Fluctuations in Local Quantum Unique Ergodicity for Generalized Wigner Matrices

Abstract

We study the eigenvector mass distribution for generalized Wigner matrices on a set of coordinates I, where $N^\varepsilon \leqslant | I | \leqslant N^{1- \varepsilon }$, and prove it converges to a Gaussian at every energy level, including the edge, as $N\rightarrow \infty$. The key technical input is a four-point decorrelation estimate for eigenvectors of matrices with a large Gaussian component. Its proof is an application of the maximum principle to a new set of moment observables satisfying parabolic evolution equations. Additionally, we prove high-probability Quantum Unique Ergodicity and Quantum Weak Mixing bounds for all eigenvectors and all deterministic sets of entries using a novel bootstrap argument.

Appendices

Resolvent Estimates

The following lemma was proved in [10].

Lemma A.1

[10, Lemma 4.8]. Fix $\omega , \delta _1, \varepsilon _1, \delta _2, \varepsilon _2>0$ such that $\delta _1 > \varepsilon _1$ and $\ell \in [\![1,N]\!]$. Let H be a generalized Wigner matrix, and let ${\mathcal {B}}= {\mathcal {B}}(\omega ,\delta _1,\varepsilon _1,\delta _2, \varepsilon _2)$ be the set from Definition 5.5. Let $\eta _\ell =\frac{N^{-\varepsilon _2}}{N^{2/3} {\widehat{\ell }}^{1/3}}$. Then there exists a constant $C= C(\varepsilon _2)>0$ such that for all $E \in I_{\delta _2}(\lambda _\ell ) \cup I_{\delta _2}({\widetilde{\lambda }}_\ell )$,

$$\begin{aligned} \sup _{a,b,c,d \in [\![1,N]\!]} \sup _{w \in [0,1]} {\mathbb {1}}_{{\mathcal {B}}}{{\,\mathrm{Im}\,}}\Theta ^{(a,b)}_w G_{cd}(E+\mathrm {i}\eta _\ell ) \leqslant C N^{4\varepsilon _2} \left( \frac{{\widehat{\ell }}}{N}\right) ^{1/3} \end{aligned}$$

(A.1)

and

$$\begin{aligned} \sup _{a,b,c,d \in [\![1,N]\!]} \sup _{w \in [0,1]} {\mathbb {1}}_{{\mathcal {B}}}{{\,\mathrm{Re}\,}}\Theta ^{(a,b)}_w G_{cd}(E+\mathrm {i}\eta _\ell ) \leqslant C N^{4\varepsilon _2}. \end{aligned}$$

(A.2)

Here ${\widetilde{E}} = {\widetilde{E}} (1, \omega )$.

In what follows we write H for $\Theta ^{(a,b)}_w H$, and similarly for G and $m_N$, omitting the rank one perturbation from the notation.

Proof of Lemma 5.17

We first observe by the spectral theorem that

$$\begin{aligned} | I _N | N^{-2} {{\,\mathrm{Tr}\,}}{{\,\mathrm{Im}\,}}G (z_1) {{\,\mathrm{Im}\,}}G (z_2) = \frac{1}{N^2} \sum _{j=1}^N \frac{ | I _N | \eta _k\eta _\ell }{ \left( ( \lambda _j - E_1)^2 + \eta ^2_k\right) \left( ( \lambda _j - E_2)^2 + \eta ^2_\ell \right) } ,\nonumber \\ \end{aligned}$$

(A.3)

and

$$\begin{aligned} \frac{2}{N} \sum _{\alpha \in I_N} \langle {\mathbf {q}}_\alpha ,{{\,\mathrm{Im}\,}}G(z_1){{\,\mathrm{Im}\,}}G(z_2){\mathbf {q}}_\alpha \rangle = \frac{2}{N} \sum _{j=1}^N \frac{ \eta _k\eta _\ell \left( p_{jj}+\frac{\vert I\vert }{N}\right) }{ \left( ( \lambda _j - E_1)^2 + \eta ^2_k\right) \left( ( \lambda _j - E_2)^2 + \eta ^2_\ell \right) }.\nonumber \\ \end{aligned}$$

(A.4)

For (5.70), we compute

$$\begin{aligned} \frac{1 }{ | I_N|}\sum _{ \alpha , \beta \in I_N}&{{\,\mathrm{Im}\,}}\langle {\mathbf {q}}_\alpha ,G(z_1){\mathbf {q}}_\beta \rangle {{\,\mathrm{Im}\,}}\langle {\mathbf {q}}_\alpha ,G(z_2){\mathbf {q}}_\beta \rangle \end{aligned}$$

(A.5)

$$\begin{aligned}&= \frac{1}{|I_N|} \sum _{\alpha , \beta \in I_N} \left( \sum _{j=1}^N \frac{ \eta _k \langle {\mathbf {q}}_\alpha ,{\mathbf {u}}_j\rangle \langle {\mathbf {q}}_\beta ,{\mathbf {u}}_j\rangle }{ ( \lambda _j - E_1)^2 + \eta ^2_k }\right) \left( \sum _{m=1}^N \frac{\eta _\ell \langle {\mathbf {q}}_\alpha ,{\mathbf {u}}_m\rangle \langle {\mathbf {q}}_\beta ,{\mathbf {u}}_m\rangle }{( \lambda _m - E_2)^2 + \eta ^2_\ell }\right) \end{aligned}$$

(A.6)

$$\begin{aligned}&= \frac{1}{N^2} \sum _{j\ne m}^N \frac{ \eta _k\eta _\ell {\widehat{p}}^2_{jm}(I_N) }{ \left( ( \lambda _j - E_1)^2 + \eta ^2_k\right) \left( ( \lambda _m - E_2)^2 + \eta ^2_\ell \right) } \end{aligned}$$

(A.7)

$$\begin{aligned}&\quad +\,\frac{1}{|I_N|} \sum _{j=1}^N \frac{ \eta _k\eta _\ell \left( p_{jj} + \frac{ | I_N | }{N} \right) ^2 }{ \left( ( \lambda _j - E_1)^2 + \eta ^2_k\right) \left( \lambda _j - E_2)^2 + \eta ^2_\ell \right) }. \end{aligned}$$

(A.8)

The second term is canceled by (A.3) and (A.4), except for the $p_{jj}^2$ term, and

$$\begin{aligned} \frac{1}{N^2} \sum _{j,m=1}^N \frac{ \eta _k\eta _\ell {\widehat{p}}^2_{jm}(I_N) }{ \left( ( \lambda _j - E_1)^2 + \eta ^2_k\right) \left( ( \lambda _m - E_2)^2 + \eta ^2_\ell \right) }&\leqslant N^{2{\mathfrak {s}} + 2{\mathfrak {a}}} \left( {{\,\mathrm{Im}\,}}m_N(E_1 + \mathrm {i}\eta _k) \right) \left( {{\,\mathrm{Im}\,}}m_N(E_2 + \mathrm {i}\eta _\ell ) \right) \end{aligned}$$

(A.9)

$$\begin{aligned}&\leqslant N^{2{\mathfrak {s}} + 2{\mathfrak {a}} + 8 \varepsilon _2 }\left( \frac{{\widehat{k}}}{N}\right) ^{1/3}\left( \frac{{\widehat{\ell }}}{N}\right) ^{1/3}. \end{aligned}$$

(A.10)

In (A.9), we used the definition of the boostrap set ${\mathcal {E}}$. In the last line, we averaged (A.1) over all indices to bound the imaginary part of $m_N$. This proves the first claim.

The first derivative of (A.5) is, where $({\mathbf {e}}_\alpha )_{\alpha \in [\![1,N]\!]}$ is the standard basis,

$$\begin{aligned} \partial _{ab}&\left( \frac{1 }{ | I_N|}\sum _{ \alpha , \beta \in I_N}{{\,\mathrm{Im}\,}}\langle {\mathbf {q}}_\alpha ,G(z_1){\mathbf {q}}_\beta \rangle {{\,\mathrm{Im}\,}}\langle {\mathbf {q}}_\alpha ,G(z_2){\mathbf {q}}_\beta \rangle \right) \end{aligned}$$

(A.11)

$$\begin{aligned}&\quad = -\frac{1}{|I_N|} \sum _{\alpha , \beta \in I_N} {{\,\mathrm{Im}\,}}\langle {\mathbf {q}}_\alpha ,G(z_1){\mathbf {e}}_a\rangle \langle {\mathbf {e}}_b,G(z_1){\mathbf {q}}_\beta \rangle {{\,\mathrm{Im}\,}}\langle {\mathbf {q}}_\alpha ,G(z_2){\mathbf {q}}_\beta \rangle \end{aligned}$$

(A.12)

$$\begin{aligned}&\qquad -\, \frac{1}{|I_N|} \sum _{\alpha , \beta \in I_N} {{\,\mathrm{Im}\,}}\langle {\mathbf {q}}_\alpha ,G(z_1){\mathbf {e}}_b\rangle \langle {\mathbf {e}}_a,G(z_1){\mathbf {q}}_\beta \rangle {{\,\mathrm{Im}\,}}\langle {\mathbf {q}}_\alpha ,G(z_2){\mathbf {q}}_\beta \rangle \end{aligned}$$

(A.13)

$$\begin{aligned}&\qquad -\, \frac{1}{|I_N|} \sum _{\alpha , \beta \in I_N} \langle {\mathbf {q}}_\alpha ,G(z_1){\mathbf {q}}_\beta \rangle {{\,\mathrm{Im}\,}}\langle {\mathbf {q}}_\alpha ,G(z_2){\mathbf {e}}_a\rangle \langle {\mathbf {e}}_b,G(z_2){\mathbf {q}}_\beta \rangle \end{aligned}$$

(A.14)

$$\begin{aligned}&\qquad -\, \frac{1}{|I_N|}\sum _{\alpha , \beta \in I_N} \langle {\mathbf {q}}_\alpha ,G(z_1){\mathbf {q}}_\beta \rangle {{\,\mathrm{Im}\,}}\langle {\mathbf {q}}_\alpha ,G(z_2){\mathbf {e}}_b\rangle \langle {\mathbf {e}}_a,G(z_2){\mathbf {q}}_\beta \rangle . \end{aligned}$$

(A.15)

We compute a representative term:

$$\begin{aligned} \frac{1}{|I_N|}&\sum _{\alpha , \beta \in I_N} {{\,\mathrm{Im}\,}}\langle {\mathbf {q}}_\alpha ,G(z_1){\mathbf {e}}_a\rangle \langle {\mathbf {e}}_b,G(z_1){\mathbf {q}}_\beta \rangle {{\,\mathrm{Im}\,}}\langle {\mathbf {q}}_\alpha ,G(z_2){\mathbf {q}}_\beta \rangle \end{aligned}$$

(A.16)

$$\begin{aligned}&= \frac{1}{|I_N|} \sum _{\alpha , \beta \in I_N} {{\,\mathrm{Im}\,}}\left[ \left( \sum _{i=1}^N \frac{\langle {\mathbf {q}}_\alpha ,{\mathbf {u}}_i\rangle {\mathbf {u}}_i(a) }{\lambda _i - z_1 } \right) \left( \sum _{j=1}^N \frac{{\mathbf {u}}_j(b) \langle {\mathbf {q}}_\beta ,{\mathbf {u}}_j\rangle }{\lambda _j - z_1 } \right) \right] \nonumber \\&\quad {{\,\mathrm{Im}\,}}\left( \sum _{m=1}^N \frac{\langle {\mathbf {q}}_\alpha ,{\mathbf {u}}_m\rangle \langle {\mathbf {q}}_\beta ,{\mathbf {u}}_m\rangle }{\lambda _m - z_2 }\right) \end{aligned}$$

(A.17)

$$\begin{aligned}&=\frac{1}{|I_N|} \sum _{\alpha , \beta \in I_N}\sum _{i,j=1}^N \sum _{m=1}^N {{\,\mathrm{Im}\,}}\left[ \frac{ {\mathbf {u}}_i(a) {\mathbf {u}}_j(b) }{(\lambda _i - z_1 )(\lambda _j - z_1 )} \right] \nonumber \\&\quad {{\,\mathrm{Im}\,}}\left( \frac{1 }{\lambda _m - z_2 }\right) \langle {\mathbf {q}}_\alpha ,{\mathbf {u}}_i\rangle \langle {\mathbf {q}}_\alpha ,{\mathbf {u}}_m\rangle \langle {\mathbf {q}}_\beta ,{\mathbf {u}}_m\rangle \langle {\mathbf {q}}_\beta ,{\mathbf {u}}_j\rangle . \end{aligned}$$

(A.18)

When $i \ne m$ and $j \ne m$, this can be bounded similarly to the previous computation by applying the bootstrap hypothesis and eigenvector delocalization. We also observe the bound

$$\begin{aligned} \sum _{i,j} \left| {{\,\mathrm{Im}\,}}\frac{ 1 }{(\lambda _i - z_1 )(\lambda _j - z_1 )} \right|&= \sum _{i,j} \left| {{\,\mathrm{Im}\,}}\frac{ 1 }{(\lambda _i - z_1 )} {{\,\mathrm{Re}\,}}\frac{1}{(\lambda _j - z_1 )} + {{\,\mathrm{Re}\,}}\frac{ 1 }{(\lambda _i - z_1 )} {{\,\mathrm{Im}\,}}\frac{1}{(\lambda _j - z_1 )} \right| \end{aligned}$$

(A.19)

$$\begin{aligned}&\leqslant 2 \sum _{i,j} {{\,\mathrm{Im}\,}}\frac{ 1 }{(\lambda _i - z_1 )} \frac{1}{| \lambda _j - z_1 |} \end{aligned}$$

(A.20)

$$\begin{aligned}&\leqslant 2 N {{\,\mathrm{Im}\,}}m_N(z_1) \left( \sum _{i} 1 + \frac{1}{| \lambda _i - z_1|^2 }\right) \end{aligned}$$

(A.21)

$$\begin{aligned}&\leqslant 2 N {{\,\mathrm{Im}\,}}m_N(z_1) \left( N \big ( 1 + {{\,\mathrm{Im}\,}}m_N(z_1) \big ) \right) , \end{aligned}$$

(A.22)

and we can again use (A.1) to bound the imaginary part of $m_N$.

We now consider the other cases. When $i=j=m$ we get

$$\begin{aligned}&\frac{1}{|I_N|} \sum _{\alpha , \beta \in I_N} \sum _{i=1}^N {{\,\mathrm{Im}\,}}\left[ \frac{ {\mathbf {u}}_i(a) {\mathbf {u}}_i(b) }{(\lambda _i- z_1 )(\lambda _i - z_1 )} \right] {{\,\mathrm{Im}\,}}\left( \frac{1 }{\lambda _i - z_2 }\right) \langle {\mathbf {q}}_\alpha ,{\mathbf {u}}_i\rangle ^2 \langle {\mathbf {q}}_\beta ,{\mathbf {u}}_i\rangle ^2 \end{aligned}$$

(A.23)

$$\begin{aligned}&\quad = \frac{1}{|I_N|} \sum _{i=1}^N {{\,\mathrm{Im}\,}}\left[ \frac{ {\mathbf {u}}_i(a) {\mathbf {u}}_i(b) }{(\lambda _i- z_1 )(\lambda _i - z_1 )} \right] {{\,\mathrm{Im}\,}}\left( \frac{1 }{\lambda _i - z_2 }\right) \left( p_{ii}(I_N) + \frac{ | I_N |}{N} \right) ^2. \end{aligned}$$

(A.24)

The $p_{ii}(I_N)^2$ term is negligible, and the $\left( \frac{ | I_N |}{N} \right) ^2$ term in the expansion is canceled by the derivative of (A.3). (Note that the cross-term $2 p_{ii}(I_N) | I_N | N^{-1}$ contributes to the $\left( \frac{ | I_N |}{N} \right) ^2$ term by the definition of $p_{ii}$.) We are left with

$$\begin{aligned} \frac{2}{N} \sum _{i=1}^N {{\,\mathrm{Im}\,}}\left[ \frac{ {\mathbf {u}}_i(a) {\mathbf {u}}_i(b) }{(\lambda _i- z_1 )(\lambda _i - z_1 )} \right] {{\,\mathrm{Im}\,}}\left( \frac{1 }{\lambda _i - z_2 }\right) \left( p_{ii} + \frac{ | I_N |}{N} \right) . \end{aligned}$$

(A.25)

With $i=m, j \ne m$ or $j =m, i \ne m$, we get cross-terms

$$\begin{aligned}&\frac{1}{N} \sum _{i\ne j }^N {{\,\mathrm{Im}\,}}\left[ \frac{ {\mathbf {u}}_i(a) {\mathbf {u}}_j(b) }{(\lambda _i - z_k )(\lambda _j - z_k )} \right] {{\,\mathrm{Im}\,}}\left( \frac{1 }{\lambda _i - z_\ell }\right) p_{ij},\nonumber \\&\quad \frac{1}{N} \sum _{i\ne j }^N {{\,\mathrm{Im}\,}}\left[ \frac{ {\mathbf {u}}_i(a) {\mathbf {u}}_j(b) }{(\lambda _i - z_k )(\lambda _j - z_k )} \right] {{\,\mathrm{Im}\,}}\left( \frac{1 }{\lambda _j - z_\ell }\right) p_{ij}. \end{aligned}$$

(A.26)

Together with the previous sum, these are canceled by the derivative of (A.4). The other three terms in (A.15) are similar.

The higher derivatives in (5.70) are controlled in the same way. One thing important to note is that differentiating only adds resolvent terms and, in particular, does not add any sums over eigenvector entries. Thus, there can only be a quadratic term for $p_{k\ell }$ which is bounded by $N^{2{\mathfrak {a}}+2{\mathfrak {s}}}$ regardless of the order of the derivatives. We omit the straightforward but tedious computations.

The remaining two almost sure bounds are trivial consequences of the inequality $\left| G_{ij}(z) \right| \leqslant ({{\,\mathrm{Im}\,}}z)^{-1}$. $\quad \square$

Proof of Lemma 7.3

We denote $z=E+\mathrm {i}\eta _\ell$ and we use the spectral theorem to see that

$$\begin{aligned}&\frac{1}{ \sqrt{| I_N|}} \left| \sum _{ \alpha \in I_N} {{\,\mathrm{Im}\,}}\langle {\mathbf {q}}_\alpha ,G(z){\mathbf {q}}_\alpha \rangle - {{\,\mathrm{Im}\,}}m_N(z) \right| = \frac{1}{ \sqrt{| I_N|}} \left| \sum _{k=1} \sum _{ \alpha \in I_N}\frac{ \eta _\ell \left( \langle {\mathbf {q}}_\alpha ,{\mathbf {u}}_k\rangle ^2 - N^{-1} \right) }{ ( E - \lambda _k)^2 + \eta ^2_\ell } \right| \end{aligned}$$

(A.27)

$$\begin{aligned}&\leqslant \frac{1}{N} \sum _{k=1} \frac{ \eta _\ell \left| {\widehat{p}}_{kk}(I_N)\right| }{ ( E - \lambda _k)^2 + \eta ^2_\ell } \end{aligned}$$

(A.28)

$$\begin{aligned}&\leqslant N^{-1 + \sigma + \omega } \sum _{k=1} \frac{ \eta _\ell }{ ( E - \lambda _k)^2 + \eta ^2_\ell } \end{aligned}$$

(A.29)

$$\begin{aligned}&= CN^{ \sigma + \omega } {{\,\mathrm{Im}\,}}m_N(z) \end{aligned}$$

(A.30)

$$\begin{aligned}&\leqslant CN^{\sigma + \omega + 4 \varepsilon _2} \left( \frac{{\widehat{\ell }}}{N}\right) ^{1/3}. \end{aligned}$$

(A.31)

In the last line we used (A.1). Here $C = C(\varepsilon _2) >0$ is a constant depending on $\varepsilon _2$.

For the first derivative, we have

$$\begin{aligned}&\partial _{ab} \left( \frac{1}{ \sqrt{| I_N|}} \sum _{ \alpha \in I_N} {{\,\mathrm{Im}\,}}\langle {\mathbf {q}}_\alpha ,G(z){\mathbf {q}}_\alpha \rangle - {{\,\mathrm{Im}\,}}m_N(z)\right) \nonumber \\&\quad = \frac{1}{ \sqrt{| I_N|}} \sum _{ \alpha \in I_N} \left( {{\,\mathrm{Im}\,}}\langle {\mathbf {q}}_\alpha ,G(z){\mathbf {e}}_a\rangle \langle {\mathbf {e}}_b,G(z){\mathbf {q}}_\alpha \rangle - {{\,\mathrm{Im}\,}}\frac{1}{N} \sum _{i=1}^N G_{i a }(z) G_{bi}(z) \right) .\nonumber \\ \end{aligned}$$

(A.32)

Using the spectral theorem, we compute

$$\begin{aligned} \langle {\mathbf {q}}_\alpha ,G(z){\mathbf {e}}_a\rangle \langle {\mathbf {e}}_b,G(z){\mathbf {q}}_\alpha \rangle&= \left( \sum _{k=1}^N \frac{\langle {\mathbf {q}}_\alpha ,{\mathbf {u}}_k\rangle {\mathbf {u}}_k(a) }{\lambda _k - z }\right) \left( \sum _{j=1}^N \frac{ {\mathbf {u}}_j(b) \langle {\mathbf {q}}_\alpha ,{\mathbf {u}}_j\rangle }{\lambda _j - z }\right) \end{aligned}$$

(A.33)

$$\begin{aligned}&= \sum _{k=1}^N \frac{\langle {\mathbf {q}}_\alpha ,{\mathbf {u}}_k\rangle ^2 {\mathbf {u}}_k(a) {\mathbf {u}}_k(b) }{(\lambda _k - z )^2} + \sum _{k \ne j} \frac{ \langle {\mathbf {q}}_\alpha ,{\mathbf {u}}_k\rangle \langle {\mathbf {q}}_\alpha ,{\mathbf {u}}_j\rangle {\mathbf {u}}_k(a) {\mathbf {u}}_j(b)}{(\lambda _k - z )(\lambda _j - z)}. \end{aligned}$$

(A.34)

We also have

$$\begin{aligned} \frac{1}{N} \sum _{i=1}^N G_{i a }(z) G_{bi}(z)&= \frac{1}{N} \sum _{i=1}^N \left( \sum _{k=1}^N \frac{{\mathbf {u}}_k(i) {\mathbf {u}}_k(a) }{\lambda _k - z }\right) \left( \sum _{j=1}^N \frac{ {\mathbf {u}}_j(b) {\mathbf {u}}_j(i) }{\lambda _j - z }\right) \end{aligned}$$

(A.35)

$$\begin{aligned}&= \frac{1}{N} \sum _{k=1}^N \frac{{\mathbf {u}}_k(a) {\mathbf {u}}_k(b) }{(\lambda _k - z )^2}. \end{aligned}$$

(A.36)

We used that the terms in the product with $k\ne j$ vanish due to the sum over i and the orthogonality of ${\mathbf {u}}_k$ and ${\mathbf {u}}_j$.

Subtracting (A.36) from the first term of (A.34), we have

$$\begin{aligned}&{{\,\mathrm{Im}\,}}\frac{1}{\sqrt{|I_N | } }\sum _{k=1}^N \sum _{\alpha \in I_N} \frac{ ( \langle {\mathbf {q}}_\alpha ,{\mathbf {u}}_k\rangle ^2 - N^{-1}) {\mathbf {u}}_k(a) {\mathbf {u}}_k(b) }{(\lambda _k - z )^2} = \frac{1}{N} {{\,\mathrm{Im}\,}}\sum _{k=1}^N \frac{ {\widehat{p}}_{kk}(I_N) {\mathbf {u}}_k(a) {\mathbf {u}}_k(b) }{(\lambda _k - z )^2} \end{aligned}$$

(A.37)

$$\begin{aligned}&\leqslant N^{ - 2 + \sigma + 3 \omega } \sum _{k=1}^N \frac{ 1 }{|\lambda _k - z |^2} \end{aligned}$$

(A.38)

$$\begin{aligned}&\leqslant N^{ - 2 + \sigma + 3 \omega } \frac{1}{\eta _\ell } {{\,\mathrm{Im}\,}}\sum _{k=1}^N \frac{ 1 }{\lambda _k - z } \end{aligned}$$

(A.39)

$$\begin{aligned}&\leqslant N^{ \sigma + 3 \omega + 5 \varepsilon _2 } \left( \frac{{\widehat{\ell }}}{N}\right) ^{1/3}. \end{aligned}$$

(A.40)

In the last inequality, we used (A.1). It remains to control the second term in (A.34). Using (A.19), this is

$$\begin{aligned}&{{\,\mathrm{Im}\,}}\sum _{k \ne j} \frac{1}{\sqrt{|I_N | } } \sum _{\alpha \in I_N} \frac{ \langle {\mathbf {q}}_\alpha ,{\mathbf {u}}_k\rangle \langle {\mathbf {q}}_\alpha ,{\mathbf {u}}_j\rangle {\mathbf {u}}_k(a) {\mathbf {u}}_j(b)}{(\lambda _k - z )(\lambda _j - z)} =\frac{1}{N} {{\,\mathrm{Im}\,}}\sum _{k \ne j} \frac{ {\widehat{p}}_{kj} {\mathbf {u}}_k(a) {\mathbf {u}}_j(b)}{(\lambda _k - z )(\lambda _j - z)} \end{aligned}$$

(A.41)

$$\begin{aligned}&\leqslant N^{-2 + \sigma +3 \omega } \sum _{k \ne j} {{\,\mathrm{Im}\,}}\frac{1}{(\lambda _k - z )(\lambda _j - z)} \end{aligned}$$

(A.42)

$$\begin{aligned}&\leqslant N^{ \sigma +3 \omega + 8 \varepsilon _2} \left( \frac{{\widehat{\ell }}}{N}\right) ^{1/3}. \end{aligned}$$

(A.43)

For the second derivative, we compute

$$\begin{aligned} \partial ^2_{ab}&\left( \frac{1}{ \sqrt{| I_N|}} \sum _{ \alpha \in I_N} {{\,\mathrm{Im}\,}}\langle {\mathbf {q}}_\alpha ,G(z){\mathbf {q}}_\alpha \rangle - {{\,\mathrm{Im}\,}}m_N(z)\right) \end{aligned}$$

(A.44)

by noting that

$$\begin{aligned}&\partial ^2_{ab} \langle {\mathbf {q}}_\alpha ,G(z){\mathbf {q}}_\alpha \rangle = 2 \langle {\mathbf {q}}_\alpha ,G(z){\mathbf {e}}_a\rangle G_{b a}(z) \langle {\mathbf {e}}_b,G(z){\mathbf {q}}_\alpha \rangle \end{aligned}$$

(A.45)

$$\begin{aligned}&\quad +\, \langle {\mathbf {q}}_\alpha ,G(z){\mathbf {e}}_b\rangle G_{a a}(z) \langle {\mathbf {e}}_b,G(z){\mathbf {q}}_\alpha \rangle \end{aligned}$$

(A.46)

$$\begin{aligned}&\quad +\, \langle {\mathbf {q}}_\alpha ,G(z){\mathbf {e}}_a\rangle G_{b a}(z) \langle {\mathbf {e}}_a,G(z){\mathbf {q}}_\alpha \rangle . \end{aligned}$$

(A.47)

One representative term is

$$\begin{aligned}&\frac{2}{ \sqrt{| I_N|}} \sum _{ \alpha \in I_N} \left( {{\,\mathrm{Im}\,}}\langle {\mathbf {q}}_\alpha ,G(z){\mathbf {e}}_a\rangle G_{b a}(z) \langle {\mathbf {e}}_b,G(z){\mathbf {q}}_\alpha \rangle - {{\,\mathrm{Im}\,}}\frac{1}{N} \sum _{i=1}^N G_{i a }(z) G_{ba}(z) G_{bi}(z) \right) \end{aligned}$$

(A.48)

$$\begin{aligned}&= \frac{2}{ \sqrt{| I_N|}} \sum _{ \alpha \in I_N} \left( {{\,\mathrm{Im}\,}}\left[ G_{b a}(z) \left( \langle {\mathbf {q}}_\alpha ,G(z){\mathbf {e}}_a\rangle \langle {\mathbf {e}}_b,G(z){\mathbf {q}}_\alpha \rangle - \frac{1}{N} \sum _{i=1}^N G_{i a }(z) G_{bi}(z) \right) \right] \right) \end{aligned}$$

(A.49)

$$\begin{aligned}&= \frac{2}{ \sqrt{| I_N|}}{{\,\mathrm{Im}\,}}G_{b a}(z) \sum _{ \alpha \in I_N} {{\,\mathrm{Re}\,}}\left( \langle {\mathbf {q}}_\alpha ,G(z){\mathbf {e}}_a\rangle \langle {\mathbf {e}}_b,G(z){\mathbf {q}}_\alpha \rangle - \frac{1}{N} \sum _{i=1}^N G_{i a }(z) G_{bi}(z) \right) \end{aligned}$$

(A.50)

$$\begin{aligned}&\quad +\,\frac{2}{ \sqrt{| I_N|}}{{\,\mathrm{Re}\,}}G_{b a}(z) \sum _{ \alpha \in I_N} {{\,\mathrm{Im}\,}}\left( \langle {\mathbf {q}}_\alpha ,G(z){\mathbf {e}}_a\rangle \langle {\mathbf {e}}_b,G(z){\mathbf {q}}_\alpha \rangle - \frac{1}{N} \sum _{i=1}^N G_{i a }(z) G_{bi}(z) \right) \end{aligned}$$

(A.51)

$$\begin{aligned}&\leqslant C N^{ \sigma +3 \omega + 12 \varepsilon _2}\left( \frac{{\widehat{\ell }}}{N}\right) ^{1/3}. \end{aligned}$$

(A.52)

In the last line, we used (A.1), (A.2), and repeated the computations for (A.32) to obtain

$$\begin{aligned} \frac{2}{ \sqrt{| I_N|}} \sum _{ \alpha \in I_N} {{\,\mathrm{Re}\,}}\left( \langle {\mathbf {q}}_\alpha ,G(z){\mathbf {e}}_a\rangle \langle {\mathbf {e}}_b,G(z){\mathbf {q}}_\alpha \rangle - \frac{1}{N} \sum _{i=1}^N G_{i a }(z) G_{bi}(z) \right) \leqslant C N^{\sigma +3 \omega + 8 \varepsilon _2}.\nonumber \\ \end{aligned}$$

(A.53)

Differentiating further, we find terms such as

$$\begin{aligned} \frac{(k+1)!}{ \sqrt{| I_N|}} \sum _{ \alpha \in I_N} {{\,\mathrm{Im}\,}}\left( \langle {\mathbf {q}}_\alpha ,G(z){\mathbf {e}}_a\rangle G_{b a}(z)^k \langle {\mathbf {e}}_b,G(z){\mathbf {q}}_\alpha \rangle - \frac{1}{N} \sum _{i=1}^N G_{i a }(z) G_{ba}(z)^k G_{bi}(z) \right) \end{aligned}$$

(A.54)

These higher derivatives are bounded exactly as the second derivative was. For example, in (A.54), we separate out the $G_{ba}^k$ term and bounding it using (A.1). $\quad \square$

Symmetrized Moment Observables

In this section, we define a generalization of the moment observables given in Definition 4.3, which facilitated the proof of decorrelation estimates for entries of one or two eigenvectors. These generalized observables lead to decorrelation estimates for more than two eigenvectors, which may be of independent interest. For brevity, we only verify that the observables follow a parabolic equation. We omit the proofs of the corresponding decorrelation estimates, since they are straightforward applications of the maximum principle, as in Sect. 4.

Consider a configuration of n particles $\varvec{\xi }$ as in Sect. 3.1. We denote by $i_1,\dots ,i_p$ the sites where there is at least one particle and define the set of vertices

$$\begin{aligned} {\mathscr {V}}_{\varvec{\xi }} = \{(i_q,a)\in {\mathbb {N}}^2,\, q\in [\![1,p]\!],\, a\in [\![1,\xi _{i_q}]\!] \}. \end{aligned}$$

Note that ${\mathscr {V}}_{\varvec{\xi }}$ differs from ${\mathcal {V}}_{\varvec{\xi }}$ (defined in (3.1)) since we do not double the number of particles here. We define now a set of matchings ${\mathfrak {M}}_{\varvec{\xi }}$ as the set of functions $\sigma$ such that if $v\in {\mathscr {V}}_{\varvec{\xi }}$, then $\sigma (v)=(\sigma _1(v),\sigma _2(v))\in [\![1,2n]\!]^2$ with $\sigma _1(v)<\sigma _2(v)$ and $\bigcup _{v\in {\mathscr {V}}_{\varvec{\xi }}}\{\sigma _1(v),\sigma _2(v)\}=[\![1,2n]\!]$. This matches each particle to a pair of distinct indices in $[\![1,2n]\!]$. Note that $\vert {\mathfrak {M}}_{\varvec{\xi }}\vert =2^{-n}(2n)!$ (Fig. 4).

Fig. 4

Examples

We then define the generalized moment observable as

Proposition 1

For all $s\in (0,1)$, the perfect matching observable $f_s$ defined in (3.2) satisfies the equation

$$\begin{aligned} \partial _s g_s(\varvec{\xi }) = \sum _{k\ne \ell } \xi _k(1+2\xi _\ell ) \frac{g_s(\varvec{\xi }^{k,\ell })-g_s(\varvec{\xi })}{N(\lambda _k(s)-\lambda _\ell (s))^2}, \end{aligned}$$

(B.1)

where $\varvec{\xi }^{k,\ell }$ is the configuration where we move a particle from k to $\ell$.

Before giving the proof of Proposition 1, we recall the generator for the dynamics (2.7).

Lemma B.1

The generator acting on smooth functions of the eigenvector diffusion (2.7) is

$$\begin{aligned} {\mathcal {L}}_s = \frac{1}{2}\sum _{1\leqslant k<\ell \leqslant N}\frac{1}{N(\lambda _k-\lambda _\ell )^2}X_{k\ell }^2 \end{aligned}$$

with

$$\begin{aligned} X_{k\ell }=\sum _{\alpha =1}^N({\mathbf {u}}_k(\alpha )\partial _{{\mathbf {u}}_\ell (\alpha )}-{\mathbf {u}}_\ell (\alpha )\partial _{{\mathbf {u}}_k(\alpha )}). \end{aligned}$$

Proof of Proposition 1

To simplify the notation we define ${\mathbf {v}}_k(p)= \langle {\mathbf {q}}_{\alpha _p},{\mathbf {u}}_k\rangle$ so that

Consider $k,\ell \in [\![1,N]\!]$ such that there is at least one particle of $\varvec{\xi }$ on sites k and $\ell$. Then we can write

$$\begin{aligned} \prod _{\begin{array}{c} v\in {\mathscr {V}}_{\varvec{\xi }}\\ v=(p,a) \end{array}}{\mathbf {v}}_p(\sigma _1(v)){\mathbf {v}}_p(\sigma _2(v)) = P(k,\ell ) Q(k)Q(\ell ) \end{aligned}$$

with

$$\begin{aligned} Q(j)=\prod _{\begin{array}{c} v\in {\mathscr {V}}_{\varvec{\xi }}\\ v=(j,a) \end{array}}{\mathbf {v}}_j(\sigma _1(v)){\mathbf {v}}_j(\sigma _2(v))\quad \text {and}\quad P(k,\ell )=\prod _{\begin{array}{c} v\in {\mathscr {V}}_{\varvec{\xi }}\\ v=(p,a)\\ p\ne k,\ell \end{array}}{\mathbf {v}}_p(\sigma _1(v)){\mathbf {v}}_p(\sigma _2(v)). \end{aligned}$$

Note that $P(k,\ell )$ does not involve any eigenvectors ${\mathbf {u}}_k$ or ${\mathbf {u}}_\ell$. We have

$$\begin{aligned} X_{k\ell }^2[P(k,\ell ) Q(k)Q(\ell )]=P(k,\ell )X_{k\ell }\left[ Q(\ell )X_{k\ell }Q(k)+Q(k)X_{k\ell }Q(\ell )\right] . \end{aligned}$$

(B.2)

We now consider the first term $X_{k\ell }Q(\ell )X_{k\ell }Q(k)+Q(\ell )X_{k\ell }^2Q(k)$; the second term can be computed similarly. We have

$$\begin{aligned} X_{k\ell }Q(k)= & {} \sum _{v=(k,a)\in {\mathscr {V}}_{\varvec{\xi }}}(-{\mathbf {v}}_\ell (\sigma _1(v)){\mathbf {v}}_k(\sigma _2(v))-{\mathbf {v}}_k(\sigma _1(v)){\mathbf {v}}_\ell (\sigma _2(v)))\nonumber \\&\quad \prod _{\begin{array}{c} v'=(k,b)\in {\mathscr {V}}_{\varvec{\xi }}\\ b\ne a \end{array}}{\mathbf {v}}_k(\sigma _1(v')){\mathbf {v}}_k(\sigma _2(v')). \end{aligned}$$

(B.3)

After symmetrization, we obtain

(B.4)

where the 4 comes from unfolding the products and $\xi _k\xi _\ell$ comes from choosing the particles in the sum in (B.3). There are two terms when computing $X_{k\ell }^2Q(k)$ coming from either differentiating the first term or the product in (B.3). When differentiating the first term we obtain

$$\begin{aligned} (I)= & {} \sum _{v=(k,a)\in {\mathscr {V}}_{\varvec{\xi }}}(-2{\mathbf {v}}_k(\sigma _1(v)){\mathbf {v}}_k(\sigma _1(v))+2{\mathbf {v}}_\ell (\sigma _1(v)){\mathbf {v}}_\ell (\sigma _2(v)))\nonumber \\&\quad \prod _{\begin{array}{c} v'=(k,b)\in {\mathscr {V}}_{\varvec{\xi }}\\ b\ne a \end{array}}{\mathbf {v}}_k(\sigma _1(v')){\mathbf {v}}_k(\sigma _2(v')). \end{aligned}$$

(B.5)

When differentiating the second term, we obtain a term with a double sum

$$\begin{aligned}&(II) =\sum _{\begin{array}{c} v=(k,a)\in {\mathscr {V}}_{\varvec{\xi }}\\ w=(k,b)\in {\mathscr {V}}_{\varvec{\xi }}\\ a\ne b \end{array}}\nonumber \\&\quad (-{\mathbf {v}}_\ell (\sigma _1(v)){\mathbf {v}}_k(\sigma _2(v))-{\mathbf {v}}_k(\sigma _1(v)){\mathbf {v}}_\ell (\sigma _2(v)))(-{\mathbf {v}}_\ell (\sigma _1(w)){\mathbf {v}}_k(\sigma _2(w))-{\mathbf {v}}_k(\sigma _1(w)){\mathbf {v}}_\ell (\sigma _2(w)))\nonumber \\&\quad \times \prod _{\begin{array}{c} v'=(k,c)\in {\mathscr {V}}_{\varvec{\xi }}\\ c\ne a,b \end{array}}{\mathbf {v}}_k(\sigma _1(v')){\mathbf {v}}_k(\sigma _2(v')). \end{aligned}$$

(B.6)

After symmetrization, we have

(B.7)

where the $\xi _k$ comes from the sum in (B.5) and the second term stems from the identity ${\mathcal {M}}(\varvec{\xi }^{k,\ell })=\frac{2\xi _\ell +1}{2\xi _k-1}{\mathcal {M}}(\varvec{\xi })$. Similarly, the symmetrization gives

(B.8)

Finally, when summing (B.4), (B.7), (B.8), and adding the second term coming from (B.2), we obtain

$$\begin{aligned} X_{k\ell }^2g_s(\varvec{\xi }) = 2\xi _k(1+2\xi _\ell )(g_s(\varvec{\xi }^{k,\ell })-g_s(\varvec{\xi })) + 2\xi _\ell (1+2\xi _k)(g_s(\varvec{\xi }^{\ell ,k})-g_s(\varvec{\xi })). \end{aligned}$$

If there is a particle on site k and not on site $\ell$, the computation is similar but easier and if there are no particles in site k and $\ell$ then $X_{k\ell }g_s(\varvec{\xi })=0$. This completes the proof of Proposition 1. $\quad \square$