdős and Schröder18, Proposition 4.5] (see also [Reference Cipolloni, Erdős and Schröder20, Eq. (4.40)]), we conclude that

(5.46) $$ \begin{align} \partial_t\lVert h_t\rVert_2^2\le -\frac{C_1(n)}{2\eta}\langle h_t\rangle_2^2+\frac{C_3(n)}{\eta}\mathcal{E}^2K^n, \end{align} $$

which implies ∥h _{T 1} ∥₂ ² ≤ C(n) $\mathcal{E}$ ² K ⁿ , by a simple Gronwall inequality, using that T ₁ ≫ η.

We point out that to go from (5.45) to (5.46), the proof is completely analogous to [Reference Cipolloni, Erdős and Schröder18, Proof of Proposition 4.5], with the only exception being the proof of [Reference Cipolloni, Erdős and Schröder18, Eqs. (4.41), (4.43)]. We thus now explain how to obtain the analog of [Reference Cipolloni, Erdős and Schröder18, Eqs. (4.41), (4.43)] in the current case as well. The fact that we now have the bound (5.37) rather than the stronger bound $N^{-1/3}$ as in [Reference Cipolloni, Erdős and Schröder20, Eq. (4.31)] does not cause any difference in the final estimate. We thus focus on the main new difficulty in the current analysis (i.e., that in (5.31), we choose the initial condition depending on $r_0$ rather than $g_0$ ). We recall that the difference between $r_0$ and $g_0$ is that $r_0$ is defined in such a way that all the eigenvector overlaps are precisely centred and normalised in an i–dependent way, whereas for $g_0$ , we can choose the i–independent constant $C_0,C_1$ so that only the overlap corresponding to a certain base point $i_0$ is exactly centred and normalised, while the nearby overlaps are centred and normalised only modulo a negligible error $K/N$ (see also the paragraph below (5.14) for a detailed explanation). This additional difficulty requires that to prove the analog of [Reference Cipolloni, Erdős and Schröder18, Eqs. (4.41), (4.43)], we need to estimate the error produced by this mismatch.

Using that the function f(x) ≡ 1(n even) is in the kernel of $\mathcal{L}$ (t) for any fixed x ∈Γ and for any fixed i, a, b, we conclude (recall the notation from (5.29))

(5.47) $$ \begin{align} \begin{aligned} &h_t(\mathbf{x}_{\mathbf{a}\mathbf{b}}^{\mathbf{i} \mathbf{j}})\\[4pt] &=\mathcal{U}_{\mathcal{S}}(0,t)\big((\mathrm{Av} r_0)(\mathbf{x}_{\mathbf{a}\mathbf{b}}^{\mathbf{i}\mathbf{j}})-(\mathrm{Av}\mathbf{1}({n} \,\, \mathrm{even}))(\mathbf{x}_{\mathbf{a}\mathbf{b}}^{\mathbf{i}\mathbf{j}})\big) \\[4pt] &=\mathrm{Av}(\mathbf{x}_{\mathbf{a}\mathbf{b}}^{\mathbf{i}\mathbf{j}})\big(\mathcal{U}_{\mathcal{S}}(0,t)r_0(\mathbf{x}_{\mathbf{a}\mathbf{b}}^{\mathbf{i}\mathbf{j}})-\mathbf{1}({n}\,\, \mathrm{even})\big)+\mathcal{O}\left(\frac{{N}^{\epsilon+{n}\xi} \ell}{{K}}\right) \\[4pt] &=\left(\mathrm{Av}(\mathbf{x})+\mathcal{O}\left(\frac{\ell}{K}\right)\right)\left(\mathcal{U}(0,t)r_0(\mathbf{x}_{\mathbf{a}\mathbf{b}}^{\mathbf{i}\mathbf{j}})-\mathbf{1}({n}\,\, \mathrm{even})+\mathcal{O}\left(\frac{{N}^{1+{n}\xi}{t}}{\ell}\right)\right)+\mathcal{O}\left(\frac{N^{\epsilon+n\xi} \ell}{K}\right) \\[4pt] &=\mathrm{Av}(\mathbf{x})\big(g_t(\mathbf{x}_{\mathbf{a}\mathbf{b}}^{\mathbf{i}\mathbf{j}})-\mathbf{1}({n}\,\, \mathrm{even})\big)+\mathcal{O}\left(\frac{{N}^{\epsilon+{n}\xi} \ell}{{K}}+\frac{{N}^{1+{n}\xi}{t}}{\ell}+\frac{{N}^\xi {K}}{\sqrt{{N}}}\right), \end{aligned} \end{align} $$

where in the definition of $g_t$ from (5.7) and (5.19), we chose

(5.48) $$ \begin{align} C_0:=\frac{\langle A\Im M(\gamma_{i_0})\rangle }{\langle \Im M(\gamma_{i_0})\rangle}, \qquad C_1:=\mathrm{Var}_{\gamma_{i_0}}(A), \end{align} $$

and the error terms are uniform in x ∈Γ. Here, $i_0$ is the index defined below (5.39). The first three inequalities are completely analogous to [Reference Cipolloni, Erdős and Schröder18, Eq. (4.41)]. We now explain how to obtain the last inequality at the price of the additional negligible error $KN^{-1/2}$ . Recall the definition of $r_t$ from (5.13) and (5.20); we now show that for any $\textbf {x}$ supported in the bulk, it holds

(5.49) $$ \begin{align} \lVert r_0(\textbf{x})-g_0(\textbf{x})\rVert_\infty\lesssim \frac{N^\xi K}{\sqrt{N}} \end{align} $$

for $C_0,C_1$ chosen as in (5.48). Using (5.49), together with $\mathcal {U}(0,t)g_0=g_t$ , this proves the last equality in (5.47). The main input in the proof of (5.49) is the following approximation result:

(5.50) $$ \begin{align} \mathring{A}^{\gamma_{i_0}}-\mathring{A}^{\gamma_{i_r}}=\mathcal{O}(|\gamma_{i_0}-\gamma_{i_r}|)I=\mathcal{O}(KN^{-1})I. \end{align} $$

We now explain the proof of (5.49); for simplicity, we present the proof only in the case $n=2$ . To prove (5.49), we see that

$$\begin{align*}2|\langle \mathbf{u}_i,\mathring{A}^{\gamma_{i_0}} \mathbf{u}_j\rangle|^2+\langle \mathbf{u}_i,\mathring{A}^{\gamma_{i_0}} \mathbf{u}_i\rangle\langle \mathbf{u}_j,\mathring{A}^{\gamma_{i_0}} \mathbf{u}_j\rangle=2|\langle \mathbf{u}_i,\mathring{A}^{\gamma_i} \mathbf{u}_j\rangle|^2+\langle \mathbf{u}_i,\mathring{A}^{\gamma_i} \mathbf{u}_i\rangle\langle \mathbf{u}_j,\mathring{A}^{\gamma_j} \mathbf{ u}_j\rangle+\mathcal{O}\left(\frac{1}{N}\cdot\frac{N^\xi K}{\sqrt{N}}\right), \end{align*}$$

where we used (5.50) to replace the ‘wrong’ $\mathring {A}^{\gamma _{i_0}}$ with the ‘correct’ $\mathring {A}^{\gamma _i}$ together with the a priori a bound $\langle \mathbf {u}_i,\mathring {A}^{\gamma _i} \mathbf {u}_j \rangle \le N^\xi N^{-1/2}$ . Then, multiplying this relation by N, we obtain (5.49). Additionally, since $r_0$ and $g_0$ contain a different rescaling in terms of $\mathrm {Var}_{\gamma _{i_0}}(A)$ and $\mathrm {Var}_{\gamma _i}(A)$ , we also used that by similar computations,

$$\begin{align*}\mathrm{Var}_{\gamma_{i_0}}(A)=\mathrm{Var}_{\gamma_i}(A)+\mathcal{O}\left(\frac{N^\xi K}{\sqrt{N}}\right). \end{align*}$$

In particular, we used this approximation to compensate the mismatch that only the diagonal overlaps corresponding to the index $i_0$ are properly centred and normalised in the definition of $g_0$ , whereas for nearby indices, we use this approximation to replace the approximate centering $C_0$ and normalisation $C_1$ from (5.48) with the correct one, which is the one in the definition of $r_0$ .

Then, proceeding as in the proof of [Reference Cipolloni, Erdős and Schröder20, Eq. (4.41)], we conclude the analog of [Reference Cipolloni, Erdős and Schröder18, Eq. (4.43)]:

(5.51) $$ \begin{align} \begin{aligned} &\sum_{\mathbf{j}}^*\left(\prod_{r=1}^n a_{i_r j_r}^{\mathcal{S}}(t)\right)\big(g_t(\mathbf{x}_{\mathbf{a}\mathbf{b}}^{\mathbf{i} \mathbf{j}})-\mathbf{1}({n}\,\, \mathrm{even})\big) \\ & \quad= \sum_{\mathbf{j}}\left(\prod_{r=1}^n a_{i_r j_r}(t)\right)\left(\frac{N^{n/2}}{\mathrm{Var}_{\gamma_{i_0}}(A)^{n/2} 2^{n/2}(n-1)!!}\sum_{G\in \mathcal{G}_{\boldsymbol{\eta}^{\mathbf{j}}}}P(G)-\mathbf{1}({n}\,\, \mathrm{even})\right) \\ &\qquad+\mathcal{O}\left(\frac{N^{n\xi}}{N\eta}+\frac{N^{1+n\xi}\eta}{\ell}+\frac{N^\xi K}{\sqrt{N}}\right). \end{aligned} \end{align} $$

Given (5.51), the remaining part of the proof is completely analogous to [Reference Cipolloni, Erdős and Schröder18, Eqs. (4.44)–(4.51)], except for the slightly different computation

(5.52) $$ \begin{align} &\frac{\langle \Im G(\lambda_{i_{r_1}}+\mathrm{i} \eta)\mathring{A}^{\gamma_{i_0}}\Im G (\lambda_{i_{r_2}}+\mathrm{i} \eta)\mathring{A}^{\gamma_{i_0}}\rangle}{\mathrm{Var}_{\gamma_{i_0}}(A)} \nonumber\\ &\qquad\qquad=-\frac{1}{4\mathrm{Var}_{\gamma_{i_0}}(A)}\sum_{\sigma,\tau\in \{+,-\}}\langle G(\lambda_{i_{r_1}}+\sigma\mathrm{i} \eta)\mathring{A}^{\gamma_{i_0}} G (\lambda_{i_{r_2}}+\tau\mathrm{i} \eta)\mathring{A}^{\gamma_{i_0}}\rangle\nonumber\\ &\qquad \qquad= -\frac{1}{4\mathrm{Var}_{\gamma_{i_0}}(A)}\sum_{\sigma,\tau\in \{+,-\}}\left\langle G(\lambda_{i_{r_1}}+\mathrm{i} \sigma \eta)\mathring{A}^{\gamma_{i_{r_1}}+\mathrm{i} \sigma\eta,\gamma_{i_{r_2}}+\mathrm{i}\tau\eta} G (\lambda_{i_{r_2}}+\mathrm{i} \tau \eta)\mathring{A}^{\gamma_{i_{r_2}}+\mathrm{i} \tau\eta, \gamma_{i_{r_1}}+\mathrm{i} \sigma\eta}\right\rangle \nonumber\\ &\qquad\qquad\quad+\mathcal{O}\left(\frac{K}{N^2\eta^{3/2}}+\frac{K^2}{N^2\eta} +\frac{1}{N\sqrt{\eta}}\right)\nonumber\\ &\qquad\qquad=\langle \Im M(\gamma_{i_0})\rangle^2+\mathcal{O}\left(\frac{K}{N^2\eta^{3/2}}+\frac{K^2}{N^2\eta}+\frac{1}{\sqrt{N\eta}}\right), \end{align} $$

which replaces [Reference Cipolloni, Erdős and Schröder18, Eqs. (4.47)]. Here, $\mathring {A}^{\gamma _{i_{r_1}}\pm \mathrm {i} \eta ,\gamma _{i_{r_2}}\pm \mathrm {i}\eta }$ is defined as in Definition 4.2. We also point out that in the second equality, we used the approximation (see (4.12))

(5.53) $$ \begin{align} \mathring{A}^{\gamma_{i_0}}=\mathring{A}^{\gamma_{i_{r_1}}\pm\mathrm{i}\eta,\gamma_{i_{r_2}}\pm \mathrm{i} \eta} +\mathcal{O}\big(|\gamma_{i_{r_1}}-\gamma_{i_0}| +|\gamma_{i_{r_2}}-\gamma_{i_0}|+\eta\big)I =\mathring{A}^{\gamma_{i_{r_1}},\gamma_{i_{r_2}}}+\mathcal{O}\left(\frac{K}{N}+\eta\right)I, \end{align} $$

together with (here we present the estimate only for one representative term – the other being analogous)

(5.54) $$ \begin{align} \langle G(\lambda_{i_{r_1}}+\mathrm{i} \eta) G (\lambda_{i_{r_2}}+\mathrm{i} \eta)\mathring{A}^{\gamma_{i_0}}\rangle&=\langle G(\lambda_{i_{r_1}}+\mathrm{i} \eta)G (\lambda_{i_{r_2}}+\mathrm{i} \eta)\mathring{A}^{\gamma_{i_{r_2}}+\mathrm{i} \eta,\gamma_{i_{r_2}}+\mathrm{i}\eta}\rangle+\mathcal{O}\left(1+\frac{K}{N\eta}\right) \nonumber\\ &=\langle M(\gamma_{i_{r_2}}+\mathrm{i} \eta, \mathring{A}^{\gamma_{i_{r_2}}+\mathrm{i} \eta, \gamma_{i_{r_1}}+\mathrm{i}\eta}, \gamma_{i_{r_1}}+\mathrm{i} \eta) \rangle+\mathcal{O}\left( 1+\frac{1}{N\eta^{3/2}}+\frac{K}{N\eta}\right)\nonumber\\ & =\mathcal{O}\left(1+\frac{1}{N\eta^{3/2}}+\frac{K}{N\eta}\right), \end{align} $$

which follows by (5.34) and Lemma 4.3 to estimate the deterministic term, together with the integral representation from [Reference Cipolloni, Erdős, Henheik and Schröder21, Lemma 5.1] (see also (6.17) later), to bound the error terms arising from the replacement (5.53). Additionally, in the third equality, we used the local for two resolvents from (5.34):

(5.55) $$ \begin{align} &-\frac{1}{4}\sum_{\sigma,\tau\in \{+,-\}}\left\langle G(\lambda_{i_{r_1}}+\mathrm{i} \sigma \eta)\mathring{A}^{\gamma_{i_{r_1}}+\mathrm{i} \sigma\eta,\gamma_{i_{r_2}}+\mathrm{i}\tau\eta} G (\lambda_{i_{r_2}}+\mathrm{i} \tau \eta)\mathring{A}^{\gamma_{i_{r_2}}+\mathrm{i} \tau\eta, \gamma_{i_{r_1}}+\mathrm{i} \sigma\eta}\right\rangle \nonumber\\ &\qquad\quad =-\frac{1}{4}\sum_{\sigma,\tau\in \{+,-\}}\left\langle M(\gamma_{i_{r_1}}+\mathrm{i} \sigma \eta, \mathring{A}^{\gamma_{i_{r_1}}+\mathrm{i} \sigma\eta,\gamma_{i_{r_2}}+\mathrm{i}\tau\eta}, \gamma_{i_{r_2}}+\mathrm{i} \tau\eta)\mathring{A}^{\gamma_{i_{r_2}}+\mathrm{i} \tau\eta, \gamma_{i_{r_1}}+\mathrm{i} \sigma\eta}\right\rangle+\mathcal{O}\left(\frac{1}{\sqrt{N\eta}}\right) \nonumber\\ &\qquad\quad = \langle \Im M(\gamma_{i_0})\rangle^2\mathrm{Var}_{\gamma_{i_0}}(A)+\mathcal{O}\left(\frac{1}{\sqrt{N\eta}}+\frac{K}{N}\right), \end{align} $$

with the deterministic term defined in (5.35). In the second equality, we used the approximation (5.53) again. We remark that here we presented this slightly different (compared to [Reference Cipolloni, Erdős and Schröder18, Reference Cipolloni, Erdős and Schröder20]) computation only for chain of length $k=2$ ; the computation for longer chains is completely analogous, and so it is omitted. This concludes the proof of (5.38).

We conclude this section with the proof of Proposition 5.1.

6.1 Master inequalities and reduction lemma: Proof of Lemma 6.2

We now state the relevant part of a nonlinear infinite hierarchy of coupled master inequalities for $\Psi ^{\mathrm {av}}_k$ and $\Psi ^{\mathrm {iso}}_k$ . In fact, for our purposes, it is sufficient to have only the inequalities for $k \in [2]$ . Slightly simplified versions of these master inequalities will be used in Appendix A for general $k \in \mathbf {N}$ . The proof of Proposition 6.3 is given in Section 6.2.

Proposition 6.3 (Master inequalities; see Proposition 4.9 in [Reference Cipolloni, Erdős, Henheik and Schröder21]).

Assume that for some deterministic control parameters $ \psi _j^{\mathrm {av/iso}}$ , we have that

(6.7) $$ \begin{align} \Psi_j^{\mathrm{av/iso}} \prec \psi_j^{\mathrm{av/iso}}\,, \quad j \in [4]\, \end{align} $$

holds uniformly in regular matrices. Then we have

(6.8a) $$ \begin{align} \Psi_1^{\mathrm{av}} &\prec 1 + \frac{\psi_1^{\mathrm{av}}}{N \eta} + \frac{\psi_1^{\mathrm{iso}} + (\psi_2^{\mathrm{av}})^{1/2} }{(N \eta)^{1/2}} + \frac{(\psi_2^{\mathrm{iso}})^{1/2}}{(N \eta)^{1/4}} \,,\qquad\qquad\qquad\qquad\qquad\qquad\qquad \end{align} $$

(6.8b) $$ \begin{align} \Psi_1^{\mathrm{iso}} &\prec 1 + \frac{\psi_1^{\mathrm{iso}} + \psi_1^{\mathrm{av}} }{(N \eta)^{1/2}} + \frac{(\psi_2^{\mathrm{iso}})^{1/2}}{(N \eta)^{1/4}} \,,\qquad\qquad\qquad \qquad\qquad\qquad \qquad\ \ \,\qquad\qquad\end{align} $$

(6.8c) $$ \begin{align} \Psi_2^{\mathrm{av}} &\prec 1 + \frac{(\psi_1^{\mathrm{av}})^{2} + {(\psi_1^{\mathrm{iso}})^2} + \psi_2^{\mathrm{av}}}{N \eta}+ \frac{\psi_2^{\mathrm{iso}} + (\psi_4^{\mathrm{av}})^{1/2} }{(N \eta)^{1/2}} + \frac{(\psi_3^{\mathrm{iso}})^{1/2} + (\psi_4^{\mathrm{iso}})^{1/2}}{(N \eta)^{1/4}} \,,\qquad \end{align} $$

(6.8d) $$ \begin{align} \Psi_2^{\mathrm{iso}} &\prec 1 + \psi_1^{\mathrm{iso}} + \frac{ \psi_1^{\mathrm{av}} \psi_1^{\mathrm{iso}} + (\psi_1^{\mathrm{iso}})^2 }{N \eta}+ \frac{\psi_2^{\mathrm{iso}} + (\psi_1^{\mathrm{iso}} \psi_3^{\mathrm{iso}})^{1/2}}{(N \eta)^{1/2}} + \frac{(\psi_3^{\mathrm{iso}})^{1/2} + \hspace{-2pt}(\psi_4^{\mathrm{iso}})^{1/2}}{(N \eta)^{1/4}}\,, \end{align} $$

again uniformly in regular matrices.

As shown in the above proposition, resolvent chains of length $k=1,2$ are estimated by resolvent chains up to length $2k$ . To avoid the indicated infinite hierarchy of master inequalities with higher and higher k indices, we will need the following reduction lemma.

Lemma 6.4 (Reduction inequalities; see Lemma 4.10 in [Reference Cipolloni, Erdős, Henheik and Schröder21]).

As in (6.7), assume that $\Psi _j^{\mathrm {av/iso}} \prec \psi _j^{\mathrm {av/iso}}$ holds for $1 \le j \le 4$ uniformly in regular matrices. Then we have

(6.9) $$ \begin{align} \Psi_4^{\mathrm{av}} \prec (N\eta)^2 + (\psi_2^{\mathrm{av}})^2\,, \end{align} $$

uniformly in regular matrices, and

(6.10) $$ \begin{align} \begin{aligned} \Psi_3^{\mathrm{iso}} &\prec N \eta \left( 1+ \frac{\psi_2^{\mathrm{iso}}}{\sqrt{N\eta}} \right) \left( 1 + \frac{\psi_2^{\mathrm{av}}}{N\eta} \right)^{1/2}\,,\\ \Psi_4^{\mathrm{iso}} &\prec (N \eta)^{3/2}\left( 1+ \frac{\psi_2^{\mathrm{iso}}}{\sqrt{N\eta}} \right) \left( 1 + \frac{\psi_2^{\mathrm{av}}}{N\eta} \right)\,, \end{aligned} \end{align} $$

again uniformly in regular matrices.

Now the estimates (6.6) follow by combining Proposition 6.3 and Lemma 6.4 in an iterative scheme, which has been carried out in detail in [Reference Cipolloni, Erdős, Henheik and Schröder21, Section 4.3]. This completes the proof of Lemma 6.2.

6.2 Proof of the master inequalities in Proposition 6.3

The proof of Proposition 6.3 is very similar to the proof of the master inequalities in [Reference Cipolloni, Erdős, Henheik and Schröder21, Proposition 4.9]. Therefore, we shall only elaborate on (6.8a) as a showcase in some detail and briefly discuss (6.8b)-(6.8d) afterwards.

First, we notice that [Reference Cipolloni, Erdős, Henheik and Schröder21, Lemma 5.2] also holds for deformed Wigner matrices (see Lemma 6.5 below). In order to formulate it, recall the definition of the second order renormalisation, denoted by underline, from [Reference Cipolloni, Erdős, Henheik and Schröder21, Equation (5.3)]. For a function $f(W)$ of the Wigner matrix W, we define

(6.11) $$ \begin{align} \underline{W f(W)} := W f(W) - \widetilde{\mathbb{E}} \big[ \widetilde{W} (\partial_{\widetilde{W}}f)(W) \big]\,, \end{align} $$

where $\partial _{\widetilde {W}}$ denotes the directional derivative in the direction of $\widetilde {W}$ , which is a GUE matrix that is independent of W. The expectation is taken w.r.t. the matrix $\widetilde {W}$ . Note that if W itself a GUE matrix, then $\mathbf {E} \underline {Wf(W)} = 0$ , whereas for W with general single entry distributions, this expectation is independent of the first two moments of W. In other words, the underline renormalises the product $W f(W)$ to second order.

We note that $\widetilde {\mathbf {E}} \widetilde {W} R \widetilde {W} = \langle R \rangle $ and furthermore, that the directional derivative of the resolvent is given by $\partial _{\widetilde {W}} G = -G \widetilde {W} G$ . For example, in the special case $f(W) = (W + D -z)^{-1} = G$ , we thus have

$$ \begin{align*} \underline{WG} = WG + \langle G \rangle G \end{align*} $$

by definition of the underline in (6.11).

Lemma 6.5. Under the assumption (6.7), for any regular matrix $A = \mathring {A}$ , we have that

(6.12) $$ \begin{align} \langle (G-M)\mathring{A}\rangle = -\langle\underline{WG\mathring{A}'}\rangle+\mathcal{O}_{\prec}\left(\mathcal{E}_1^{\mathrm{av}}\right)\, \end{align} $$

for some other regular matrix $A'=\mathring {A}'$ , which linearly depends on A (see (6.21) for an explicit formula). Here, $G=G(z)$ and $\eta :=|\Im z|$ . For the error term, we used the shorthand notation

(6.13) $$ \begin{align} \mathcal{E}_1^{\mathrm{av}} :=\frac{1}{N\eta^{1/2}}\left(1+\frac{\psi_1^{\mathrm{av}}}{N\eta}\right)\,. \end{align} $$

By simple complex conjugation of (6.12), we may henceforth assume that $z = e + \mathrm {i} \eta $ with $\eta> 0$ . The representation (6.12) will be verified later. Now, using (6.12), we compute the even moments of $\langle (G-M)\mathring {A}\rangle $ as

(6.14) $$ \begin{align} \mathbf{E}\left\vert\langle(G-M)A\rangle\right\vert{}^{2p}=\left\vert -\mathbf{E} \langle \underline{WG}A'\rangle\langle (G-M)A\rangle^{p-1}\langle (G-M)^*A^*\rangle^p\right\vert+\mathcal{O}_\prec \left(\left(\mathcal{E}_1^{\mathrm{av}}\right)^{2p}\right) \end{align} $$

and then apply a so-called cumulant expansion to the first summand. More precisely, we write out the averaged traces and employ an integration by parts (see, for example, [Reference Cipolloni, Erdős and Schröder16, Eq. (4.14)])

(6.15) $$ \begin{align} \mathbf{E} w_{ab} f(W) = \mathbf{E} |w_{ab}|^2 \mathbf{E} \partial_{w_{ba}}f(W) + ... \quad \text{with} \quad \mathbf{E} |w_{ab}|^2 = \frac{1}{N}\,, \end{align} $$

indicating higher derivatives and an explicit error term, which can be made arbitrarily small, depending on the number of involved derivatives (see, for example, [Reference Erdős, Krüger and Schröder27, Proposition 3.2]). We note that, if W were a GUE matrix, the relation (6.15) would be exact without higher derivatives, which shall be discussed below.

Considering the explicitly written Gaussian term in (6.15) for the main term in (6.14), we find that it is bounded from above by (a p-dependent constant times)

(6.16) $$ \begin{align} \mathbf{E}\left[\frac{\vert \langle GGA'GA\rangle\vert + \vert\langle G^*GA'G^*A^*\rangle\vert}{N^2}\vert \langle (G-M)A\rangle\vert^{2p-2} \right]\,. \end{align} $$

The main technical tool to estimate (6.16) is the following contour integral representation for the square of resolvent (see [Reference Cipolloni, Erdős, Henheik and Schröder21, Lemma 5.1]). This is given by

(6.17) $$ \begin{align} G(z)^2=\frac{1}{2\pi i}\int\limits_{\Gamma}\frac{G(\zeta)}{(\zeta-z)^2}\mathrm{d} \zeta\,, \end{align} $$

where the contour $\Gamma =\Gamma (z)$ is the boundary of a finite disjoint union of half bands which are parametrised counterclockwise. Here, J is a finite disjoint unionFootnote ¹¹ of closed intervals, which we take as $\mathbf {B}_{\kappa '}$ for a suitable $ \kappa ' \in (0, \kappa )$ – to be chosen below – and hence contains $e=\Re z$ ; the parameter is chosen to be smaller than say, $\eta /2$ . Applying the integral identity (6.17) to the product $GG$ in the term $\langle GGA'GA\rangle $ yields that

(6.18) $$ \begin{align} \left\vert\langle GGA'GA\rangle\right\vert \lesssim \left\vert\int\limits_{\Gamma}\frac{\langle G(\zeta)A'GA\rangle}{(\zeta-z)^2}\mathrm{d}\zeta\right\vert\,. \end{align} $$

Now, we split the contour $\Gamma $ in three parts; i.e.,

(6.19) $$ \begin{align} \Gamma = \Gamma_1 + \Gamma_2 + \Gamma_3\,. \end{align} $$

As depicted in Figure 1, the first part, $\Gamma _1$ , of the contour consists of the entire horizontal part of $\Gamma $ . The second part, $\Gamma _2$ , covers the vertical components up to $|\Im \zeta | \le N^{100}$ . Finally, $\Gamma _3$ consists of the remaining part with $|\Im \zeta |> N^{100}$ . The contribution coming from $\Gamma _3$ can be estimated with a trivial norm bound on G. To estimate the integral over $\Gamma _2$ , we choose the parameter $\kappa '$ in the definition of $J = \mathbf {B}_{\kappa '}$ in such a way that the distance between z and $\Gamma _2$ is greater than $\delta> 0$ from Definition 4.2. Hence, for $\zeta \in \Gamma _2$ , every matrix is considered regular w.r.t. $(z,\zeta )$ .

Figure 1

The contour $\Gamma $ is split into three parts (see (6.19)). Depicted is the situation, where the bulk $\mathbf {B}_{\kappa }$ consists of two components. The boundary of the associated domain $\mathbf {D}^{(\epsilon , \kappa )}$ is indicated by the two U-shaped dashed lines. Modified version of [Reference Cipolloni, Erdős, Henheik and Schröder21, Figure 4].

Therefore, after splitting the contour and estimating each contribution as just described, we find, with the aid of Lemma 4.3,

In this integral over $J= \mathbf {B}_{\kappa '}$ , the horizontal part $\Gamma _1$ , we decompose A and $A'$ in accordance to the spectral parameters of the adjacent resolvents and use the following regularity property:

Now the integral over J is represented as a sum of four integrals: one of them contains two regular matrices, and the rest have at least one identity matrix with a small error factor. For the first one, we use the same estimates as for the integral over the vertical part $\Gamma _2$ . For the other terms, thanks to the identity matrix, we use a resolvent identity (e.g., for

) and note that the $\big (\vert x-e\vert +\eta \big )$ -error improves the original $1/\eta $ -blow up of $\int _J \frac {1}{(x-e)^2+\eta ^2}\mathrm {d} x$ to an only $|\log \eta |$ -divergent singularity, which is incorporated into ‘ $\prec $ ’.

For the term $\langle G^*GA'G^*A^*\rangle $ from (6.16), we use a similar strategy: after an application of the Ward identity $G^* G = \Im G/\eta $ , we decompose the deterministic matrices $A, A'$ according to the spectral parameters of their neighbouring resolvents in the product. This argument gives us that each of the terms $\langle GGA'GA\rangle $ and $\langle G^*GA'G^*A^*\rangle $ is stochastically dominated by

$$ \begin{align*} \frac{1}{\eta}\left(1+\frac{\psi_2^{\mathrm{av}}}{N\eta}\right). \end{align*} $$

Contributions stemming from higher order cumulants in (6.16) are estimated exactly in the same way as in [Reference Cipolloni, Erdős, Henheik and Schröder21, Section 5.5]. The proof of (6.8a) is concluded by applying Young inequalities to (6.16) (see [Reference Cipolloni, Erdős, Henheik and Schröder21, Section 5.1]).

Finally, we show that (6.12) holds:

Proof of Lemma 6.5.

The proof of this representation is simpler than the proof of its analogue [Reference Cipolloni, Erdős, Henheik and Schröder21, Lemma 5.2] because in the current setting, all terms in [Reference Cipolloni, Erdős, Henheik and Schröder21, Lemma 5.2] containing the particular chiral symmetry matrix $E_{-}$ are absent. In the same way as in [Reference Cipolloni, Erdős, Henheik and Schröder21], we arrive at the identity

(6.20) $$ \begin{align} \langle (G-M)A\rangle = -\langle \underline{WG} \mathcal{X}\left[A\right]M \rangle+\langle G-M\rangle \langle (G-M)\mathcal{X}[A]M \rangle\,, \end{align} $$

where we introduced the bounded linear operator $\mathcal {X}[B] := \big (1 - \langle M \cdot M \rangle \big )^{-1}[B]$ . Indeed, boundedness follows from the explicit formula

$$ \begin{align*} \mathcal{X}[B] = B + \frac{\langle MBM\rangle}{1 - \langle M^2 \rangle} \end{align*} $$

by means of the lower bound

$$ \begin{align*} \vert 1 - \langle M^2 \rangle \vert = \vert (1 - \langle M M^* \rangle) - 2\mathrm{i} \langle M \Im M \rangle \vert \ge 2 \langle \Im M \rangle^2 \gtrsim 1\,, \end{align*} $$

obtained by taking the imaginary part of the MDE (2.8), in combination with $\Vert M \Vert \lesssim 1$ .

Next, completely analogous to [Reference Cipolloni, Erdős, Henheik and Schröder21, Lemma 5.4], we find the decomposition

(6.21) $$ \begin{align} \mathcal{X}[A]M = \big(\mathcal{X}[A]M\big)^\circ + \mathcal{O}(\eta) I = \mathring{A}' + \mathcal{O}(\eta)I \,, \qquad A':=\mathcal{X}[A]M\,. \end{align} $$

Plugging this into (6.20), we thus infer

(6.22) $$ \begin{align} \langle (G-M)A\rangle = -\langle \underline{WG}A'\rangle + \langle G-M\rangle \langle (G-M)\mathring{A}'\rangle + \left(-\langle \underline{WG}\rangle + \langle G-M\rangle^2\right) \mathcal{O}(\eta). \end{align} $$

The second term in the rhs. of (6.22) is obviously bounded by $\psi _1^{\mathrm {av}}/(N^2\eta ^{3/2})$ , and in the third term, we use the usual local law (4.1) to estimate it by $\mathcal {O}_\prec \big (N^{-1}\big )$ . Combining this information gives (6.12).

Notice that the above arguments leading to (6.8a) are completely identical to the ones required in the proof of the analogous master inequality in [Reference Cipolloni, Erdős, Henheik and Schröder21, Proposition 4.9] with one minor but key modification: every term involving the chiral symmetry matrix $E_-$ in [Reference Cipolloni, Erdős, Henheik and Schröder21] is simply absent, and hence, with the notation of [Reference Cipolloni, Erdős, Henheik and Schröder21], all sums over signs $\sum _{\sigma = \pm }\cdots $ collapse to a single summand with $E_+ \equiv I$ . With this recipe, the proofs of (6.8b)–(6.8d) are completely analogous to the ones given in [Reference Cipolloni, Erdős, Henheik and Schröder21, Sections 5.2–5.5] and hence omitted.