Optimal local laws and CLT for the circular
Riesz gas

In this paper, we study the one-dimensional Riesz gas on the circle. We denote $\mathbb{T}:=\mathbb{R}/\mathbb{Z}$. For a parameter $s\in(0,1)$, let us consider the Riesz $s$-kernel on $\mathbb{T}$, defined by

$$g(x)=\lim_{n\to\infty}\Bigr{(}\sum_{k=-n}^{n}\frac{1}{|k+x|^{s}}-\frac{2}{1-s}n^{1-s}\Bigr{)}=\zeta(s,x)+\zeta(s,1-x),$$

(1.1)

where $\zeta(s,x)$ stands for the Hurwitz zeta function [Ber72]. Note that $g$ is the fundamental solution of the fractional Laplace equation on the circle

$$\begin{cases}(-\Delta)^{\frac{1-s}{2}}g=c_{s}(\delta_{0}-1)\\ \int g=0,\end{cases}$$

(1.2)

with $c_{s}$ given by

$$c_{s}=\frac{\Gamma(\frac{1-s}{2})}{\Gamma(\frac{s}{2})}\frac{\sqrt{\pi}}{2^{1-s}}.$$

(1.3)

We endow $\mathbb{T}$ with the natural order $x<y$ if $x=x^{\prime}+k$, $y=y^{\prime}+k^{\prime}$ with $k,k^{\prime}\in\mathbb{Z}$, $x^{\prime},y^{\prime}\in[0,1)$ and $x^{\prime}<y^{\prime}$ and work on the set of ordered configurations

$$D_{N}=\{X_{N}=(x_{1},\ldots,x_{N})\in\mathbb{T}^{N}:x_{2}-x_{1}<\ldots<x_{N}-x_{1}\}.$$

On $D_{N}$ let us consider the energy

$$\mathcal{H}_{N}:X_{N}\in D_{N}\mapsto N^{-s}\sum_{i\neq j}g(x_{i}-x_{j}),$$

(1.4)

where $g$ is given by (1.2). The circular Riesz gas, at the inverse temperature $\beta>0$, is defined by the Gibbs measure

$$\mathrm{d}\mathbb{P}_{N,\beta}=\frac{1}{Z_{N,\beta}}\exp(-\beta\mathcal{H}_{N}(X_{N}))\mathds{1}_{D_{N}}(X_{N})\mathrm{d}X_{N},$$

where $Z_{N,\beta}$ is the normalizing constant, called the partition function, given by

$$Z_{N,\beta}=\int_{D_{N}}e^{-\beta\mathcal{H}_{N}(X_{N})}\mathrm{d}X_{N}.$$

Throughout the paper, $s$ is a fixed parameter in $(0,1)$.

The choice of the normalization in the definition of the energy (1.4) appears to be a natural choice, making $\beta$ the effective inverse temperature governing the microscopic scale behavior.

The model described above belongs to a family of interacting particle systems named Riesz gases. On $\mathbb{R}^{\mathrm{d}}$, these are associated to a kernel of the form $|x|^{-s}$ with $s>0$. The Riesz family also contains in dimensions $1$ and $2$ the so-called log-gases with kernel $-\log|x|$. For $\mathrm{d}\geq 3$ and $s=\mathrm{d-2}$, $|x|^{-s}$ is the fundamental solution of the Laplace equation on $\mathbb{R}^{d}$ and therefore corresponds to the Coulombian interaction. The parameter $s$ determines the singularity as well as the range of the interaction. When $s>\mathrm{d}$, the interaction is short-range and the system, referred to as hypersingular Riesz gas, resembles a nearest-neighbour model. For $s\in(0,\mathrm{d})$ or $s=0$ and $\mathrm{d}=1,2$, Riesz gases are long-range particle systems, which have, as such, attracted much attention in both mathematical and physical contexts.

The 1D log-gas, also called the $\beta$-ensemble has been extensively studied in the last few decades, partly for its connection to random matrix theory. Indeed, it corresponds, in the cases $\beta\in\{1,2,4\}$, to the distribution of the eigenvalues of $N\times N$ symmetric/Hermitian/symplectic random matrices with independent Gaussian entries (see the original paper of Dyson [Dys62]). The 1D log-gas also appears in many other contexts such as zeros of random polynomials, zeros of the Riemann function and is conjectured to be related to the eigenvalues of random Schrödinger operators [Lew22]. The 2D Coulomb gas is another fundamental model. Among many other examples, it is connected to non-unitary random matrices, Ginzburg-Landau vertices, Fekete points, complex geometry, the XY model and the KT transition [Ser18]. For other values of $s$, let us mention that the case $s=2$ in dimension $1$ is an integrable system, called the classical Calogero-Sutherland model. The study of minimizers of Riesz interactions is also a dynamic topic [HS05, CSW21] and is the object of long-standing conjectures related to sphere packing problems [BL15]. From a statistical physics perspective, even in dimension $1$, the Riesz gas is not fully elucidated since the classical theory of the 60-70s [Rue74] cannot be applied due the long-range nature of the interaction. The reader may refer to the nice review [Lew22], where an account of the literature and many open problems on Riesz gases are given.

As the number of particles $N$ tends to infinity, the empirical measure

$$\mu_{N}:=\frac{1}{N}\sum_{i=1}^{N}\delta_{x_{i}}$$

converges almost surely under $\mathbb{P}_{N,\beta}$ (in a suitable topology) to the uniform measure on the circle. This result can be obtained through standard large deviations techniques (see for instance [Ser14, Ch. 2] for the case of Riesz gases on the real line, which adapts readily to the periodic setting). In the large $N$ limit, particles tend to spread uniformly on the circle, which suggests that particles spacing (or gaps) $N(x_{i+k}-x_{i})$ concentrate around the value $k$. The first goal of this paper is to quantify the fluctuations of the gaps around their mean. We establish the optimal size of the fluctuations of $N(x_{i+k}-x_{i})$, which turns out to be in $O(\beta^{-\frac{1}{2}}k^{\frac{s}{2}})$, as conjectured in the recent physics paper [SKA⁺21]. This type of result is referred to in the literature as a rigidity estimate. It was intensively investigated for $\beta$-ensembles (see for instance [BEY12, BEY⁺14b, BEY14a, BMP22]), but the correct observable in that case is $x_{i}-\gamma_{i}$, where $x_{i}$ is the $i$-th particle and $\gamma_{i}$ the classical location of the $i$-th particle, that is the corresponding quantile of the equilibrium measure arising in the mean-field limit.

A complementary way to study the rigidity of the system is to investigate the fluctuations of linear statistics of the form

$$\mathrm{Fluct}_{N}[\xi(\ell_{N}^{-1}\cdot)]:=\sum_{i=1}^{N}\xi(\ell_{N}^{-1}x_{i})-N\ell_{N}\int_{\mathbb{T}}\xi,$$

(1.5)

where $\xi:\mathbb{T}\to\mathbb{R}$ is a given measurable test-function and $\{\ell_{N}\}$ a sequence of numbers in $(0,1]$. For smooth test-functions, many central limit theorem (CLT) results are available in the literature on 1D-log gases, including [J⁺98, Shc13, BG13, BEY12, BEY⁺14b, BLS⁺18, BMP22, HL21]. For 1D Riesz gases with $s\in(0,1)$, to our knowledge, no prior results on CLT for linear statistics are known. In this paper we obtain a quantitative CLT for (1.5), which is valid at all scales $\{\ell_{N}\}$ down to microscopic scales $\ell_{N}\gg\frac{1}{N}$. A major direction in random matrix theory is to establish CLTs for (1.5) allowing test-functions which are as singular as possible. Indeed it is a natural question to capture the fluctuations of the number of points and of the logarithmic potential, which are key observables for the log-gas. The question of the optimal regularity of the test-function has also drawn a lot of interest because it encapsulates non-universality features in the context of Wigner matrices. One of the goals of this paper is to provide a robust method which allows singular test-functions to be treated in a systematic way. The main question we investigate is therefore a regularity issue. Using new concentration inequalities we are able to treat singular test-functions, including characteristic functions of intervals and inverse power functions up to the critical power $\frac{s}{2}$. In particular we obtain a CLT for the number of points, thus extending to a Riesz (periodic and fully convex) setting some of the recent results of [BMP22].

Let us now introduce the main tools and objects used in the proofs. For any reasonable Gibbs measure on $D_{N}$ (or $\mathbb{R}^{N}$), the fluctuations of any (smooth) statistic $F:D_{N}\to\mathbb{R}$ are related to the properties of a partial differential equation called the Helffer-Sjöstrand (H.-S.) equation, which is sometimes referred to as a Witten Laplacian (on $1$-forms). This equation appears in [Sjo93a, Sjo93b, HS94]. It is more substantially studied in [Hel98b, Hel98a, NS97], where it is used to establish correlation decay, uniqueness of the limiting measure and log-Sobolev inequalities for models with convex interactions. The purpose of the present paper is to show how the analysis of Helffer-Sjöstrand equations provides powerful tools to study the fluctuations of linear statistics with singular test-functions.

The proof of the near-optimal rigidity is essentially similar to [BEY12]. It exploits the convexity of the interaction and is thus very specific to 1D systems. The method is mainly based on a concentration inequality for divergence-free functions and on a key convexity result due to Brascamp [BL02].

The method of proof of the CLT for linear statistics starts by performing the mean-field transport argument usually attributed to Johansson [J⁺98]. When studying the Laplace transform of linear statistics $\mathrm{Fluct}_{N}[\xi]$, this consists in applying a well-chosen change of variables on each point, depending only on its position, to transport the uniform measure on the circle to the perturbed equilibrium measure (perturbed by the effect of adding $t\mathrm{Fluct}_{N}[\xi]$ to the energy). In this paper, one computes variances instead of Laplace transforms and the implementation of the transport of [J⁺98] takes the form of an integration by parts. This argument is a variation of the so-called loop equations (see [BEY12] for various comments on this topic). It is the starting point of many CLTs on $\beta$-ensembles and Coulomb gases but it received a more systematic analysis in the series of works [LS18, BLS⁺18, Leb18, LZ20, Ser20]. One can also interpret this transport as a mean-field approximation of the H.-S. equation associated to (the gradient of) linear statistics.

Since the transport is an approximate solution (a mean-field approximation) of the H.-S. equation, it creates an error term, sometimes called loop equation term, which is essentially a local weighted energy and the heart of the problem is to estimate its fluctuations. In contrast, in [LS18, BLS⁺18, Leb18, Ser20] the typical size in a large deviation sense of this error term is evaluated, rather than the size of its fluctuations. Let us point out however that the latter seems intractable in dimension $\mathrm{d}\geq 2$ because of the lack of convexity. The core of this paper is about the control on the fluctuations of this loop equation term through the analysis of the related H.-S. equation. Our proof is based on the Brascamp-Lieb inequality, the near-optimal rigidity estimates on gaps and nearest-neighbour distances and a comparison principle for the Helffer-Sjöstrand equation.

The use of the comparison principle mentioned above (also known in [Car74, HS94]) is one of the main novelties of the paper. This is the key technical tool to be able to treat linear statistics with singular test-functions. Indeed, after performing loop equations techniques, we will study singular local quantities, for which standard concentration inequalities, such as the Brascamp-Lieb or log-Sobolev inequalities, do not give the right order of fluctuations.

The central limit theorem is then obtained from a rather straightforward application of Stein’s method. We show how the mean-field transport naturally leads to an approximate Gaussian integration by parts formula. As a result, quantifying normality boils down to controlling the variance of the loop equation term. The CLT then follows from the variance bound discussed in the above comments.

The following results are valid for $s\in(0,1)$, thus covering the entire long-range regime.

Our first result concerns the fluctuations of gaps and discrepancies. We establish the following near-optimal decay estimate:

Theorem 1 is the natural extension of the rigidity result of [BEY12, Th. 3.1] to the Riesz setting. Since the probability of deviations are exponentially small, the estimates of Theorem 1 allow one to reduce the phase space to an event where all gaps are close to their standard value. Theorem 1 is proved in Section 4.

We next prove that $k^{\frac{s}{2}}$ is the exact fluctuation scale of $N(x_{i+k}-x_{i})$. This is equivalent to proving that $\sum_{i=1}^{N}\mathds{1}_{(0,\ell_{N})}(x_{i})$ fluctuates at scale $(N\ell_{N})^{\frac{s}{2}}$. For any map $\xi:\mathbb{T}\to\mathbb{R}$, consider the linear statistic

$$\mathrm{Fluct}_{N}[\xi]:=\sum_{i=1}^{N}\xi(x_{i})-N\int_{\mathbb{T}}\xi.$$

We prove sharp variance estimates valid at any scale for possibly singular test-functions satisfying the following assumptions:

Let us first comment upon the above assumptions.

Before stating the theorem, recall the definition of the fractional Sobolev seminorm on the circle $|\cdot|_{H^{\alpha}}$, for $\alpha>0$. Let $h:\mathbb{T}\to\mathbb{R}$ in $L^{2}(\mathbb{T},\mathbb{R})$ with Fourier coefficients $\hat{h}(k)$, $k\in\mathbb{Z}$. Whenever it is finite, we call $|h|_{H^{\alpha}}^{2}$ the quantity

$$|h|_{H^{\alpha}}^{2}:=\sum_{k\in\mathbb{Z}}|k|^{2\alpha}|\hat{h}|^{2}(k).$$

Similarly, the fractional Sobolev seminorm of a function $h\in L^{2}(\mathbb{R},\mathbb{R})$, also denoted $|h|_{H^{\alpha}}$, is defined by

$$|h|_{H^{\alpha}}^{2}:=\int|\xi|^{2\alpha}|\hat{h}|^{2}(\xi)\mathrm{d}\xi,$$

(1.9)

where $\hat{h}$ stands for the Fourier transform of $h$.

Let $\{\ell_{N}\}$ be a sequence taking values in $(0,1]$. For any function $\xi\in H^{\frac{1-s}{2}}(\mathbb{T},\mathbb{R})$, define

$$\sigma_{\ell_{N}}^{2}(\xi):=\ell_{N}^{-s}|\xi(\ell_{N}^{-1}\cdot)|_{H^{\frac{1-s}{2}}}^{2}.$$

(1.10)

If $\xi\in H^{\frac{1-s}{2}}(\mathbb{T},\mathbb{R})$ is supported on $(-\frac{1}{2},\frac{1}{2})$, let $\xi_{0}:\mathbb{R}\to\mathbb{R}$ given for all $x\in\mathbb{R}$ by

$$\xi_{0}(x)=\begin{cases}\xi(x)&\text{if $|x|\leq\frac{1}{2}$}\\ 0&\text{if $|x|>\frac{1}{2}$}.\end{cases}$$

(1.11)

For all $\xi\in H^{\frac{1-s}{2}}(\mathbb{T},\mathbb{R})$, we then let

$$\sigma_{\infty}^{2}(\xi):=\frac{1}{2\beta c_{s}}\begin{cases}|\xi|_{H^{\frac{1-s}{2}}}^{2}&\text{if $\ell_{N}=1$ for each $N$}\\ |\xi_{0}|^{2}_{H^{\frac{1-s}{2}}}&\text{if $\ell_{N}\to 0$, with $\xi_{0}$ as in (\ref{eq:defxi0})}.\end{cases}$$

(1.12)

The variance of $\mathrm{Fluct}_{N}[\xi(\cdot\ell_{N}^{-1})]$ under $\mathbb{P}_{N,\beta}$ may be expanded as follows:

Since $\alpha<\frac{s}{2}$, note that $\xi\in H^{\frac{1-s}{2}}(\mathbb{T},\mathbb{R})$ and that the remaining term in (1.13) is $o((N\ell_{N})^{s})$.

By Remark 1.2, the number-variance (i.e variance of the number of points) of the Riesz gas grows in $O(N^{s})$, like the variance of smooth linear statistics. In comparison, for the 1D log-gas, smooth linear statistics fluctuate in $O(1)$ with an asymptotic variance proportional to the squared Sobolev norm $|\cdot|_{H^{\frac{1}{2}}}^{2}$ (see [BLS⁺18] for instance) whereas the number of points in an interval $(-a,a)$ fluctuates in $O(\log N)$ since $\mathcal{1}_{(-a,a)}\notin H^{\frac{1}{2}}(\mathbb{T},\mathbb{R})$. Theorem 2 shows that the Riesz gas with $s\in(0,1)$ interpolates between the 1D log-gas case $s=0$ and the Poisson-type case $s=1$. Moreover Theorem 2 makes the 1D long-range Riesz gas a hyperuniform particle system in the sense of [Tor16] (meaning that the number-variance is much smaller than for i.i.d variables).

As mentioned in the beginning of the introduction, the next-order term in the expansion (1.13) corresponds to the variance of a local energy arising from the mean-field transport of [J⁺98], sometimes referred in the literature as the loop equation term. One could extract the leading-order of this variance and relate it to the second derivative of the free energy of the infinite Riesz gas.

The next question we address concerns the asymptotic behavior of rescaled linear statistics. We show that if $\xi$ satisfies Assumptions 1.1 and provided $\ell_{N}\gg\frac{1}{N}$, $\mathrm{Fluct}_{N}[\xi(\ell_{N}^{-1}\cdot)]$ converges after suitable rescaling to a Gaussian random variable. For any probability measures $\mu$ and $\nu$ on $\mathbb{R}$ let us denote $\mathsf{d}(\mu,\nu)$ the distance

$$\mathsf{d}(\mu,\nu):=\sup_{f\in\mathcal{C}^{2}(\mathbb{R},\mathbb{R})}\Bigr{\{}\int f\mathrm{d}(\mu-\nu):|f|_{\infty}\leq 1,|f^{\prime}|_{\infty}\leq 1,|f^{\prime\prime}|_{\infty}\leq 1\Bigr{\}}.$$

We establish the following result:

Theorem 3 can be interpreted as the convergence of the field

$$\sum_{i=1}^{N}g(x_{i}-\cdot)-N\int g(x-\cdot)\mathrm{d}x$$

to a fractional Gaussian field. Observe that if $\xi$ and $\chi$ are smooth test-functions with disjoint support, then the asymptotic covariance is, in general, not equal to $0$, meaning that the corresponding fractional field does not exhibit spatial independence. This reflects the non-local nature of the fractional Laplacian $(-\Delta)^{\frac{1-s}{2}}$ for $s\in(0,1)$.

Following Remark 1.2, Theorem 3 gives a CLT for gaps and discrepancies:

Corollary 1.1 is an extension of the results on the fluctuations of single particles in the bulk for $\beta$-ensembles, see [Gus05, CFLW21] for the GUE case and [BMP22]. Theorem 3 can also be applied to singular function having singularity around $0$ in $|x|^{-\alpha}$ for $\alpha\in(0,\frac{s}{2})$.

The test-function $x\in\mathbb{T}\mapsto|x|^{-\frac{s}{2}}$ is the critical inverse power which does not lie in $H^{\frac{1-s}{2}}$. This should be compared in the case $s=0$ to the test-functions $\mathds{1}_{(-a,a)}$ and $-\log|x|$, for which the associated linear statistics satisfy a log-correlated central limit theorem as shown for instance in [BMP22].