Overcrowding and Separation Estimates for the Coulomb Gas

For $d\geq 2$, the $d$-dimensional Coulomb gas (or one-component plasma) at inverse temperature $\beta\in(0,\infty)$ is a probability measure on point configurations $X_{N}=(x_{1},\ldots,x_{N})\in(\mathbb{R}^{d})^{N}$ given by

where $dX_{N}$ is Lebesgue measure on $(\mathbb{R}^{d})^{N}$, $\mathcal{Z}$ is a normalizing constant, and

is the Coulomb energy of $X_{N}$ with confining potential $W$. The kernel $\mathsf{g}$ is the Coulomb interaction given by

While we will rarely require it, we have in mind the scaling $W=V_{N}:=N^{2/d}V(N^{-1/d}\cdot)$ for a potential $V$ satisfying certain conditions. The normalization in $V_{N}$ is chosen so that the typical interstitial distance is of size $O(1)$, i.e. the Coulomb gas $\mathbb{P}^{V_{N}}_{N,\beta}$ is on the “blown-up” scale. However, unless otherwise stated, we will work only under the assumption that $\Delta W$ exists and is bounded from above on $\mathbb{R}^{d}$ and (1.1) is well-defined, though see Remark 1.9 for comments on how this can be loosened significantly. For some results, we will need additional assumptions on $W$.

Up to normalization, the kernel $\mathsf{g}$ gives the repulsive interaction between two positive point charges, and so the Coulomb gas exhibits a competition between particle repulsion, given by the first sum in (1.2), and particle confinement, given by the second sum in (1.2). The behavior of $X_{N}$ at the macroscopic scale (i.e. in a box of side length $O(N^{1/d})$) is largely dictated by the equilibrium measure $\mu_{\mathrm{eq}}$, a compactly supported probability measure on $\mathbb{R}^{d}$ solving a variational problem involving $V$, in the sense that the empirical measure $N^{-1}\sum_{i=1}^{N}\delta_{N^{-1/d}x_{i}}$ is well-approximated weakly by $\mu_{\mathrm{eq}}$ with high probability as $N\to\infty$. In particular, the rescaled points condense on the droplet, i.e. the support of $\mu_{\mathrm{eq}}$. Even on mesoscales $O(N^{\alpha})$, $0<\alpha<1/d$, the equilibrium measure gives a good approximation for particle density. Letting $\mu_{\mathrm{eq}}^{N}$ be defined by $\mu_{\mathrm{eq}}^{N}(A)=N\mu_{\mathrm{eq}}(N^{-\frac{1}{d}}A)$ for measurable $A\subset\mathbb{R}^{d}$, one can form the random fluctuation measure

which, despite being of size $O(N)$ in total variation, is typically of size $O(1)$ when acting on smooth functions (e.g. [Ser20, LS18, AS21, BBNY19]).

Most of the time, we will work with the more general probability measure $\mathbb{P}^{W,U}_{N,\beta}$ defined by

where $U=U_{N}:(\mathbb{R}^{d})^{N}\to\mathbb{R}$ is symmetric, superharmonic and locally integrable in each variable $x_{i}$, and such that the measure $\mathbb{P}^{W,U}_{N,\beta}$ is well-defined. Measures of this form capture behavior of the gas under conditioning. For example, the Coulomb gas (1.1) disintegrates along $x_{n}$ to $\mathbb{P}^{W,U}_{N-1,\beta}$ with $U(X_{N-1})=\sum_{i=1}^{N-1}\mathsf{g}(x_{i}-x_{N})$. They also play an important role in the study of the fractional quantum Hall effect; see Section 1.6 for further discussion as well the surveys [Rou22b, Rou22a].

We will apply a transport-type argument, called isotropic averaging, to give upper bounds for $\mathbb{P}^{W,U}_{N,\beta}$ on a variety of events, all concerning the overcrowding of particles. This terminology and a similar method was first used in [Leb21], but a technical issue limited its applicability. Our main contribution is to demonstrate that the method has wide-ranging applicability by giving relatively short and intuitive solutions to several open problems. We believe that it will be an important tool in future studies of the Coulomb gas.

The basic idea behind isotropic averaging will be motivated through the following model computation. We will refer to this computation, in more general forms, throughout the paper.

We start by defining certain isotropic averaging operators. Given an index set $\mathcal{I}\subset\{1,2,\ldots,N\}$ and a rotationally symmetric probability measure $\nu$ on $\mathbb{R}^{d}$, we define

for any nice enough function $F:(\mathbb{R}^{d})^{\mathcal{I}}\to\mathbb{R}$. We also consider the operator $\mathrm{Iso}_{\mathcal{I},\nu}$ acting on functions of $X_{N}$, or more generally any set of labeled coordinates, by convolution with $\nu$ on the coordinates with labels in $\mathcal{I}$. For example, we have

by convention, and $\mathrm{Iso}_{\mathcal{I},\nu}F(x_{1})=F(x_{1})$ if $1\not\in\mathcal{I}$, otherwise $\mathrm{Iso}_{\mathcal{I},\nu}F(x_{1})=F\ast\nu(x_{1})$.

An important observation is that the kernel $\mathsf{g}$ is superharmonic everywhere and harmonic away from the origin, and thus we have the mean value inequality

In our physical context, the isotropic averaging operator replaces each point charge $x_{i}$, $i\in\mathcal{I}$, by a charge distribution shaped like $\nu$ centered at $x_{i}$. Newton’s theorem implies that the electric interaction between two disjoint, radial, unit charge distributions is the same as the interaction between two point charges located at the respective centers. More generally, if the charge distributions are not disjoint, then the interaction is more mild than that of the point charge system (this is because $\mathsf{g}(r)$ is decreasing in $r$ and the electric field generated by a uniform spherical charge is $0$ in the interior).

Consider an event $E$ which we wish to show to be unlikely. For definiteness, we let $E$ be the event “$B_{r}(z)$ contains at least $2$ particles” for some fixed $r\ll 1$ and $z\in\mathbb{R}^{d}$. By a union bound, we have

We can bound the likelihood of $E_{\{1,2\}}$ by comparing each $X_{N}\in E_{\{1,2\}}$ to the weighted family of configurations generated by replacing $x_{1}$ and $x_{2}$ by unit charged annuli of inner radius $1/2$ and outer radius $1$. Letting $\nu$ be the uniform probability measure supported on the centered annulus $\mathrm{Ann}_{[1/2,1]}(0)\subset\mathbb{R}^{d}$, we have by Jensen’s inequality

for

We can then consider the $L^{2}((\mathbb{R}^{d})^{\{1,2\}})$-adjoint of $\mathrm{Iso}_{\{1,2\},\nu}$, which we call $\mathrm{Iso}^{\ast}_{\{1,2\},\nu}$, to bound

We call the above calculation, namely (1.8) and (1.9), the model computation. There are now two tasks: (1) give a lower bound for $\Delta$ and (2) give an upper bound for the expectation of $\mathrm{Iso}^{\ast}_{\{1,2\},\nu}\mathbf{1}_{E_{\{1,2\}}}$.

Regarding task (1), we expect $\Delta$ will be large: two particles initially clustered in $B_{r}(z)$ are replaced by annular charges of microscopic length scale, which interact mildly. It is a simple calculation to see the pairwise interaction between the charged annuli is bounded by $\mathsf{g}(1/2)$ (with the abuse of notation $\mathsf{g}(x)=\mathsf{g}(|x|)$). Regarding the potential term $\sum_{i=1}^{N}W(x_{i})$ within $\mathcal{H}^{W,U}(X_{N})$, it will increase by at most a constant after isotropic averaging since $\Delta W\leq C$. The superharmonic term $U$ does not increase. Therefore, we have $\Delta\geq\mathsf{g}(2r)-C.$

Regarding task (2), since $\mathrm{Iso}_{\{1,2\},\nu}$ is a convolution by $\nu^{\otimes 2}$, we have

Moreover, we have $\mathrm{Iso}^{\ast}_{\{1,2\},\nu}\mathbf{1}_{E_{\{1,2\}}}(X_{N})=0$ if $x_{1}$ or $x_{2}$ is not in $B_{1+r}(z)\subset B_{2}(z)$. Thus

Assembling the above, starting with (1.7), we find

The probability appearing in the RHS will be bounded by $CN^{-2}$ by our microscopic local law Theorem 1, which is proved using a separate isotropic averaging argument, and we see that the probability of $E$ is bounded by $Cr^{2d}e^{-\beta\mathsf{g}(2r)}$. This is optimal in $d=2$, but can be improved to $Cr^{3d-2}e^{-\beta\mathsf{g}(2r)}$ in $d\geq 3$ (see Theorem 3). The $CN^{-2}r^{2d}$ bound for the probability of $E_{\{1,2\}}$ comes from the decrease in phase space volume available to $x_{1}$ and $x_{2}$ from the full macroscopic scale of $O(N)$ volume per particle to a specific sub-microscopic ball of $O(r^{d})$ volume upon restricting to $E_{\{1,2\}}$. In $d\geq 3$, the polynomial singularity of $\mathsf{g}$ generates additional effective constraints on $x_{1}$ and $x_{2}$ within $B_{r}(z)$.

We remark that our technique exhibits perfect localization and gives quantitative estimates with computable constants. In particular, it is robust to certain types of conditioning and randomization of the ball $B_{r}(z)$, as well as allowing to prove disparate phenomena on vastly different scales. It can also be generalized to use operators other than $\mathrm{Iso}_{\mathcal{I},\nu}$, as in the proof of Theorem 4 where we give both upper and lower bounds on the minimal inter-particle difference. For the lower bound, we must apply our model computation with a “mimicry” operator defined in Proposition 4.3. The method, in particular techniques for estimating $\Delta$, can be made very precise, as in Theorem 5. Our model computation bears resemblance to the Mermin-Wagner argument from statistical physics [MW66]. It is also similar to an argument of Lieb, which applies only to ground states ($\beta=\infty$) and was generously shared and eventually generalized and published in [NS15, RS16, PS17].

Introduced in [JLM93], Jancovici-Lebowitz-Manificat (JLM) laws give the probability of large charge discrepencies in the Coulomb gas. The authors considered an infinite volume jellium and approximated the probability of an absolute net charge of size much larger than $R^{(d-1)/2}$ in a ball of radius $R$ as $R\to\infty$. The jellium is a Coulomb gas with a uniform negative background charge, making the whole system net neutral in an appropriate sense. Since the typical net charge in $B_{R}(0)$ is expected to be of order $R^{(d-1)/2}$ (see [MY80]), the JLM laws are moderate to large deviation results and exhibit tail probabilities with very strong decay in the charge excess. The arguments of [JLM93] are based on electrostatic principles and consider several different regimes of the charge discrepancy size.

We are interested in a rigorous proof of the high density versions of the JLM laws. These versions apply when $X(B_{R}(z))$ exceeds the expected charge $\mu_{\mathrm{eq}}(B_{R}(z))$ by a large multiplicative factor $C$. They predict that

for $Q_{0}=|B_{R}(z)|$. The prediction applies for $Q\gg R^{d}$ as $R\to\infty$

Our main results prove the high density JLM law upper bounds in all dimensions in the ultra-high positive charge excess regime. We do so for $\mathbb{P}^{W,U}_{N,\beta}$, a Coulomb gas with potential confinement and superharmonic perturbation, though the result holds also for the jellium mutatis mutandis. We note that our result does not require $R\to\infty$. Indeed, we have found it very useful at $R=1$ as a local law upper bound valid on all of $\mathbb{R}^{d}$, an extension of the microscale local law in [AS21] which is only proved for $z$ sufficiently far into the interior of the droplet and under other more restrictive assumptions. Note that although we do not obtain a sharp coefficient on $Q^{2}$ in the exponent of the $d\geq 3$ case, it could be improved with additional effort in Proposition 2.1.

Theorem 1 immediately allows us to generalize [AS21, Corollary 1.1], which established the existence of limiting microscopic point processes for $(x_{1},\ldots,x_{N})$ re-centered around a point $z$. Previous to the work of Armstrong and Serfaty, the existence of such a process was only known in $d=2$ and $\beta=2$, where it is the Ginibre point process with an explicit correlation kernel. In [AS21], the point $z$ must be in the droplet $\Sigma_{N}$ and a mesoscopic distance $CN^{\frac{1}{d+2}}$ distance from the edge of the droplet $\partial\Sigma_{N}$. We are able to lift that restriction, and in particular we can take $W=V_{N}$ and $z=z_{N}$ near or in $\partial\Sigma_{N}$, in which case one would expect a genuinely different limit than the bulk case.

We have already seen in Section 1.2 that isotropic averaging can be applied to the description of the gas below the microscale, and we now state our full results. We are interested in pointwise bounds for the $k$-point correlation function $\rho_{k}$, defined by

for measurable sets $A_{1},A_{2},\ldots,A_{k}\subset\mathbb{R}^{d}$.

The functions $\rho_{k}$, and their truncated versions, are objects of intense interest. For example, in the physics literature, they are known to capture the charge screening behavior of the gas and satisfy sum rules and BBGKY equations [GLM80, Mar88]. For $d=2$, spatial oscillations of $\rho_{1}$ are expected to occur for large enough $\beta$ [CSA20, Cif06, CW03]. Starting at $\beta>2$, the oscillations occur near the edge of the droplet, and as $\beta$ increases the oscillations penetrate the bulk of the droplet (numerically, it is present at $\beta=200$) [CSA20]. This phenomenon is part of a debated freezing or crystallization transition in the two-dimensional Coulomb gas [KK16].

Many results on $\rho_{k}$ are known when integrated on the microscale or higher, though these results are often stated in terms of integration of the empirical measure $N^{-1}X$ against test functions. We will not comprehensively review previous results, but only mention that [AS21] proves that $\int_{B_{1}(z)}\rho_{1}(y)dy$ is uniformly bounded in $N$ for $z$ sufficiently far inside the droplet.

We will be interested in pointwise bounds on $\rho_{k}(y_{1},\ldots,y_{k})$, particularly when some of the $y_{i}$ within sub-microscopic distances of each other. One should see $\rho_{k}\to 0$ as $y_{1}\to y_{2}$ due to the repulsion between particles. There are no previously existing rigorous results for pointwise values for general $\beta$; even boundedness of $\rho_{1}$ was until now unproved.

The following bound on sub-microscopic particle clusters is easily derived by integrating Theorem 2. We point out the enhanced $r^{2d-2}$ factor in (1.22), which will be crucial for Theorem 4.

We remark that the $k=1$ and $Q=1$ cases of Theorem 2 and Theorem 3, respectively, are instances of Wegner estimates. In the context of $\beta$-ensembles on the line, Wegner estimates were proved in [BMP21] and for Wigner matrices in [ESY10]. The $Q=2$ cases of Theorem 3 are particle repulsion estimates. These estimates, as well as eigenvalue minimal gaps, have been considered for random matrices in many articles, e.g. [NTV17, Tao13, TV11, EKYY12].

We will also study the law of the $k$th smallest particle gap $\eta_{k}$, i.e.

Note that the particle gaps $|x_{i}-x_{j}|$, $i\neq j$, are almost surely unique under $\mathbb{P}^{W,U}_{N,\beta}$.

Previously, the order of $\eta_{1}$ was investigated dimension two in [Ame18] and [AR22]. The latter article proves that $\eta_{1}\geq(N\log N)^{-\frac{1}{\beta}}$ with high probability as $N\to\infty$ for all $\beta>1$. It is also proved that $\eta_{1}>C^{-1}$ with high probability if $\beta=\beta_{N}$ grows like $\log N$. This suggests that the gas “freezes”, even at the level of the extremal statistic $\eta_{1}$, in this temperature regime.

We remark that Theorem 3 already improves this bound.

Furthermore, an examination of the proof’s dependence on $\beta$ shows we can let $\beta=\beta_{N}=c_{0}\log N$ and take a constant $C$ equal to $e^{C\beta}=N^{C}$ on the RHS of (1.24). Letting $\gamma=N^{1/(2+\beta)}\gamma^{\prime}$ for a small enough $\gamma^{\prime}>0$ in Corollary 1.6 shows that $\eta_{1}$ is bounded below by a constant independent of $N$ with high probability, offering an alternative proof of the freezing regime identified in [Ame18, AR22]. An identical idea using (1.21) in place of (1.20) proves that the gas freezes in dimension $d\geq 3$ in the $\log N$ inverse temperature regime as well.

It is natural to wonder whether Corollary 1.6 is sharp in some sense and whether there are versions for $\eta_{k}$ and $d\geq 3$. We will give an affirmative answer to these questions.

For the eigenvalues of random matrices, the law of $\eta_{k}$ has been of great interest. While one dimensional (for the most studied cases), these models are particularly relevant to the $d=2$ case because the interaction between eigenvalues is also given by $\mathsf{g}=-\log$ for certain ensembles. In [AB13], the authors prove for the CUE and GUE ensembles that $N^{4/3}\eta_{1}$ is tight and has limiting law with density proportional to $x^{3k-1}e^{-x^{3}}$. Note that the interstitial distance is order $N^{-1}$ for these ensembles, whereas for us it will be order $1$. The proof uses determinantal correlation kernel methods. Recently, the extremal statistics of generalized Hermitian Wigner matrices was studied by Bourgade in [Bou22] with dynamical methods, proving that $\eta_{1}$ is also of order $N^{-4/3}$ and the rescaled limiting law is identical to the GUE case. For a symmetric Wigner matrix, [Bou22] proves that the minimal gap is of order $N^{-3/2}$.

The articles [FW21, FTW19] prove even more detailed results on the minimal particle gaps for certain random matrix models. Specifically, for the GOE ensemble and the circular $\beta$ ensemble with positive integer $\beta$, the joint limiting law of the minimal particle gaps is determined. Actually, convergence of a point process containing the data of the minimal particle gaps and their location is proved, showing in particular that the gap locations are asymptotically Poissonian. The methods crucially rely on certain exact identities unavailable in our case.

The only optimal $d=2$ result is in [SJ12]. Here, the authors consider the $k$th smallest eigenvalue gap of certain normal random matrix ensembles. The Ginibre ensemble, after rescaling the eigenvalues by $N^{1/2}$, corresponds to $\mathbb{P}^{V_{N}}_{N,\beta}$ for a certain quadratic $V$ and $\beta=2$. Using determinantal correlation kernel methods, [SJ12] proves that, for the Ginibre ensemble, $N^{\frac{1}{4}}\eta_{k}$ is tight as $N\to\infty$ (with interstitial distance $O(1)$), and its limiting law on $\mathbb{R}$ has density proportional to $x^{4k-1}e^{-x^{4}}dx$. Theorem 4 extends the tightness result to all $\beta$ and general potential $V$.

For our main result, we will take $W=V_{N}$ and require some extra assumptions on $V$. These assumptions are only used to prove Lemma 4.2, which is used in proving lower bounds on $\eta_{k}$. In particular, our upper bounds on $\eta_{k}$ below hold for $\mathbb{P}^{W,U}_{N,\beta}$ in full generality. For Theorem 4, we require

The proof of the upper bounds on $\eta_{k}$ in Theorem 4 is an application of the ideas from the model computation in Section 1.2, except with some more precision. The proof of the lower bounds can be understood via an idea related to isotropic averaging that we term “mimicry.” See the beginning of Section 4 for a high-level discussion of the proof.

Our previously mentioned results all concerned either sub-microscopic length scales or high particle densities, but we can in fact effectively use isotropic averaging on mesoscopic length scales and under only slight particle density excesses. We are interested in the discrepancy

which measures the deficit or excess of particles with reference to the equilibrium measure in a measurable set $\Omega$. It is also useful to define the compression

where $c_{d}$ is such that $\Delta\mathsf{g}=-c_{d}\delta_{0}$. Note that $\mathrm{Disc}$ and $\mathrm{Disc}_{W}$ agree when $\Omega\subset\Sigma_{N}:=N^{\frac{1}{d}}\mathrm{supp}\ \mu_{\mathrm{eq}}$ and the equilibrium measure exists.

In [AS21], it is proved that the discrepancy in $\Omega=B_{R}(z)$ is typically not more than $O(R^{d-1})$ in size whenever $z$ is sufficiently far in the interior of the droplet and $R$ is sufficiently large. The proof involves a multiscale argument inspired by stochastic homogenization [AKM19], as well as a technical screening procedure. More generally, [AS21] gives local bounds on the electric energy, which provide a technical basis for central limit theorems [Ser20] among other things. The upper bound of $O(R^{d-1})$ represents that the dominant error contribution within the argument comes from surface terms appearing in the multiscale argument. The JLM laws [JLM93] predict that the discrepancy in $B_{R}(z)$ is actually typically of size $O(R^{(d-1)/2})$; in particular, the Coulomb gas is hyperuniform. This motivates the search for other methods to prove discrepancy bounds that, with enough refinement, may be able to overcome surface error terms.

A motivation for studying the compression $\mathrm{Disc}_{W}$ comes from fractional quantum Hall effect (FQHE) physics (see [Rou22b, Rou22a] and references therein for a more comprehensive discussion). In FQHE physics, one considers wave functions $\Psi_{F}:\mathbb{C}^{N}\to\mathbb{C}$ of the form (considering $x_{i}\in\mathbb{C}\cong\mathbb{R}^{2}$)

for integers $\ell\geq 1$, symmetric, analytic functions $F:\mathbb{C}^{N}\to\mathbb{C}$, and magnetic field strengths $B>0$. All wave functions are assumed to be $L^{2}(\mathbb{C}^{N})$ normalized. This means that $|\Psi_{F}(X_{N})|^{2}dX_{N}=\mathbb{P}^{W,U}_{N,\beta}(dX_{N})$ for special values $W=W_{B,\ell}$, $\beta=\ell$, and

In particular, the function $U$ is superharmonic in each variable since $F$ is analytic. The connection between the Laughlin wave function $\Psi_{\mathrm{Lau}}$ and the Coulomb gas is termed the plasma analogy.

When studying the robustness of FQHE under e.g. material impurities, a physically relevant variational problem is to find $F$ minimizing the functional

for a certain $\mathcal{O}:\mathbb{R}^{2}\to\mathbb{R}$ giving a one-body energy associated to material impurities and trapping. Here $\mathbb{P}^{W,U}_{N,\beta}(dX_{N})=|\Psi_{F}(X_{N})|^{2}dX_{N}$ as in the plasma analogy and $\rho_{1,F}$ is the $1$-point function of $\mathbb{P}^{W,U}_{N,\beta}$. It is expected that the infimum within $E_{\mathcal{O}}(N,B,\ell)$ is approximately achieved as $N\to\infty$ with $F$ of the form $F(X_{N})=\prod_{i=1}^{N}f(x_{i})$ for an analytic function $f$. Such a factorization is important physically as it indicates the presence of uncorrelated “quasi-holes” at the zeros of $f$, a remarkably simple system when compared to the those with nontrivial couplings between particles and quasi-holes.

Lieb, Rougerie, and Yngvason, in the main result of [LRY19], proved $\rho_{1,F}\leq c_{d}^{-1}\Delta W(1+o(1))$ as $N\to\infty$ when integrated on scales of size $N^{1/4+\varepsilon}$ for $\varepsilon>0$. In other words, we have