Fermionic eigenvector moment flow

Abstract

We exhibit new functions of the eigenvectors of the Dyson Brownian motion which follow an equation similar to the Bourgade-Yau eigenvector moment flow (Bourgade and Yau in Commun Math Phys 350(1):231–278, 2017). These observables can be seen as a Fermionic counterpart to the original (Bosonic) ones. By analyzing both Fermionic and Bosonic observables, we obtain new correlations between eigenvectors: (i) The fluctuations $\sum _{\alpha \in I}\vert u_k(\alpha )\vert ^2-{\vert I\vert }/{N}$ decorrelate for distinct eigenvectors as the dimension N grows. (ii) An optimal estimate on the partial inner product $\sum _{\alpha \in I}u_k(\alpha )\overline{u_\ell }(\alpha )$ between two eigenvectors is given. These static results obtained by integrable dynamics are stated for generalized Wigner matrices and should apply to wide classes of mean field models.

Appendices

Combinatorial proof of Theorem 1.2

In this appendix, we give another proof of Theorem 1.2 where we use the generator of the dynamics (1.7) without any consideration of Grassmann variables. The proof is based on the expansion of the determinant and a careful bookkeeping on the action of the generator on permutations.

Proof of Theorem 1.2

First define

$$\begin{aligned} g(\varvec{\xi })=\sum _{\sigma \in {\mathfrak {S}}_n}\epsilon (\sigma )\prod _{i=1}^np_{i_ki_{\sigma (k)}}\quad \text {so that}\quad f_s^{\text {Fer}}(\varvec{\xi })=\mathbb {E}\left[ g(\varvec{\xi })\vert \varvec{\lambda }\right] . \end{aligned}$$

We therefore need to show the following two equality

$$\begin{aligned} X^2_{i_kj}g(\varvec{\xi })&=2(g(\varvec{\xi }^{i_kj})-g(\varvec{\xi }))\quad \text {for}\quad k\in \{1\dots ,n\},\,j\notin \{i_1,\dots ,i_n\}, \end{aligned}$$

(A.1)

$$\begin{aligned} X^2_{i_ki_\ell }g(\varvec{\xi })&=0\quad \text {for}\quad k,\ell \in \{1,\dots ,n\}. \end{aligned}$$

(A.2)

Part of the reasoning is done via induction. We first describe the proof for two particles. For simplicity, we describe one of the joint moments of the family $\left( p_{k\ell }\right)$ as a graph corresponding to a permutation in the determinant. For instance, for two particles, we have two distinct graphs.

Thus we can write our fermionic observable for these two particles as

$$\begin{aligned} g_t(\varvec{\xi })=p_{i_1i_1}p_{i_2i_2}-p_{i_1i_2}^2=\textcircled {1}-\textcircled {2}. \end{aligned}$$

Note that we have a sign difference between the terms because of the changing signature between these two permutations. Now, see that the generator X operates on our family of overlaps $(p_{i_ki_\ell })$ in the following way

$$\begin{aligned} \begin{aligned}&X_{i_ki_\ell }p_{i_ki_k}=-2p_{i_ki_\ell }=-X_{i_ki_\ell }p_{i_\ell i_\ell },\,X_{i_ki_\ell }p_{i_ki_\ell }=p_{i_ki_k}-p_{i_\ell i_\ell },\\&X_{i_ki_\ell }p_{i_kj}=-p_{i_\ell j},\,X_{i_ki_\ell }p_{i_\ell j}=p_{kj}. \end{aligned} \end{aligned}$$

(A.3)

With these algebraic relations, one can easily see that we have (A.1) for $g_t(\varvec{\xi })$. One can also easily deduce (A.2) with these simple relations, however, in order to explain the more detailed approach of the case of n particles, we disclose the proof in more details. We can first operate X on both of the terms in $g_t(\varvec{\xi })$ and see that

$$\begin{aligned} X^2_{i_1i_2}\left( p_{i_1i_1}p_{i_2i_2} \right) = 2\left( p_{i_1i_1}^2+p_{i_2i_2}^2-2(p_{i_1i_1}p_{i_2i_2}+2p_{i_1i_2}^2) \right) =X_{i_1i_2}^2p_{i_1i_2}^2. \end{aligned}$$

(A.4)

First see that (A.4) gives us that $X_{i_1i_2}^2g_t(\varvec{\xi })=0$ and the proof for two particles is clear. However, to introduce the notations we use in the case of n particles, we write (A.4) as the following, using the graphical representation from the previous table,

Consider now the case of n particles on $\{i_1,\dots ,i_n\}$. By induction, we can only look at the permutations in the sum where either $\ell (i_k)+\ell (i_\ell )= n$ or $\ell (i_k)=\ell (i_\ell )=n$ where $\ell (j)$ is the length of the cycle containing j. Note that the second condition is there to take in account the fact that $i_k$ and $i_\ell$ can be in the same cycle. Also see that by definition of $X_{i_ki_\ell }$, we are only interested in the sites $i_k,\,i_{\sigma (k)},\,i_{\sigma ^{-1}(k)},\,i_\ell ,\,i_{\sigma (\ell )}$ and $i_{\sigma ^{-1}(\ell )}.$

First consider the permutations such that $\ell (i_k)$ or $\ell (i_\ell )$ is equal to 1, such a permutation wil be represented by ① or ② in the following table.

These four graphs are the one involved when applying $X^2$ to ①. Note that while we display $i_{\sigma ^{-1}(\ell )}$ and $i_{\sigma (\ell )}$ as distinct points, they could potentially be the same. The dashed red line represent the rest of the permutation, note also that we have two distinct cycles for the graphs ① and ② while there is a single cycle for the graphs ③ and ④ so that .For simplicity, consider the notations

$$\begin{aligned} P^{(1)}_\ell&=p_{i_\ell i_\ell }p_{i_{\sigma ^{-1}(\ell )}i_\ell }p_{i_\ell i_{\sigma (\ell )}},\\ P^{(1)}_k&=p_{i_k i_k}p_{i_{\sigma ^{-1}(\ell )}i_k}p_{i_k i_{\sigma (\ell )}}. \end{aligned}$$

Now, using the relations (A.3) we obtain

So that finally, taking in account the different signatures, we finally have

Note that we have a coefficient of 2 in front of each graph because both $\sigma$ and $\sigma ^{-1}$ follows the same graph. Now, we consider permutations where $\ell (i_k)$ and $\ell (i_\ell )$ are greater than 1. Thus we consider such a permutation as the graph ⑤ in the following table.

In this table, we represented all the permutations which are relevant when applying $X_{i_ki_\ell }$ to a general permutation of type ⑤. The different colors explains the different behavior of the permutation on the rest of the sites that are not seen by the operator X but are relevant when counting the signatures on the different graphs. We first introduce the following notations as earlier

$$\begin{aligned} P_\ell ^{(2)}&= p_{i_{\sigma ^{-1}(k)}i_\ell } p_{i_\ell i_{\sigma (k)}} p_{i_{\sigma ^{-1}(\ell )}i_\ell } p_{i_\ell i_{\sigma (\ell )}},\\ P_k^{(2)}&= p_{i_{\sigma ^{-1}(k)}i_k} p_{i_ki_{\sigma (k)}} p_{i_{\sigma ^{-1}(\ell )}i_k} p_{i_k i_{\sigma (\ell )}}. \end{aligned}$$

Now, if we apply $X_{i_ki_\ell }$ to a permutation such that the cycle of $i_k$ and of $i_\ell$ are greater than 1 we obtain the following set of equations:

Now, in order to put all these equations together, one needs to see the number of permutations following these graphs and their respective signature. Both of these values depend on the number of cycles, which is equal to 1 or 2 in these cases, of the permutations and thus depend on the corresponding color in the previous table. We finally have

Combining this result with the case where $\ell (i_k)$ or $\ell (i_\ell )$ is equal to 1 gives us the result for any permutation which finally gives

$$\begin{aligned} X^2_{i_ki_\ell }g_t(\varvec{\xi })=0. \end{aligned}$$

$\square$

The Hermitian case

In this paper, we focused and developed the proof for symmetric random matrices, but the proof holds for Hermitian matrices as well. While the maximum principle technique can clearly be directly applied to the Hermitian case, we focused here in the definition of the Fermionic observable for the Hermitian Dyson Brownian motion. The Dyson vector flow in this case has a different generator and it is not necessarily clear that the determinant is still the correct one. Indeed, the Bosonic observable has a different form for Hermitian matrices [10, Appendix] since we obtain the permanent of a matrix instead of a Hafnian. We now give the Dyson flow of eigenvalues and eigenvectors for Hermitian matrices.

Definition B.1

Let B be a Hermitian $N\times N$ matrix such that $\hbox {Re}B_{ij},\hbox {Im}\,B_{ij}$ for $i<j$ and $B_{ii}/\sqrt{2}$ are standard independent brownian motions. The Hermitian Dyson Brownian motion is given by the stochastic differential equation

$$\begin{aligned} \text {d}H_s=\frac{\text {d}B_s}{\sqrt{2N}}-\frac{1}{2} H_s\text {d}t. \end{aligned}$$

(B.1)

Besides, it induces the following dynamics on eigenvalues and eigenvectors,

$$\begin{aligned} \text {d}\lambda _k&=\frac{\text {d}{\widetilde{B}}_{kk}}{\sqrt{2N}}+\left( \frac{1}{N}\sum _{\ell \ne k}\frac{1}{\lambda _k-\lambda _\ell }-\frac{\lambda _k}{2} \right) \text {d}s, \end{aligned}$$

(B.2)

$$\begin{aligned} \text {d}u_k&=\frac{1}{\sqrt{2N}}\sum _{\ell \ne k}\frac{\text {d}{\widetilde{B}}_{k\ell }}{\lambda _k-\lambda _\ell }u_\ell -\frac{1}{2N}\sum _{\ell \ne k}\frac{\text {d}s}{(\lambda _k-\lambda _\ell )^2}u_k \end{aligned}$$

(B.3)

where ${\widetilde{B}}$ is distributed as B.

The generator for the Hermitian Dyson vector flow is also known and given in the following proposition.

Proposition B.2

([9]) The generator acting on smooth functions of the diffusion (B.3) is given by

$$\begin{aligned} L_t = \frac{1}{2}\sum _{1\leqslant k<\ell \leqslant N}\frac{1}{N(\lambda _k-\lambda _\ell )^2} \left( X_{k\ell }{\overline{X}}_{k\ell }+{\overline{X}}_{k\ell }X_{k\ell } \right) \end{aligned}$$

(B.4)

with the operator $X_{k\ell }$ defined by

$$\begin{aligned} X_{k\ell }=\sum _{\alpha =1}^N \left( u_k(\alpha )\partial _{u_\ell (\alpha )} - {\overline{u}}_\ell (\alpha )\partial _{{\overline{u}}_\ell (\alpha )} \right) \quad \text {and} \quad {\overline{X}}_{k\ell }=\sum _{\alpha =1}^N \left( {\overline{u}}_k(\alpha )\partial _{{\overline{u}}_\ell (\alpha )} - {u}_\ell (\alpha )\partial _{{u}_\ell (\alpha )} \right) . \end{aligned}$$

We see that the determinant of fluctuations is again an observable which follows the Fermionic eigenvector moment flow. In the Hermitian case, if one considers $(u_1,\dots ,u_N)$ the eigenvectors associated to the eigenvalues $\lambda _1\leqslant \dots \leqslant \lambda _N$ of $H_s$ given by (B.1), we define the fluctuations and mixed overlap by, for a family $( {\mathbf {q}}_\alpha )_{\alpha \in I}\in ({\mathbb {R}}^N)^{\vert I\vert }$,

$$\begin{aligned} p_{kk}=\sum _{\alpha \in I}\vert \langle {\mathbf {q}}_\alpha ,u_k\rangle \vert ^2-\frac{\vert I\vert }{2N} \quad \text {and}\quad p_{k\ell }=\sum _{\alpha \in I}\langle {\mathbf {q}}_\alpha ,u_k\rangle \langle {\mathbf {q}}_\alpha ,{\overline{u}}_\ell \rangle \text {for} k\ne \ell . \end{aligned}$$

Note in particular that we have $p_{k\ell }\ne p_{\ell k}$ but $p_{k\ell }=\overline{p_{\ell k}}.$ Now, we define the same observable, for ${\mathbf {k}}=(k_1,\dots ,k_n)$, with $k_i\ne k_j$,

$$\begin{aligned} f^{\text {Fer}}_s({\mathbf {k}})=\mathbb {E}\left[ \det P_s({\mathbf {k}}) \vert \varvec{\lambda } \right] \end{aligned}$$

(B.5)

with $P_s({\mathbf {k}})$ given by (1.9), note that it becomes a Hermitian matrix instead of a symmetric matrix in the symmetric case. We then have the same fact that $f_s^{\text {Fer}}$ follows the eigenvector moment flow.

Theorem B.3

Let $({\mathbf {u}},\varvec{\lambda })$ be the solution to the coupled flows as in Definition B.1 and let $f_{s}^{\text {Fer}}$ be as in (B.5), it satisfies the following equation, for ${\mathbf {k}}$ a pairwise distinct set of indices such that $\vert {\mathbf {k}}\vert =n$,

$$\begin{aligned} \partial _s f_s^{\text {Fer}}({\mathbf {k}})=\sum _{i=1}^n\sum _{\begin{array}{c} \ell \in [\![1,N]\!]\\ \ell \notin \{k_1,\dots ,k_n\} \end{array}} \frac{f_s^{\text {Fer}}({\mathbf {k}}^i(\ell ))-f_s^{\text {Fer}}({\mathbf {k}})}{N(\lambda _{k_i}-\lambda _\ell )^2}. \end{aligned}$$

(B.6)

The proof of Theorem B.3 can also be done using Grassmann variables and a Fermionic Wick theorem as in Sect. 2 or by carefully expanding the determinant and following the contribution of each permutation as in Appendix A. We do not develop the proof here as it is very similar but it is interesting to note that the determinant and the Fermionic eigenvector moment flow is universal regarding the symmetry of the system contrary to the Bosonic observable. Indeed, we saw the definition of the Bosonic observable via (1.12) for the symmetric Dyson flow, but the Bosonic observable in the Hermitian case is different.

While we can also define it as a sum over (colored) graphs similarly to (1.12) another possible definition can be given in the following way: Let $\varvec{\xi }$ be a configuration of n particles, denote the position of the sites where each particle is situated as $(k_1,\dots ,k_n)$ (note that we can have $k_i=k_j$ for some i’s and j’s) then we can define

$$\begin{aligned} f^{\text {Bos}}(\varvec{\xi })=\frac{1}{{\mathcal {M}}(\varvec{\xi })}\mathbb {E}\left[ {{\,\mathrm{per}\,}}P_s(\varvec{\xi }) \vert \varvec{\lambda }\right] \quad \text {with}\quad P_s(\varvec{\xi })=\left( p_{k_ik_j} \right) _{1\leqslant i,j\leqslant n} \quad \text {and}\quad {\mathcal {M}}(\varvec{\xi })=\prod _{i=1}^N \eta _i! \end{aligned}$$

where ${{\,\mathrm{per}\,}}$ denote the permanent of the matrix,

$$\begin{aligned} {{\,\mathrm{per}\,}}A=\sum _{\sigma \in {\mathfrak {S}}_n} \prod _{i=1}^nA_{i,\sigma (i)}. \end{aligned}$$