Rigidity and a mesoscopic central limit theorem for Dyson Brownian motion for general $\beta$ and potentials

Abstract

We study Dyson Brownian motion with general potential V and for general $\beta \ge 1$. For short times $t = o (1)$ and under suitable conditions on V we obtain a local law and corresponding rigidity estimates on the particle locations; that is, with overwhelming probability, the particles are close to their classical locations with an almost-optimal error estimate. Under the condition that the density of states of the initial data is bounded below and above down to the scale $\eta _* \ll t \ll 1$, we prove a mesoscopic central limit theorem for linear statistics at all scales $\eta$ with $N^{-1}\ll \eta \ll t$.

A Universality

In this appendix we roughly sketch the following universality theorem.

Theorem A.1

Suppose V satisfies Assumption 2.1. Fix $N^{-1}\le \eta _*\ll r\le 1$, and assume that the initial data ${\varvec{}}\lambda (0)$ satisfies Assumption 4.1. For any time t with $\eta _*\ll t\ll (\log N)^{-1}r\wedge (\log N)^{-2}$, and index k such that $E-r/2\le \lambda _{k} (0) \le E+r/2$, there is a constant ${\mathfrak c}>0$ so that the following estimate holds for any smooth test function $O: \mathbb {R}\rightarrow \mathbb {R}$ and N large enough,

$$\begin{aligned} \left| \mathbb {E} \left[ O ( \rho N ( \lambda _{k+1} ( t) - \lambda _{k} (t) ) )\right] - \mathbb {E}_\beta \left[ O ( N ( \lambda _{\ell +1} - \lambda _{\ell } )) \right] \right| \le N^{-{\mathfrak c}}, \end{aligned}$$

(A.1)

where the second expectation is with respect to the Gaussian $\beta$-ensemble, the index $\ell$ satisfies $\varepsilon N \le \ell \le (1- \varepsilon ) N$ for any $\varepsilon >0$ and the scaling factor $\rho$ is defined by

$$\begin{aligned} \rho := \frac{ \mu _t ( \gamma _{k} (t) )}{ \rho _{\mathrm {sc}} ( \gamma ^{\mathrm {sc}}_{\ell } )} \end{aligned}$$

(A.2)

where $\mu _t$ is the limiting measure-valued process appearing earlier, $\gamma _i (t)$ are its classical eigenvalue locations, $\rho _{\mathrm {sc}}$ is the semicircle density and $\gamma ^{\mathrm {sc}}_i$ are its classical eigenvalue locations.

The proof of this theorem is a modification of the proof of universality in [38]. In this work, the above theorem was proven in the case that $\beta = 1, 2$ or 4 and V is quadratic. These assumptions ensured that the process $\lambda _i (t)$ was equal in distribution to the eigenvalues of a random matrix ensemble evolving according to a matrix Brownian motion. The matrix structure was used to prove the rigidity estimates (3.4) in this special case. Therefore the proof given there carries over without change to the case that $\beta \ge 1$ is general and V is quadratic. In order to prove the above theorem we need only show how to deal with general V. The case of general V was carried out also in the work [38], but in the case of equilibrium initial data. This is the case when the DBM with general potential has initial data the equilibrium measure of the $\beta$-ensemble with the potential V. Short-time universality was proved, which then implied universality for the initial data as it is stationary. The approach to $\beta$-ensembles of [38] is somewhat different than the approach to matrix Brownian motion given in the same work. This is due to the fact that the matrix Brownian motion is handled using a certain family of interpolating ensembles for which the rigidity was not known in the case of general $\beta$ (the matrix case being handled via matrix estimates). Therefore, the proof of the above theorem comes down to a modification of the methods in [38]. In the interest of brevity, we mention the important changes.

k be as above, and fix a time $\eta _*\ll s\ll (\log N)^{-1}r \wedge (\log N)^{-2}$. Thanks to Theorem 3.1, the eigenvalue rigidity holds at time s. After appropriate re-scalings and translations of the initial data and the potential, let us assume that

$$\begin{aligned} \gamma _{k} (s) = \gamma _{k}^{\mathrm {sc}}, \qquad \mu _{t} (\gamma _{k} (s) ) = \rho _{\mathrm {sc}} ( \gamma _{k}^{\mathrm {sc}} ). \end{aligned}$$

(A.3)

Note that due to gap universality of the Gaussian $\beta$-ensemble [29], it suffices to assume $k = \ell$. We use a coupling idea from [12] which was used again in [38]. For $t \ge 0$, denote now

$$\begin{aligned} \mathrm {d}x_i (t) = \sqrt{\frac{2}{\beta N}}\mathrm{d}B_i(t) + \frac{1}{N} \sum _{j \ne i } \frac{1}{ x_j (t) - x_i (t) } \mathrm {d}t - \frac{1}{2} V (x_i (t) ) \mathrm {d}t, \qquad x_i (0) = \lambda _i (s) \end{aligned}$$

(A.4)

and

$$\begin{aligned} \mathrm {d}y_i (t) =\sqrt{\frac{2}{\beta N}}\mathrm{d}B_i(t) + \frac{1}{N} \sum _{j \ne i } \frac{1}{ y_j (t) - y_i (t) } \mathrm {d}t - \frac{y_i}{2} \mathrm {d}t \end{aligned}$$

(A.5)

where the initial data $y_i (0)$ is an independent Gaussian $\beta$-ensemble. Above, the Brownian motion terms are identical. We define the following interpolating processes for $0 \le \alpha \le 1$.

$$\begin{aligned} \mathrm{d}z_i (t, \alpha )= & {} \sqrt{\frac{2}{\beta N}} \mathrm{d}B_i(t) +\frac{1}{N}\sum _{j:j\ne i}\frac{\mathrm{d}t}{z_i (t, \alpha )-z_j (t, \alpha )}\nonumber \\&-\frac{1}{2}V_\alpha '(z_i (t, \alpha ))\mathrm{d}t,\quad i=1,2,\ldots , N, \end{aligned}$$

(A.6)

with the potential

$$\begin{aligned} V_\alpha =\alpha V +(1-\alpha )W, \end{aligned}$$

(A.7)

and the initial data

$$\begin{aligned} z_i (0, \alpha ):= \alpha x_i (0) + (1 - \alpha ) y_i (0), \end{aligned}$$

(A.8)

for $i=1,2,\ldots ,N$. We can write the difference of the $x_i(t)$ and $y_i(t)$ as an integral in terms of the interpolating process $z_i(t,\alpha )$,

$$\begin{aligned} x_i(t)-y_i(t)=\int _{0}^1 (\partial _\alpha z_i(t,\alpha ))\mathrm{d}\alpha . \end{aligned}$$

(A.9)

Thanks to the rigidity and the smoothness of the potential V, the contribution of the potential is to leading order deterministic in the vicinity of $x_{k}(t)$ and contributes only a drift which is summarized by the movement of the classical eigenvalue location $\gamma _{k} (t)$. The following “homogenization result” follows from a careful analysis of the differential equation of $\partial _\alpha z_i(t,\alpha )$ essentially the same as in [38]. And the Gap universality Theorem A.1 is a direct consequence.

Lemma A.2

Let time s, index k, $x_i(t)$ and $y_i (t)$ be as above. There are small constants ${\mathfrak c},{\mathfrak e}>0$, For any time $1/N\ll t$, such that $t=(sN)^{{\mathfrak e}}/N$ the following estimate holds with overwhelming probability: for any indices $|i-k| \le (sN)^{{\mathfrak e}}$

$$\begin{aligned} ( x_i (t) - \gamma _{k} (t) ) -( y_i (t) - \gamma _{k}^{\mathrm {sc}} ) ) = \zeta (i)_x - \zeta (i)_y +{{\mathrm{O}}}\left( \frac{1}{N^{1+{\mathfrak c}}}\right) . \end{aligned}$$

(A.10)

The functions $\zeta (i)_x$ and $\zeta (i)_y$ are mesoscopic linear statistics of the initial data and satisfy the estimate $| \zeta (i)_x - \zeta (i+1)_x | + | \zeta (i)_y - \zeta (i+1)_y | \le C / (N^2 t)\ll 1/N$, which proves gap universality.

The above lemma is the analogue of Theorem 3.1 of [38] in our setting which is proven using the modifications described above. The function $\zeta (i)_x$ is the same as the one appearing in that theorem, and the required estimate follows from Proposition 3.2 of [38]. We remark that in the setting of the current work, we are unable to prove universality of the correlation functions at fixed energy and so the above theorem is stated in terms of the eigenvalue gaps. This is precisely due to the fact that the quantities $\zeta$ appearing in the above lemma are mesoscopic linear statistics that are not compactly supported. The result proved in the present work does not apply to these statistics and so we cannot conclude fixed energy universality from the above homogenization result as in [38].