# Overcrowding and Separation Estimates for the Coulomb Gas

Eric Thoma

(Date: February 20, 2023)

###### Abstract.

We prove several results for the Coulomb gas in any dimension $d\geq 2$ that
follow from isotropic averaging, a transport method based on Newton’s theorem.
First, we prove a high-density Jancovici-Lebowitz-Manificat law, extending the
microscopic density bounds of Armstrong and Serfaty and establishing strictly
sub-Gaussian tails for charge excess in dimension $2$. The existence of
microscopic limiting point processes is proved at the edge of the droplet.
Next, we prove optimal upper bounds on the $k$-point correlation function for
merging points, including a Wegner estimate for the Coulomb gas for $k=1$. We
prove the tightness of the properly rescaled $k$th minimal particle gap,
identifying the correct order in $d=2$ and a three term expansion in $d\geq
3$, as well as upper and lower tail estimates. In particular, we extend the
two-dimensional “perfect-freezing regime” identified by Ameur and Romero to
higher dimensions. Finally, we give positive charge discrepancy bounds which
are state of the art near the droplet boundary and prove incompressibility of
Laughlin states in the fractional quantum Hall effect, starting at large
microscopic scales. Using rigidity for fluctuations of smooth linear
statistics, we show how to upgrade positive discrepancy bounds to estimates on
the absolute discrepancy in certain regions.

###### 2020 Mathematics Subject Classification:

82B05, 60G55, 60G70, 49S05

##  1\. Introduction

###  1.1. The setting.

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

$\mathbb{P}^{W}_{N,\beta}(dX_{N})=\frac{1}{\mathcal{Z}}\exp\left(-\beta\mathcal{H}^{W}(X_{N})\right)dX_{N}$
(1.1)

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

$\mathcal{H}^{W}(X_{N})=\frac{1}{2}\sum_{1\leq i\neq j\leq
N}\mathsf{g}(x_{i}-x_{j})+\sum_{i=1}^{N}W(x_{i})$ (1.2)

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

$\mathsf{g}(x)=\begin{cases}-\log|x|\quad&\text{if }d=2\\\
\frac{1}{|x|^{d-2}}\quad&\text{if }d\geq 3.\end{cases}$ (1.3)

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

$\mathrm{fluct}(dx)=\sum_{i=1}^{N}\delta_{x_{i}}(dx)-\mu_{\mathrm{eq}}^{N}(dx),$
(1.4)

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

$\mathbb{P}^{W,U}_{N,\beta}(dX_{N})\propto\exp\left(-\beta\mathcal{H}^{W,U}(X_{N})\right)dX_{N},\quad\mathcal{H}^{W,U}(X_{N})=\mathcal{H}^{W}(X_{N})+U(x_{1},\ldots,x_{N}),$
(1.5)

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.

###  1.2. A model computation.

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

$\mathrm{Iso}_{\mathcal{I},\nu}F((x_{i})_{i\in\mathcal{I}})=\int_{(\mathbb{R}^{d})^{\mathcal{I}}}F\left((x_{i}+y_{i})_{i\in\mathcal{I}}\right)\prod_{i\in\mathcal{I}}\nu(dy_{i})$

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

$\mathrm{Iso}_{\mathcal{I},\nu}F(X_{N})=\int_{(\mathbb{R}^{d})^{\mathcal{I}}}F\left(X_{N}+(y_{i}\mathbf{1}_{\mathcal{I}}(i))_{i=1}^{N}\right)\prod_{i\in\mathcal{I}}\nu(dy_{i})$

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

$\mathrm{Iso}_{\mathcal{I},\nu}\mathsf{g}(x_{i}-x_{j})\leq\mathsf{g}(x_{i}-x_{j})\quad\forall
i,j.$ (1.6)

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

$\mathbb{P}^{W,U}_{N,\beta}(E)\leq\sum_{i<j}\mathbb{P}^{W,U}_{N,\beta}(E_{\\{i,j\\}})=\binom{N}{2}\mathbb{P}^{W,U}_{N,\beta}(E_{\\{1,2\\}}),\quad
E_{\\{i,j\\}}:=\\{x_{i}\in B_{r}(z)\\}\cap\\{x_{j}\in B_{r}(z)\\}.$ (1.7)

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

$\displaystyle\mathbb{P}^{W,U}_{N,\beta}(E_{\\{1,2\\}})$
$\displaystyle=\frac{1}{\mathcal{Z}}\int_{E_{\\{1,2\\}}}e^{-\beta\mathcal{H}^{W,U}(X_{N})}dX_{N}\leq\frac{e^{-\beta\Delta}}{\mathcal{Z}}\int_{E_{\\{1,2\\}}}e^{-\beta\mathrm{Iso}_{\\{1,2\\},\nu}\mathcal{H}^{W,U}(X_{N})}dX_{N}$
(1.8)
$\displaystyle\leq\frac{e^{-\beta\Delta}}{\mathcal{Z}}\int_{E_{\\{1,2\\}}}\mathrm{Iso}_{\\{1,2\\},\nu}e^{-\beta\mathcal{H}^{W,U}(X_{N})}dX_{N}$

for

$\Delta=\inf_{X_{N}\in
E_{\\{1,2\\}}}\mathcal{H}^{W,U}(X_{N})-\mathrm{Iso}_{\\{1,2\\},\nu}\mathcal{H}^{W,U}(X_{N}).$

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

$\displaystyle\mathbb{P}^{W,U}_{N,\beta}(E_{\\{1,2\\}})$
$\displaystyle\leq\frac{e^{-\beta\Delta}}{\mathcal{Z}}\int_{(\mathbb{R}^{d})^{N}}\mathrm{Iso}^{\ast}_{\\{1,2\\},\nu}\mathbf{1}_{E_{\\{1,2\\}}}(X_{N})e^{-\beta\mathcal{H}^{W,U}(X_{N})}dX_{N}$
(1.9)
$\displaystyle=e^{-\beta\Delta}\mathbb{E}^{W,U}_{N,\beta}[\mathrm{Iso}^{\ast}_{\\{1,2\\},\nu}\mathbf{1}_{E_{\\{1,2\\}}}].$

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

$\mathrm{Iso}^{\ast}_{\\{1,2\\},\nu}\mathbf{1}_{E_{\\{1,2\\}}}(X_{N})=\mathrm{Iso}_{\\{1,2\\},\nu}\mathbf{1}_{E_{\\{1,2\\}}}(X_{N})\leq\|\nu\|^{2}_{L^{\infty}}\|\mathbf{1}_{E_{\\{1,2\\}}}(\cdot,\cdot,x_{3},\ldots,x_{N})\|_{L^{1}(\mathbb{R}^{2})}\leq
Cr^{2d}.$

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

$\mathbb{E}^{W,U}_{N,\beta}[\mathrm{Iso}^{\ast}_{\\{1,2\\},\nu}\mathbf{1}_{E_{\\{1,2\\}}}]\leq
Cr^{2d}\mathbb{P}^{W,U}_{N,\beta}(\\{x_{1},x_{2}\in B_{2}(z)\\}).$

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

$\mathbb{P}^{W,U}_{N,\beta}(E)\leq
Ce^{-\beta\mathsf{g}(2r)}r^{2d}N^{2}\mathbb{P}^{W,U}_{N,\beta}(\\{x_{1},x_{2}\in
B_{2}(z)\\}).$ (1.10)

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].

######  Notation.

We identify $\mathbb{P}^{W,U}_{N,\beta}$ with the law of a point process $X$,
with the translation between $X_{N}$ and $X$ given by
$X=\sum_{i=1}^{N}\delta_{x_{i}}$. All point processes will be assumed to be
simple. We also define the “index” process $\mathbb{X}$ given by
$\mathbb{X}(A)=\\{i:x_{i}\in A\\}$ for measurable sets $A$. For example, we
have $E=\\{X(B_{r}(z))\geq 2\\}$ and
$E_{\\{1,2\\}}=\\{\\{1,2\\}\subset\mathbb{X}(B_{r}(z))\\}$ for the events $E$
and $E_{\\{1,2\\}}$ considered in this subsection.

###  1.3. JLM laws.

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

$\mathbb{P}_{\mathrm{jell}}(\\{X(B_{R}(z))\geq
Q\\})\sim\begin{cases}\exp\left(-\frac{\beta}{4}Q^{2}\log\frac{Q}{Q_{0}}\right)\quad\text{if
}d=2,\\\ \exp\left(-\frac{\beta}{4R}Q^{2}\right)\quad\text{if
}d=3,\end{cases}$ (1.11)

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 (High Density JLM Law).

For any $R\geq 1$, integer $\lambda\geq 100$, and integer $Q$ satisfying

$Q\geq\begin{cases}\frac{C\lambda^{2}R^{2}+C\beta^{-1}}{\log(\frac{1}{4}\lambda)}\quad&\text{if
}d=2,\\\ CR^{d}+C\beta^{-1}R^{d-2}\quad&\text{if }d\geq 3,\end{cases}$ (1.12)

we have

$\mathbb{P}^{W,U}_{N,\beta}(\\{X(B_{R}(z))\geq
Q\\})\leq\begin{cases}e^{-\frac{1}{2}\beta\log(\frac{1}{4}\lambda)Q^{2}+C(1+\beta\lambda^{2}R^{2})Q}\quad&\text{if
}d=2,\\\ e^{-2^{-d}\beta R^{-d+2}Q(Q-1)}\quad&\text{if }d\geq 3,\end{cases}$
(1.13)

and the result remains true if $z$ is replaced by $x_{1}$. The constant $C$
depends only on the dimension and the upper bound for $\Delta W$. In
particular if $d=2$ and $Q\geq C_{\beta,W}R^{2}$, we may choose
$\lambda=\sqrt{\frac{Q}{R^{2}}}$ to see

$\mathbb{P}^{W}_{N,\beta}(\\{X(B_{R}(z))\geq Q\\})\leq
e^{-\frac{\beta}{4}\log\left(\frac{Q}{R^{2}}\right)Q^{2}+C\beta Q^{2}+CQ}.$

######  Remark 1.1.

The physical principles leading to the law (1.11) focus on the change of free
energy between an unconstrained Coulomb gas and one constrained to have charge
$Q$ in $B_{R}(z)$. For the constrained gas, the most likely particle
configurations involve a build up of positive charge on an inner boundary
layer of $B_{R}(z)$ and a near vacuum outside of $B_{R}(z)$ which “screens”
the excess charge. Since the negative charge density is bounded (in a jellium
by definition and in $\mathbb{P}^{W,U}_{N,\beta}$ by $\Delta W\leq C$), the
negative screening region must be extremely thick when $Q\gg R^{d}$. The self-
energy of the negative screening region is the dominant contributor to the
(1.11) bounds in [JLM93]. In our proof, we apply an isotropic averaging
operator that moves the particles within $B_{R}(z)$ to the bulk of the vacuum
region, extracting a large average energy change per particle, thus providing
a different perspective on the JLM law.

######  Remark 1.2.

Theorem 1 applies to small $\beta>0$. In particular, one sees that charge
excesses of order $TR^{d}$, $T\gg 1$, become unlikely as soon as $R\geq
C^{-1}\beta^{-1/2}$. For this particular estimate type, Theorem 1 therefore
improves the minimal effective distance given in [AS21, Theorem 1] in
dimensions $d=2$ and $d\geq 5$ ($R\geq C\beta^{-1/2}(\log\beta^{-1})^{1/2}$
and $R\geq C\beta^{\frac{1}{d-2}-1}$, respectively).

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.

######  Corollary 1.3.

For any sequence of points $z_{N}\in\mathbb{R}^{d}$, the law under
$\mathbb{P}^{W,U}_{N,\beta}$ of the point process
$\sum_{i=1}^{N}\delta_{x_{i}-z_{N}}$ converges weakly along subsequences as
$N\to\infty$ to a simple point process.

###### Proof.

Tightness of the law of the finite dimensional distributions
$(X(A_{1}),\ldots,X(A_{n}))$ for bounded Borel sets $A_{1},\ldots,A_{n}$ (or
for shifted versions of $X$) follows from Theorem 1. This implies weak
convergence of the laws of the point processes (see [DVJ08, Theorem
11.1.VII]). ∎

######  Remark 1.4.

Any limit from Corollary 1.3 will also enjoy analogs of Theorem 1, Theorem 2,
and Theorem 5.

###  1.4. Clustering and the $k$-point function.

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

$\int_{A_{1}\times A_{2}\times\cdots\times
A_{k}}\rho_{k}(y_{1},y_{2},\ldots,y_{k})dy_{1}\cdots
dy_{k}=\frac{N!}{(N-k)!}\mathbb{P}^{W,U}_{N,\beta}\left(\bigcap_{i=1}^{k}\\{x_{i}\in
A_{i}\\}\right)$ (1.14)

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.

######  Theorem 2.

We have that

$\rho_{1}(y)\leq C$ (1.15)

for some constant $C$ independent of $N$ and $y$. We also have

$\rho_{k}(y_{1},y_{2},\ldots,y_{k})\leq C\prod_{1\leq i<j\leq
k}(1\wedge|y_{i}-y_{j}|^{\beta})\quad\text{if }d=2$ (1.16)

and

$\rho_{k}(y_{1},y_{2},\ldots,y_{k})\leq
C\exp\left(-\beta\mathcal{H}^{0}(y_{1},\ldots,y_{k})\right)\quad\text{if
}d\geq 3.$ (1.17)

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.

######  Theorem 3.

Let $Q$ be a positive integer. There exists a constant $C$, dependent only on
$\beta$, $Q$, and $\sup\Delta W$, such that for all $r>0$ we have

$\mathbb{P}^{W,U}_{N,\beta}(\\{X(B_{r}(z))\geq Q\\})\leq
Cr^{2Q+\beta\binom{Q}{2}}\quad\text{if }d=2,$ (1.18)

and

$\mathbb{P}^{W,U}_{N,\beta}(\\{X(B_{r}(z))\geq Q\\})\leq
Cr^{dQ}e^{-\frac{\beta}{2^{d-2}}\cdot\frac{1}{r^{d-2}}\binom{Q}{2}}\quad\text{if
}d\geq 3.$ (1.19)

We also have for $Q\geq 2$ and $d=2$ that

$\mathbb{P}^{W,U}_{N,\beta}(\\{X(B_{r}(x_{1}))\geq Q\\})\leq
Cr^{2(Q-1)+\beta\binom{Q}{2}}$ (1.20)

and $d\geq 3$ that

$\mathbb{P}^{W,U}_{N,\beta}(\\{X(B_{r}(x_{1}))\geq Q\\})\leq
Cr^{d(Q-1)}e^{-\frac{\beta}{2^{d-2}}\cdot\frac{1}{r^{d-2}}\binom{Q-1}{2}}e^{-\beta\frac{1}{r^{d-2}}(Q-1)}.$
(1.21)

In the case of $Q=2$ and $d\geq 3$ this can be improved to

$\mathbb{P}^{W,U}_{N,\beta}(\\{X(B_{r}(x_{1}))\geq 2\\})\leq
Cr^{2d-2}e^{-\frac{\beta}{r^{d-2}}}.$ (1.22)

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].

######  Remark 1.5.

We claim that our results in Theorem 2 are essentially optimal and that
Theorem 3 is optimal if $d=2$ or $Q=2$. For $d,Q\geq 3$, one can improve
Theorem 3 by more carefully integrating Theorem 2, though an optimal, explicit
solution for all $Q$ would be difficult. Our claim is evidenced by the
tightness results we prove in Theorem 4 and by computations for merging points
with fixed $N$.

###  1.5. Minimal particle gaps.

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

$\eta_{k}(X_{N})=\text{the}\ k\text{th smallest element of}\ \\{|x_{i}-x_{j}|\
:\ i,j\in\\{1,\ldots,N\\},i\neq j\\}.$ (1.23)

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.

######  Corollary 1.6.

Let $d=2$. We have

$\mathbb{P}^{W}_{N,\beta}(\\{\eta_{1}\leq\gamma N^{-\frac{1}{2+\beta}}\\})\leq
C\gamma^{2+\beta}\quad\forall\gamma>0$ (1.24)

for a constant $C$ independent of $N$.

###### Proof.

This follows from Theorem 3, specifically (1.20) with $Q=2$ and $r=\gamma
N^{-\frac{1}{2+\beta}}$, and a union 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

$\lim_{|x|\to\infty}V(x)+\mathsf{g}(x)=+\infty\quad\text{and}\quad\int_{\mathbb{R}^{d}}e^{-\beta(V(x)-\log(1+|x|))}dx<\infty\quad\text{if
}d=2,$ (A1)
$\exists\varepsilon>0\quad\liminf_{|x|\to\infty}\frac{V(x)}{|x|^{\varepsilon}}>0\quad\text{if
}d\geq 3.$ (A2)

######  Theorem 4.

Let $W=V_{N}$ with $\Delta W\leq C$, in $d=2$ condition (A1) satisfied, and in
$d\geq 3$ condition (A2) satisfied. Then, in $d=2$ the law of
$N^{\frac{1}{2+\beta}}\eta_{k}$ is tight as $N\to\infty$. Moreover, we have

$\limsup_{N\to\infty}\mathbb{P}^{V_{N}}_{N,\beta}(\\{N^{\frac{1}{2+\beta}}\eta_{k}\leq\gamma\\})\leq
C\gamma^{k(2+\beta)},$
$\limsup_{N\to\infty}\mathbb{P}^{V_{N}}_{N,\beta}(\\{N^{\frac{1}{2+\beta}}\eta_{k}\geq\gamma\\})\leq
C\gamma^{-\frac{4+2\beta}{4+\beta}}$

for all $\gamma>0$.

In $d\geq 3$, let $Z_{k}$ be defined by

$\eta_{k}=\left(\frac{\beta}{\log
N}\right)^{\frac{1}{d-2}}\left(1+\frac{2d-2}{(d-2)^{2}}\frac{\log\log N}{\log
N}+\frac{Z_{k}}{(d-2)\log N}\right).$

Then the law of $Z_{k}$ is tight as $N\to\infty$. We have

$\limsup_{N\to\infty}\mathbb{P}^{V_{N}}_{N,\beta}(\\{Z_{k}\leq-\gamma\\})\leq
Ce^{-k\gamma},$
$\limsup_{N\to\infty}\mathbb{P}^{V_{N}}_{N,\beta}(\\{Z_{k}\geq\gamma\\})\leq
Ce^{-\frac{1}{2}\gamma}$

for $\gamma>0$.

###### Proof.

The theorem follows easily from combining Proposition 4.1 and Proposition 4.4
from Section 4. ∎

######  Remark 1.7.

The estimates for $\eta_{k}$ in Theorem 4 can be understood by the following
Poissonian ansatz. Consider $x_{i}$, $i=1,\ldots,N$, to be an i.i.d. family
with $x_{1}$ having law with density proportional to
$e^{-\beta\mathsf{g}(x)}dx$ on $B_{1}(0)\subset\mathbb{R}^{d}$. If we let
$\eta_{k}$ be the $k$th smallest element of $\\{|x_{i}|:i=1,\ldots,N\\}$, then
the order of $\eta_{k}$ agrees with Theorem 4.

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.

###  1.6. Discrepancy and incompressibility bounds.

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

$\mathrm{Disc}(\Omega)=X(\Omega)-N\mu_{\mathrm{eq}}(N^{-\frac{1}{d}}\Omega),\quad\Omega\subset\mathbb{R}^{d}$
(1.25)

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

$\mathrm{Disc}_{W}(\Omega)=X(\Omega)-\frac{1}{c_{d}}\int_{\Omega}\Delta
W(x)dx$ (1.26)

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 $x_{i}\in\mathbb{C}\cong\mathbb{R}^{2}$)

$\Psi_{F}(X_{N})=F(X_{N})\Psi_{\mathrm{Lau}}(X_{N}),\quad\Psi_{\mathrm{Lau}}(X_{N})\propto\prod_{1\leq
i<j\leq N}(x_{i}-x_{j})^{\ell}e^{-B\sum_{i=1}^{N}|x_{i}|^{2}/4}$ (1.27)

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

$U(X_{N})=-\beta^{-1}\log|F(X_{N})|^{2}.$

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

$E_{\mathcal{O}}(N,B,\ell)=\inf_{F}\mathbb{E}^{W,U}_{N,\beta}\left[\sum_{i=1}^{N}\mathcal{O}(x_{i})\right]=\inf_{F}\int_{\mathbb{R}^{2}}\mathcal{O}(x)\rho_{1,F}(x)$
(1.28)

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

$\mathrm{Disc}_{W}(B_{R}(z))\leq o(1)\quad\text{as }N\to\infty$

for $R\geq N^{1/4+\varepsilon}$ with high probability. Such a result is called
an incompressibility estimate. A consequence is that if $\mathcal{O}$ varies
on a scale larger than $N^{1/4}$, then we have that the “bathtub” energy

$E_{\mathcal{O}}^{\mathrm{bt}}(N,B,\ell)=\inf\left\\{\int_{\mathbb{R}^{2}}\mathcal{O}(x)\rho(x)\
:\ \int_{\mathbb{R}^{2}}\rho(x)dx=N,\ 0\leq\rho\leq c_{d}^{-1}\Delta
W\right\\}$ (1.29)

is an approximate lower bound for $E_{\mathcal{O}}(N,B,\ell)$. The infimum in
(1.29) is over $\rho$ that do not necessarily come as a $1$-point function for
some $\mathbb{P}^{W,U}_{N,\beta}$. The restriction on the length scale of
$\mathcal{O}$ does not capture all physically realistic scenarios. To prove
that $F(X_{N})=\prod_{i=1}^{N}f(x_{i})$ approximately saturates the infimum in
$E_{\mathcal{O}}(N,B,\ell)$, the remaining task is to show that a set of
profiles $\rho$ saturating the infimum in the bathtub energy are well
approximated by a $1$-point function $\rho_{1,F}$ with $F$ of the factorized
form, a task that was considered in [RY18, OR20].

We present a new method to prove incompressibility down to large microscopic
scales, i.e. for $R\gg 1$, and also to give quantitative estimates for $o(1)$
terms. Our result is also interesting because it gives density upper bounds on
balls $B_{R}(z)$ with weaker restrictions on the location of $z$ than in
[AS21]. For technical reasons appearing in Proposition 5.3, we will need to
approximate the density of $\mu$ by a constant in $B_{2R}(z)$, and the
resulting error begins to dominate for $R\geq N^{3/(5d)}$. We have therefore
chosen to restrict to small enough mesoscales $R$. For FQHE applications, one
has that $\mu$ is a multiple of Lebesgue measure and so this restriction is
unnecessary.

######  Theorem 5 (Incompressibility).

Let $R\geq 1$ and $z\in\mathbb{R}^{d}$. Suppose either $\Delta W$ is constant
in $B_{2R}(z)$ or both of the following: firstly that

$\|\nabla\Delta W\|_{L^{\infty}(B_{2R}(z))}\leq CN^{-1/d},$

and secondly $R\leq N^{3/(5d)}$. Assume also $\Delta W(x)\geq C^{-1}$ for
$x\in B_{2R}(z)$. Then we have

$\mathbb{P}^{W,U}_{N,\beta}(\\{\mathrm{Disc}_{W}(B_{R}(z))\geq
TR^{d-2/3}(1+\mathbf{1}_{d=2}\log R)\\})\leq
e^{-cR^{d}T}+e^{-cR^{(d+2)/3}T^{2}}+e^{-R^{d+2}}$ (1.30)

for some $c>0$ for all $T\geq 1$ large enough.

Theorem 5 affirmatively answers an important question posed in [LRY19],
demonstrating the remarkable property that the Coulomb gas cannot be
significantly compressed beyond density $c_{d}^{-1}\Delta W$ by any choice of
superharmonic perturbation $U$. The importance of incompressibility of the
Laughlin phase was first raised in [RY15a] and first progress was made in
[RY15b]. Further progress and the previously best result appears in [LRY19].
There, the authors transfer $\beta=\infty$ incompressibility estimates to the
positive temperature system, whereas we work directly with the positive
temperature Gibbs measure. We also avoid completely the use of potential
theoretic subharmonic quadrature domains (also termed “screening regions”). We
expect our result will have significant applications toward proving the
stability of the Laughlin phase, which appears in the study of 2D electron
gases and rotating Bose gases [RSY14].

######  Remark 1.8.

Theorem 5 may, at first, seem to be in disagreement with numerical results
showing oscillations in $\rho_{1}$ at the edge of the droplet [CSA20, Cif06,
CW03] in $d=2$, with $\sup\rho_{1}$ significantly larger than
$(2\pi)^{-1}\Delta W$. The oscillation wavelength appears to be of order of
the inter-particle distance $R=1$, whereas our theorem becomes effective for
$R\gg 1$, resolving the apparent disagreement.

While Theorem 5 controls only positive discrepancies, when combined with an
estimate on the fluctuations of smooth linear statistics, it can also give
lower bounds. Indeed, when $B_{R}(z)$ has a large discrepancy, the physically
realistic scenario is that positive charge excess builds up either on the
inside or outside of $\partial B_{R}(z)$. We call a thin annulus with the
positive charge buildup a “screening region”, the existence of which is
implied by rigidity for smooth linear statistics. We use rigidity from
[Ser20], which we note does not take “heavy-lifting”, e.g. it is independent
of the multi-scale argument from [AS21]. Proposition 5.6 proves a stronger
version of Theorem 5 that applies to screening regions.

We will need some additional assumptions to apply results from [Ser20] and
[AS19]. We refer to the introduction of [Ser20] for commentary on the
conditions. While the conditions hold in significant generality, our main
purpose is to demonstrate the usefulness of Theorem 5 rather than optimize the
conditions. Assume that $W=V_{N}$ where $V\in C^{7}(\mathbb{R}^{d})$, the
droplet $\Sigma=\mathrm{supp}\ \mu_{\mathrm{eq}}$ has $C^{1}$ boundary,
$\Delta V\geq C^{-1}>0$ in a neighborhood of $\Sigma$, and $\sup\Delta V\leq
C$. Assume further

$\begin{cases}&\int_{\mathbb{R}^{d}}e^{-\frac{\beta}{2}N(V(x)-\log(1+|x|))}dx+\int_{\mathbb{R}^{d}}e^{-\beta
N(V(x)-\log(1+|x|))}(|x|\log(1+|x|))^{2}dx<\infty\quad\text{if }d=2,\\\
&\int_{\mathbb{R}^{d}}e^{-\frac{\beta}{2}V(x)}dx<\infty\quad\text{if }d=3,\\\
&\lim_{|x|\to\infty}V(x)+\mathsf{g}(x)=+\infty.\end{cases}$

Finally, assume there exists a constant $K$ such that

$\mathsf{g}\ast\mu_{\mathrm{eq}}(x)+V(x)-K\geq
C^{-1}\min(\mathrm{dist}(x,\Sigma)^{2},1)\quad\forall x\in\mathbb{R}^{d}.$

######  Theorem 6.

Let $R\in[1,N^{\frac{5}{7d}}]$ and $z\in\mathbb{R}^{d}$ be such that
$B_{2R}(z)\subset\\{x\in\Sigma_{N}\ :\ \mathrm{dist}(x,\partial\Sigma_{N})\geq
C_{0}N^{1/(d+2)}\\}$ for a large enough constant $C_{0}$. Assume that
$W=V_{N}$ with $V$ satisfying the above conditions. Then we have

$\mathbb{P}^{V_{N}}_{N,\beta}(\\{|\mathrm{Disc}(B_{R}(z))|\geq
TR^{d-4/5}(1+\mathbf{1}_{d=2}\log R)\\})\leq
e^{-cR^{d-10/15}T}+e^{-cR^{2/5+d/15}T^{2}}+e^{-cR^{(d+2)/5}}.$

for large enough $T\gg 1$ and some $c>0$.

We note that by applying isotropic averaging to a screening region, not only
do we obtain bounds on the absolute discrepancy, we also give a sharper bound
on the positive part than Theorem 5, albeit with some extra restrictions on
$R$ and $z$.

###  1.7. Notation.

We now introduce some notation and conventions used throughout the paper.
First, we recall the point process $X$ and “index” process $\mathbb{X}$
introduced at the end of Section 1.1. All point processes will be simple.

Implicit constants $C$ will change from line to line and may depend on
$\sup\Delta W$ and $d$ without further comment. In all sections except Section
2, we will also allow $C$ to depend on continuously $\beta$ and $\beta^{-1}$.
A numbered constant like $C_{0},C_{1}$ will be fixed, but may be needed to be
taken large depending on various parameters. For positive quantities $a,b$, we
write $a\gg b$ to mean that $a\geq Cb$ for a large enough constant $C>0$, and
$a\ll b$ for $a\leq C^{-1}b$ for large enough $C>0$.

For brevity we will sometimes write $\mathbb{P}$ for
$\mathbb{P}^{W,U}_{N,\beta}$ and $\mathbb{E}$ for
$\mathbb{E}^{W,U}_{N,\beta}$. This will only be done in proofs or sections
where the probability measure is fixed throughout. We will write
$\mathsf{g}(s)$ to mean $-\log s$ in dimension $2$ or $|s|^{-d+2}$ in $d\geq
3$ when $s>0$. For a measure with a Lebesgue density, we often denote the
density with the same symbol as the measure, e.g. $\nu(x)$ as the density of
$\nu(dx)$.

When it exists, we let $\mu_{\mathrm{eq}}$ be the equilibrium measure
associated to $V$, and let $\Sigma=\mathrm{supp}\ \mu_{\mathrm{eq}}$ be the
droplet. Note that $\mu_{\mathrm{eq}}$ is a probability measure and $\Sigma$
has length scale $1$. We let $\mu_{\mathrm{eq}}^{N}$ be the blown-up
equilibrium measure with
$\mu_{\mathrm{eq}}^{N}(A)=N\mu_{\mathrm{eq}}(N^{-1/d}A)$ for Borel sets $A$
and $\Sigma_{N}=\mathrm{supp}\
\mu_{\mathrm{eq}}^{N}=N^{1/d}\mathrm{supp}\mu_{\mathrm{eq}}$ be the blown-up
droplet. Finally, we define

$\mu(dx)=\frac{1}{c_{d}}\Delta W(x)dx$ (1.31)

where $c_{d}$ is such that $\Delta\mathsf{g}=-c_{d}\delta_{0}$. When we take
$W=V_{N}$, we have that $\mu=\mu_{\mathrm{eq}}^{N}$ on the blown-up droplet
$\Sigma_{N}$.

We let $B_{R}(z)\subset\mathbb{R}^{d}$ be the open Euclidean ball of radius
$R$ centered at $z$, and for a nonempty interval $I\subset\mathbb{R}^{\geq
0}$, we let $\mathrm{Ann}_{I}(z)$ be the annulus containing points $x$ with
$|x-z|\in I$. We use $|\cdot|$ to denote both Lebesgue measure for subsets of
$\mathbb{R}^{d}$ and cardinality for finite sets. It will be clear via context
which is meant.

######  Remark 1.9.

For most applications, the requirement $\Delta W\leq C$ can be loosened to
boundedness on the macroscopic scale. This is because the points $X_{N}$ are
typically confined to a vicinity of the droplet and our arguments are
localized to macroscopic neighborhoods of the regions to which they are
applied. In $d=2$, for example, Ameur [Ame21] has proved strong confinement
estimates. Furthermore, one can apply our arguments to other Coulomb-type
systems like finite volume jelliums provided our arguments do not “run into”
domain boundaries. The limiting factor is generally in iterating Proposition
2.2 as in the proof of Theorem 1.

#### Acknowledgments.

The author would like to thank his advisor, Sylvia Serfaty, for her
encouragement and comments. He also thanks Nicolas Rougerie for helpful
comments on the incompressibility estimates, especially in connection to the
fractional quantum Hall effect. The author was partially supported by NSF
grant DMS-2000205.

##  2\. High Density Estimates

In this section, we prove Theorem 1. We will write $\mathbb{P}$ for
$\mathbb{P}_{N,\beta}^{W,U}$, and implicit constants $C$ will depend only on
$d$ and $\sup\Delta W$; they are independent of $\beta$.

The idea behind the proof is as follows. We will consider two scales $r,R$
with $1\leq r=\lambda^{-1}R$ for $\lambda\geq 10$ and two concentric balls
$B_{r}(z)$ and $B_{R}(z)$. In the event that $X(B_{r}(z))\gg r^{d}$, there is
a large pairwise Coulomb energy benefit upon replacing each point charge
within $B_{r}(z)$ by annuli of scale $R$. In particular, this energy benefit
dominates any loss from the potential term $\sum_{i}W(x_{i})$ in
$\mathcal{H}^{W,U}$. We must however consider entropy factors: after applying
isotropic averaging, the particles originally confined to $B_{r}(z)$ become
indistinguishable from the particles within $B_{R}(z)$. If
$X(B_{R}(z))\leq\lambda^{d}X(B_{r}(z))$, i.e. the density of particles in
$B_{R}(z)$ is not larger than that of $B_{r}(z)$, the entropy costs are
manageable. If, on the other hand, $B_{R}(z)$ has an extremely high particle
density, we may iterate our estimate to larger scales $R$ and $\lambda R$. The
iteration terminates once $R^{d}\gg N$ and the considered overcrowding event
becomes impossible.

Proposition 2.1 will compute the energy change upon isotropic averaging and
estimate the adjoint isotropic averaging operator on the relevant event, which
Proposition 2.2 uses as in the model computation to obtain the iteration step.
Theorem 1 will then follow shortly. Note that the below computation leading up
to (2.3) and (2.4) will be used often in slightly different contexts.

######  Proposition 2.1.

Consider $0<r<\frac{1}{10}R$ and $\nu_{R}$ the uniform probability measure on
the annulus $\mathrm{Ann}_{[\frac{1}{2}R,R-2r]}(0)$. We have

$\mathrm{Iso}_{\mathbb{X}(B_{r}(z)),\nu_{R}}\mathcal{H}^{W,U}(X_{N})\leq\mathcal{H}^{W,U}(X_{N})+CR^{2}X(B_{r}(z))+\binom{X(B_{r}(z))}{2}\left(\mathsf{g}\left(\frac{R}{2}\right)-\mathsf{g}(2r)\right),$
(2.1)

and for any index sets $\mathcal{N},\mathcal{M}\subset\\{1,\ldots,N\\}$ we
have

$\mathrm{Iso}_{\mathcal{N},\nu_{R}}^{\ast}\mathbf{1}_{\\{\mathbb{X}(B_{r}(z))=\mathcal{N}\\}\cap\\{\mathbb{X}(B_{R}(z))=\mathcal{M}\\}}\leq
e^{C|\mathcal{N}|}\left(\frac{r}{R}\right)^{d|\mathcal{N}|}\mathbf{1}_{\\{\mathbb{X}(B_{R}(z))=\mathcal{M}\\}}.$
(2.2)

###### Proof.

We first prove (2.1) by considering the effect of the isotropic averaging
operator on $\mathsf{g}(x_{i}-x_{j})$ and $W(x_{i})$. Let $\sigma$ denote
$(d-1)$-dimensional Hausdorff measure. First, since
$-\Delta\mathsf{g}=c_{d}\delta_{0}$ for $c_{d}=\sigma(\partial
B_{1})(d-2+\mathbf{1}_{d=2})$, we have for any $s>0$ and $y\in\mathbb{R}^{d}$
by Green’s third identity that

$\displaystyle\frac{1}{c_{d}}\int_{B_{s}(y)}\mathsf{g}(x-y)\Delta W(x)dx+W(y)$
$\displaystyle=\frac{1}{c_{d}s}\int_{\partial
B_{s}(y)}\left(\mathsf{g}(x-y)\nabla
W(x)\cdot(x-y)-W(x)\nabla\mathsf{g}(x-y)\cdot(x-y)\right)\sigma(dx).$

Using the divergence theorem, the RHS can be simplified further to

$\frac{\mathsf{g}(s)}{c_{d}}\int_{B_{s}(y)}\Delta
W(x)dx+\frac{d-2+\mathbf{1}_{d=2}}{c_{d}s^{d-1}}\int_{\partial
B_{s}(y)}W(x)\sigma(dx).$

After a rearrangement, we see

$\frac{1}{\sigma(\partial B_{s})}\int_{\partial
B_{s}(y)}W(x)\sigma(dx)=W(y)+\frac{1}{c_{d}}\int_{B_{s}(y)}(\mathsf{g}(x-y)-\mathsf{g}(s))\Delta
W(x)dx,$

and we can then integrate against $\sigma(\partial B_{s})ds$ with $y=x_{1}$ to
see

$\displaystyle\mathrm{Iso}_{\\{1\\},\nu_{R}}W(x_{1})$
$\displaystyle=W(x_{1})+\frac{1}{c_{d}|\mathrm{Ann}_{[\frac{1}{2}R,R-2r]}(0)|}\int_{\frac{1}{2}R}^{R-2r}\sigma(\partial
B_{s})\int_{B_{s}(x_{1})}(\mathsf{g}(x-x_{1})-\mathsf{g}(s))\Delta W(x)dxds.$

If we bound $\Delta W\leq C$, one can check by explicit integration that

$\mathrm{Iso}_{\\{1\\},\nu_{R}}W(x_{1})\leq W(x_{1})+CR^{2}.$ (2.3)

A similar computation, this time using $\Delta\mathsf{g}\leq 0$ or
superharmonicity of $U$ in each variable, shows that

$\mathrm{Iso}_{\mathbb{X}(B_{r}(z)),\nu_{R}}\mathsf{g}(x_{i}-x_{j})\leq\mathsf{g}(x_{i}-x_{j}),\quad\mathrm{Iso}_{\mathbb{X}(B_{r}(z)),\nu_{R}}U(X_{N})\leq
U(X_{N})\quad\forall i,j\in\\{1,\ldots,N\\}.$ (2.4)

Finally, by Newton’s theorem, the Coulomb interaction between a sphere of unit
charge and radius $s$ and a point charge is bounded above by $\mathsf{g}(s)$.
It follows from superposition that

$\mathrm{Iso}_{\mathbb{X}(B_{r}(z)),\nu_{R}}\mathsf{g}(x_{i}-x_{j})\leq\mathsf{g}\left(\frac{R}{2}\right)$

whenever $i\in\mathbb{X}(B_{r}(z))$, particularly whenever
$i,j\in\mathbb{X}(B_{r}(z))$ in which case
$\mathsf{g}(x_{i}-x_{j})\geq\mathsf{g}(2r)$. Putting the above results
together proves (2.1).

We now consider (2.2). Using that
$\mathrm{Iso}_{\mathcal{N},\nu_{R}}=\mathrm{Iso}_{\mathcal{N},\nu_{R}}^{\ast}$
is a convolution on $(\mathbb{R}^{d})^{\mathcal{N}}$ and Young’s inequality,
we have

$\displaystyle\mathrm{Iso}_{\mathcal{N},\nu_{R}}^{\ast}\mathbf{1}_{\\{\mathbb{X}(B_{r}(z))=\mathcal{N}\\}\cap\\{\mathbb{X}(B_{R}(z))=\mathcal{M}\\}}$
(2.5)
$\displaystyle\leq\|\nu_{R}\|_{L^{\infty}(\mathbb{R}^{d})}^{|\mathcal{N}|}\|\mathbf{1}_{\\{\mathbb{X}(B_{r}(z))=\mathcal{N}\\}\cap\\{\mathbb{X}(B_{R}(z))=\mathcal{M}\\}}\|_{L^{1}((\mathbb{R}^{d})^{\mathcal{N}})}\leq
e^{C|\mathcal{N}|}\left(\frac{r}{R}\right)^{d|\mathcal{N}|}.$

For any configuration $X_{N}$ with $\mathbb{X}(B_{R}(z))\neq\mathcal{M}$, we
claim that

$\mathrm{Iso}_{\mathcal{N},\nu_{R}}^{\ast}\mathbf{1}_{\\{\mathbb{X}(B_{r}(z))=\mathcal{N}\\}\cap\\{\mathbb{X}(B_{R}(z))=\mathcal{M}\\}}(X_{N})=0.$

First, our claim follows if
$\mathbb{X}(B_{R}(z))\setminus\mathcal{N}\neq\mathcal{M}\setminus\mathcal{N}$
simply because our isotropic averaging operator leaves coordinates with labels
in $\mathcal{N}^{c}$ fixed. So we may instead assume there exists
$i\in\mathcal{N}$ with $x_{i}\not\in B_{R}(z)$. Then convolution with
$\nu_{R}$ in the $x_{i}$ coordinate considers translates $x_{i}+y$ with
$|y|\leq R-2r$, none of which can be found in $B_{r}(z)$. Our claim follows,
and together with our pointwise bound (2.5) this establishes (2.2). ∎

We are ready to prove the main iterative estimate that establishes Theorem 1.

######  Proposition 2.2.

Let $0<r<R$ be such that $\lambda:=\frac{R}{r}\geq 10$. Then we have that

$\mathbb{P}(\\{X(B_{r}(z))\geq
Q\\})\leq\mathbb{P}(\\{X(B_{R}(z))\geq\lambda^{d}Q\\})+e^{C(1+\beta\lambda^{2}r^{2})Q-\beta\binom{Q}{2}(\mathsf{g}(2r)-\mathsf{g}(\lambda
r/2))}$

for all $z\in\mathbb{R}^{d}$ and integers $Q$ with

$Q\geq\begin{cases}\frac{C\lambda^{2}r^{2}+C\beta^{-1}}{\log(\frac{1}{4}\lambda)}\quad&\text{if
}d=2,\\\ C\lambda^{2}r^{d}+C\beta^{-1}r^{d-2}\quad&\text{if }d\geq
3.\end{cases}$ (2.6)

###### Proof.

For simplicity, we consider $\lambda$ an integer. Let
$\mathcal{N}\subset\mathcal{M}$ be index sets of size $n$ and $m$,
respectively. We apply the model computation detailed in Section 1.2 and
Proposition 2.1 to see

$\displaystyle\mathbb{P}(\\{\mathbb{X}(B_{r}(z))=\mathcal{N}\\}\cap\\{\mathbb{X}(B_{R}(z))=\mathcal{M}\\})$
$\displaystyle\leq e^{C(1+\beta
R^{2})n-\beta\binom{n}{2}(\mathsf{g}(2r)-\mathsf{g}(R/2))}\lambda^{-dn}\mathbb{P}(\\{\mathbb{X}(B_{R}(z))=\mathcal{M}\\}).$

By particle exchangeability, we have

$\mathbb{P}(\\{X(B_{r}(z))=n\\}\cap\\{X(B_{R}(z)=m)\\})=\binom{N}{m}\binom{m}{n}\mathbb{P}(\\{\mathbb{X}(B_{r}(z))=\mathcal{N}\\}\cap\\{\mathbb{X}(B_{R}(z))=\mathcal{M}\\}),$
$\mathbb{P}(\\{X(B_{R}(z)=m)\\})=\binom{N}{m}\mathbb{P}(\\{\mathbb{X}(B_{R}(z))=\mathcal{M}\\}),$

whence

$\displaystyle\mathbb{P}(\\{X(B_{r}(z))=n\\}\cap\\{X(B_{R}(z)=m)\\})$
$\displaystyle\leq e^{C(1+\beta
R^{2})n-\beta\binom{n}{2}(\mathsf{g}(2r)-\mathsf{g}(R/2))}\binom{m}{n}\lambda^{-dn}\mathbb{P}(\\{X(B_{R}(z))=m\\}).$

By Stirling’s approximation, we can estimate for $1\leq n\leq m$ that

$\binom{m}{n}\leq
2\sqrt{\frac{m}{n(m-n)}}e^{n\log\frac{m}{n}+(m-n)\log\frac{m}{m-n}}\leq
e^{Cn}e^{n\log\frac{m}{n}}.$

The RHS is bounded by $e^{Cn}\lambda^{dn}$ in the case that
$m\leq\lambda^{d}n$, which we consider. We thus have

$\displaystyle\mathbb{P}(\\{X(B_{r}(z))\geq Q\\}\cap\\{X(B_{R}(z))\leq
Q\lambda^{d}\\})$ $\displaystyle\leq\sum_{n=Q}^{Q\lambda^{d}}e^{C(1+\beta
R^{2})n-\beta\binom{n}{2}(\mathsf{g}(2r)-\mathsf{g}(R/2))}\sum_{m=n}^{Q\lambda^{d}}\mathbb{P}(\\{X(B_{R}(z))=m\\})$
$\displaystyle\leq\sum_{n=Q}^{Q\lambda^{d}}e^{C(1+\beta
R^{2})n-\beta\binom{n}{2}(\mathsf{g}(2r)-\mathsf{g}(R/2))}.$

We can bound the ratio between successive terms in the last sum above as

$e^{C(1+\beta R^{2})-\beta n(\mathsf{g}(2r)-\mathsf{g}(R/2))}\leq\frac{1}{2}$

if $Q\log(\lambda/4)\geq C\beta^{-1}+Cr^{2}\lambda^{2}$ in $d=2$ and if $Q\geq
C\beta^{-1}r^{d-2}+C\lambda^{2}r^{d}$ in $d\geq 3$. We conclude

$\mathbb{P}(\\{X(B_{r}(z))\geq Q\\}\cap\\{X(B_{R}(z))\leq Q\lambda^{d}\\})\leq
e^{C(1+\beta\lambda^{2}r^{2})Q-\beta\binom{Q}{2}(\mathsf{g}(2r)-\mathsf{g}(\lambda
r/2))},$

and the proposition follows. ∎

We conclude the section with a proof of the high density JLM laws.

###### Proof of Theorem 1.

In $d=2$, we let $\lambda\geq 10$ be a free parameter, and in $d\geq 3$ we fix
$\lambda$ large enough dependent on $d$. We apply Proposition 2.2 iteratively
to a series of radii $r_{k}=\lambda r_{k-1}$, $k\geq 1$, with $r_{0}=R$ to
achieve

$\mathbb{P}(\\{X(B_{R}(z))\geq
Q\\})\leq\limsup_{k\to\infty}\mathbb{P}(\\{X(B_{r_{k}}(z))\geq\lambda^{dk}Q\\})+\sum_{k=0}^{\infty}e^{a_{k}}$

for

$a_{k}={C(1+\beta\lambda^{2k+2}R^{2})\lambda^{dk}Q-\beta{\binom{\lambda^{dk}Q}{2}}(\mathsf{g}(2\lambda^{k}R)-\mathsf{g}(\lambda^{k+1}R/2))}.$

If $d=2$, we compute

$a_{k+1}-a_{k}\leq-\beta\frac{\lambda^{4(k+1)}\log(\lambda/4)Q^{2}}{4}+C(1+\beta\lambda^{2k+4}R^{2})\lambda^{2(k+1)}Q.$

We can use $Q\log(\lambda/4)\geq C\lambda^{2}R^{2}+C\beta^{-1}$ for a large
enough $C$ to see

$a_{k+1}-a_{k}\leq-\beta\frac{\lambda^{4(k+1)}\log(\lambda/4)Q^{2}}{8}\leq-\log
2.$

In $d\geq 3$, we have

$\displaystyle a_{k+1}$ $\displaystyle\leq
C(1+\beta\lambda^{2k+2}R^{2})\lambda^{dk}Q-\beta\binom{\lambda^{dk}Q}{2}\frac{1}{2^{d-2}R^{d-2}\lambda^{(k+1)(d-2)}}\left(1-\frac{2^{2d-4}}{\lambda^{d-2}}\right)$
$\displaystyle\leq
C(1+\beta\lambda^{2k+2}R^{2})\lambda^{dk}Q-\frac{\beta\lambda^{dk+2k+2-d}Q^{2}}{2^{d}R^{d-2}}.$

By using $Q\geq CR^{d}+C\beta^{-1}R^{d-2}$ (and $\lambda$ fixed), we find

$a_{k+1}\leq-\beta\frac{\lambda^{(d+2)k-d+2}Q^{2}}{2^{d+1}R^{d-2}}.$

One can also compute $a_{k}\geq-\beta R^{-d+2}\lambda^{2dk-(d+2)k}Q^{2}$,
which is dominated by $a_{k+1}$ if $\lambda$ is large enough. It follows that
$a_{k+1}-a_{k}\leq-\log 2$.

We obtain

$\mathbb{P}(\\{X(B_{R}(z))\geq Q\\})\leq 2e^{a_{0}}\leq
e^{-\beta{\binom{Q}{2}}(\mathsf{g}(2R)-\mathsf{g}(\lambda
R/2))+C(1+\beta\lambda^{2}R^{2})Q}.$

The desired result for balls centered at a fixed $z$ follows from some routine
simplifications.

Finally, it will be useful to have versions of the overcrowding estimates for
$z=x_{1}$. For this, note that conditioning $\mathbb{P}^{W,U}_{N,\beta}$ on
$x_{1}$ gives a new measure
$\mathbb{P}^{W,U+\sum_{i=2}^{N}\mathsf{g}(x_{1}-x_{i})}_{N-1,\beta}$ on
$(x_{2},\ldots,x_{N})$. We can then apply our results to this $(N-1)$-particle
Coulomb gas with modified potential. Actually, we could even extract an extra
beneficial term in (2.1) from strict superharmonicity of
$\mathsf{g}(x_{1}-\cdot)$, but it is mostly inconsequential for the large $Q$
results. We omit the details. ∎

##  3\. Clustering Estimates

The goal of this section is to prove Theorem 2 and Theorem 3. Our idea is
similar to that of the previous section, except we will work with
submicroscopic scales and transport particles distances of order $1$. We will
precisely compute energy and volume gains associated to the transport and
control entropy costs using Theorem 1 with $R=1$. Here, it will be important
that we work with measures $\mathbb{P}^{W,U}_{N,\beta}$, since changing $U$
will effectively allow us to condition the gas without deteriorating the
estimates. In this section, we will allow implicit constants $C$ to depend
continuously on $\beta$, $\beta^{-1}$, and $\sup\Delta W$.

###  3.1. $k$-point function bounds

We will first bound the $k$-point function $\rho_{k}(y_{1},\ldots,y_{k})$,
$y_{1},\ldots,y_{k}\in\mathbb{R}^{d}$, which was defined in (1.14). In
particular, we will prove Theorem 2.

Note that

$\rho_{k}(y_{1},\ldots,y_{k})=\rho_{k-1}(y_{1},\ldots,y_{k-1})\rho_{1}(y_{k}|y_{1},\ldots,y_{k-1})$
(3.1)

where $\rho_{1}(\cdot|y_{1},\ldots,y_{k-1})$ is the $1$-point function of the
gas $\mathbb{P}^{W,U_{y_{1},\ldots,y_{k-1}}}_{N-k+1,\beta}$ with

$U_{y_{1},\ldots,y_{k-1}}(x_{k},x_{k+1},\ldots,x_{N})=U(y_{1},\ldots,y_{k-1},x_{k},\ldots,x_{N})+\sum_{i=1}^{k-1}\sum_{j=k}^{N}\mathsf{g}(y_{i}-x_{j}).$

We let $X_{N,k}=(x_{k},\ldots,x_{N})$ and the representation

$\rho_{1}(y_{k}|y_{1},\ldots,y_{k-1})=\lim_{r\to
0^{+}}\frac{N-k+1}{|B_{1}(0)|r^{d}}\mathbb{P}^{W,U_{y_{1},\ldots,y_{k-1}}}_{N-k+1,\beta}(\\{x_{k}\in
B_{r}(y_{k})\\})$ (3.2)

and bound the probability in the RHS using isotropic averaging. Theorem 2 will
follow easily from iterating our one-point function bound. Let $\nu$ be the
uniform probability measure on the annulus
$\mathrm{Ann}_{[\frac{1}{2}+2r,1-2r]}(0)$. We will replace $x_{k}$ by a charge
shaped like $\nu$, and the below lemma gives estimates on the energy change
and on the isotropic averaging operator.

######  Lemma 3.1.

With the definitions above and $0<r<\frac{1}{100}$, we have

$\displaystyle\inf_{\begin{subarray}{c}X_{N,k}\in\mathbb{R}^{d(N-k+1)}\\\
x_{1}\in
B_{r}(y_{k})\end{subarray}}\mathcal{H}^{W,U_{y_{1},\ldots,y_{k-1}}}(X_{N,k})-\mathrm{Iso}_{\\{k\\},\nu}\mathcal{H}^{W,U_{y_{1},\ldots,y_{k-1}}}(X_{N,k})$
(3.3) $\displaystyle\quad\geq-
Ck+\sum_{i=1}^{k-1}\max(\mathsf{g}(|y_{k}-y_{i}|+r),0).$

We also have

$\mathrm{Iso}^{\ast}_{\\{k\\},\nu}\mathbf{1}_{B_{r}(y_{k})}(x_{k})\leq
Cr^{d}\mathbf{1}_{B_{1}(y_{k})}(x_{k}).$ (3.4)

###### Proof.

We begin by noting that (see Proposition 2.1 for a similar computation)

$\mathrm{Iso}_{\\{k\\},\nu}\mathsf{g}(x_{i}-x_{j})\leq\mathsf{g}(x_{i}-x_{j}),\quad\mathrm{Iso}_{\\{k\\},\nu}W(x_{k})\leq
W(x_{k})+C.$

It follows that

$\mathrm{Iso}_{\\{k\\},\nu}\mathcal{H}^{W,0}(X_{N,k})\leq\mathcal{H}^{W,0}(X_{N,k})+C,$

and it remains to consider the $U_{y_{1},\ldots,y_{k-1}}$ term. For this term,
we compute

$\mathrm{Iso}_{\\{k\\},\nu}\mathsf{g}(y_{i}-x_{k})\leq
C,\quad\mathrm{Iso}_{\\{k\\},\nu}\mathsf{g}(y_{i}-x_{k})=\mathsf{g}(y_{i}-x_{k})\text{
if }|y_{i}-y_{k}|\geq 1$

using Newton’s theorem. We also have
$\mathsf{g}(y_{i}-x_{k})\geq\mathsf{g}(|y_{k}-y_{i}|+r)$ since $x_{k}\in
B_{r}(y_{k})$. Thus we have

$\displaystyle\mathrm{Iso}_{\\{k\\},\nu}U_{y_{1},\ldots,y_{k-1}}(X_{N,k})$
$\displaystyle\leq
U_{y_{1},\ldots,y_{k-1}}(X_{N,k})+\sum_{i=1}^{k-1}\mathbf{1}_{|y_{i}-y_{k}|\leq
1}(C-\mathsf{g}(y_{i}-x_{k}))$ $\displaystyle\leq
U_{y_{1},\ldots,y_{k-1}}(X_{N,k})+Ck-\sum_{i=1}^{k-1}\max(\mathsf{g}(|y_{k}-y_{i}|+r,0)).$

This finishes the proof of (3.3). The proof of (3.4) is straightforward using
that $\mathrm{Iso}^{\ast}_{\\{k\\},\nu}$ is convolution by $\nu$. ∎

######  Proposition 3.2.

We have

$\rho_{1}(y_{k}|y_{1},\ldots,y_{k-1})\leq
e^{Ck-\beta\sum_{i=1}^{k-1}\max(\mathsf{g}(y_{i}-y_{k}),0)}$ (3.5)

for the one-point function of
$\mathbb{P}^{W,U_{y_{1},\ldots,y_{k-1}}}_{N-k+1,\beta}$.

###### Proof.

An application of the model computation with the operator
$\mathrm{Iso}_{\\{k\\},\nu}$ and Lemma 3.1 shows

$\mathbb{P}^{W,U_{y_{1},\ldots,y_{k-1}}}_{N-k+1,\beta}(\\{x_{k}\in
B_{r}(y_{k})\\})\leq
r^{d}e^{Ck-\beta\sum_{i=1}^{k-1}\max(\mathsf{g}(y_{i}-y_{k}+r),0)}\mathbb{P}^{W,U_{y_{1},\ldots,y_{k-1}}}_{N-k+1,\beta}(\\{x_{k}\in
B_{1}(y_{k})\\})$ (3.6)

for all $r$ small enough. Let $Y=\sum_{i=k}^{N}\delta_{x_{i}}$. Since

$\sum_{i=k}^{N}\mathbf{1}_{\\{x_{i}\in
B_{1}(y_{k})\\}}\leq\sum_{n=1}^{N-k+1}n\mathbf{1}_{\\{Y(B_{1}(y_{k}))=n\\}}$

and by Theorem 1 we have

$P^{W,U_{y_{1},\ldots,y_{k-1}}}_{N-k+1,\beta}(\\{Y(B_{1}(y_{k}))=n\\})\leq
Ce^{-cn^{2}}$

for some $c>0$, we find by exchangeability that

$\mathbb{P}^{W,U_{y_{1},\ldots,y_{k-1}}}_{N-k+1,\beta}(\\{x_{k}\in
B_{1}(y_{k})\\})\leq\frac{C}{N-k+1}\sum_{n=1}^{\infty}ne^{-cn^{2}}\leq\frac{C}{N-k+1}.$

Combining this with (3.6) and taking $r\to 0^{+}$ in (3.2) proves the
proposition. ∎

###### Proof of Theorem 2..

The theorem follows easily by applying Proposition 3.2 iteratively to the
representation (3.1) ∎

###  3.2. Bounds on particle clusters.

We apply our $k$-point function estimates to prove the clustering estimates of
Theorem 3.

###### Proof of Theorem 3..

We will need slightly different arguments based on $d=2$ or $d\geq 3$. We
integrate the results of Theorem 2. We have

$\displaystyle\mathbb{P}^{W,U}_{N,\beta}(\\{X(B_{r}(z))\geq Q\\})$
$\displaystyle\leq\frac{1}{Q!}\sum_{i_{1},\ldots,i_{Q}\
\text{distinct}}\mathbb{P}^{W,U}_{N,\beta}\left(\bigcap_{\ell=1}^{Q}\\{x_{i_{\ell}}\in
B_{r}(z)\\}\right)$
$\displaystyle=\frac{1}{Q!}\int_{(B_{r}(z))^{Q}}\rho_{Q}(y_{1},\ldots,y_{Q})dy_{1}\cdots
dy_{Q}.$

Thus, for a $Q$ dependent constant $C$, we have

$\mathbb{P}^{W,U}_{N,\beta}(\\{X(B_{r}(z))\geq Q\\})\leq
Cr^{Qd}\|\rho_{Q}\|_{L^{\infty}((B_{r}(z))^{Q})}.$

In $d=2$ one estimates

$\|\rho_{Q}\|_{L^{\infty}((B_{r}(z))^{Q})}\leq Cr^{\beta{\binom{Q}{2}}}$

which establishes (1.18). In $d\geq 3$, we can estimate

$\|\rho_{Q}\|_{L^{\infty}((B_{r}(z))^{Q})}\leq
Ce^{-\beta{\binom{Q}{2}}\inf_{x,y\in B_{r}(z)}\mathsf{g}(x-y)}\leq
Ce^{-\frac{\beta}{2^{d-2}r^{d-2}}{\binom{Q}{2}}},$

establishing (1.19). To prove the estimates with $z=x_{1}$, we consider
$(x_{2},\ldots,x_{N})$ drawn from the Coulomb gas
$\mathbb{P}^{W,U_{x_{1}}}_{N-1,\beta}$ where
$U_{x_{1}}(x_{2},\ldots,x_{N})=U(X_{N})+\sum_{i=2}^{N}\mathsf{g}(x_{i}-x_{1})$.
The argument of Theorem 2 applies to this gas, except we have an extra term
when applying our isotropic averaging operator to $U_{x_{1}}$ coming from
particle repulsion generated by $x_{1}$. It is a straightforward modification
to include this term in Lemma 3.1 and the proof of Theorem 2. We can prove

$\rho_{Q-1}(y_{2},\ldots,y_{Q}|y_{1})\leq\begin{cases}C\prod_{1\leq i<j\leq
Q}\min(1,|y_{i}-y_{j}|^{\beta})\quad&\text{if}\ d=2,\\\
Ce^{-\beta\mathcal{H}^{0}(y_{1},\ldots,y_{Q})}\quad&\text{if}\ d\geq
3,\end{cases}$ (3.7)

where $\rho_{Q-1}(y_{2},\ldots,y_{Q}|y_{1})$ is the $(Q-1)$-point function of
$\mathbb{P}^{W,U_{x_{1}}}_{N-1,\beta}$ with $x_{1}=y_{1}$. Then since

$\mathbb{P}^{W,U}_{N,\beta}(\\{X(B_{r}(x_{1}))\geq Q\\})\leq
Cr^{(Q-1)d}\sup_{y_{1}\in\mathbb{R}^{d}}\|\rho_{Q-1}(\cdot|y_{1})\|_{L^{\infty}((B_{r}(y_{1}))^{Q-1})},$

we obtain

$\mathbb{P}^{W,U}_{N,\beta}(\\{X(B_{r}(x_{1}))\geq Q\\})\leq
Cr^{(Q-1)d+\beta{\binom{Q}{2}}}\quad\text{if}\ d=2,$

and

$\mathbb{P}^{W,U}_{N,\beta}(\\{X(B_{r}(x_{1}))\geq Q\\})\leq
Cr^{(Q-1)d}e^{-\beta\frac{Q-1}{r^{d-2}}-\beta\frac{1}{2^{d-2}r^{d-2}}{\binom{Q-1}{2}}}\quad\text{if}\
d\geq 3.$ (3.8)

In the latter estimate, we used that $|y_{i}-y_{1}|\leq r$ in our
$L^{\infty}(B_{r}(y_{1})^{Q-1})$ bound on $\rho_{Q-1}(\cdot|y_{1})$.

We expect one can obtain significant improvements to the $d\geq 3$ bound by
more accurately estimating the minimum value of
$\mathcal{H}^{0}(y_{1},\ldots,y_{Q})$ and the relative volume within
$(B_{r}(z))^{Q}$ of near-minimizers. We will now do so for the case of $Q=2$
and $z=x_{1}$ since it is relevant to the minimal separation problem.

Let $d\geq 3$. We find

$\displaystyle\mathbb{P}^{W,U}_{N,\beta}(\\{X(B_{r}(x_{1}))\geq 2\\})\leq
C\sup_{y_{1}\in\mathbb{R}^{d}}\int_{B_{r}(y_{1})}e^{-\frac{\beta}{|y_{1}-y_{2}|^{d-2}}}dy_{2}$
$\displaystyle\leq Cr^{d}\int_{B_{1}(0)}e^{-\frac{\beta}{r^{d-2}|y|^{d-2}}}dy$
(3.9) $\displaystyle\leq
Cr^{d}\int_{0}^{1}s^{d-1}e^{-\frac{\beta}{r^{d-2}}\frac{1}{s^{d-2}}}ds.$

For any $\alpha>0$, we can compute

$\int_{0}^{1}s^{d-1}e^{-\frac{\alpha}{s^{d-2}}}ds\leq\int_{0}^{1}\frac{1}{s^{d-1}}e^{-\frac{\alpha}{s^{d-2}}}ds=\frac{e^{-\alpha}}{\alpha}.$

Applying this at $\alpha=r^{-d+2}\beta$ allows us to conclude

$\mathbb{P}^{W,U}_{N,\beta}(\\{X(B_{r}(x_{1}))\geq 2\\})\leq
Cr^{2d-2}e^{-\frac{\beta}{r^{d-2}}}.$

Notice that this is significantly stronger than (3.8) with $Q=2$. The “volume
factor” has been reduced from $r^{d}$ to $r^{2d-2}$. ∎

##  4\. Minimal Separation

In this section, we prove the minimal separation theorem Theorem 4 via
Proposition 4.1 and Proposition 4.4. The proof that $\eta_{k}$ is not smaller
than expected is a relatively straightforward application of the clustering
bounds of Theorem 3. The proof the $\eta_{k}$ is not too large takes a new
idea that we term “mimicry”, for which we now give some intuition.

Consider a configuration $X_{N}$ with $\eta_{1}>r$, let $R=O(1)$ be a large
microscopic scale, and let $r<R$. Consider two particles, say $x_{1}$ and
$x_{2}$, with $|x_{1}-x_{2}|\in(r,R)$. We can move the particles closer
together by “replacing” $x_{2}$ with a charged annulus of inner radius $r/2$
and outer radius $r$ centered $x_{1}$, and applying our model computation to
associate a family of new point configurations to $X_{N}$. A key difference
from isotropic averaging is that the annulus is centered at $x_{1}$ instead of
$x_{2}$.

By Newton’s theorem, the interaction of the annulus with the particles
$x_{3},\ldots,x_{N}$ is the same or more mild than that of $x_{1}$ and the
potential term is less than $W(x_{1})+Cr^{2}$. The interaction between $x_{1}$
and $x_{2}$ has increased, in the worst case, from $\mathsf{g}(R)$ to
$\mathsf{g}(r/2)$. The particle $x_{2}$ is “mimicking” $x_{1}$. We can also
have $x_{1}$ mimic $x_{2}$; one of the two mimicking operations is favorable
energy-wise up to $O(r^{2})+\mathsf{g}(r/2)-\mathsf{g}(R)$. There is also an
entropy cost associated to mimicry. The particle $x_{2}$ originally occupies a
$O(R^{d})$ volume region around $x_{1}$ and afterwards is restricted to
$O(r^{d})$ volume. We can apply this argument to any of the $O(N)$ particle
pairs within distance $R$, creating new configurations from $X_{N}$ with
either (1) all particles separated by $r$ except a single pair or (2) with a
cluster of three or more particles within a ball of radius $r$. Situation (2)
can be proved unlikely using our clustering estimates from Theorem 3. Assuming
situation (1), we can never create the same configuration by applying a
mimicry argument to two distinct index pairs, and so we achieve a volume
benefit factor of $O(N)$. We find that $\eta_{1}>r$ is unlikely as soon as

$\frac{Nr^{d}}{R^{d}}e^{-\beta(\mathsf{g}(r/2)-\mathsf{g}(R))}\gg 1.$

In $d=2$, this happens as soon as $r\ll N^{1/(2+\beta)}$, matching our desired
result Theorem 4. In $d\geq 3$, we must be more careful and consider thinner
annuli for mimicry, but a similar intuition holds. This argument is carried
out in Proposition 4.4, but before then we must take some care to provide an
appropriate parameter $R$, which is done in Lemma 4.2 using results from
[CHM18]. The main idea is that most of the particles are contained within some
volume of size $O(N)$ with high probability.

######  Proposition 4.1.

There is an absolute constant $C_{0}>0$ such that in $d=2$ we have

$\mathbb{P}^{W,U}_{N,\beta}(\\{\eta_{k}\leq\gamma
N^{-\frac{1}{2+\beta}}\\})\leq
C\gamma^{(2+\beta)k}+CN^{-\frac{2+2\beta}{2+\beta}}\gamma^{4+3\beta}+CN^{-7k}\gamma^{C_{0}(1+\beta)k}\quad\forall\gamma>0,$
(4.1)

and in $d=3$ we have

$\mathbb{P}^{W,U}_{N,\beta}\left(\left\\{\eta_{k}\leq\left(\frac{\beta}{\log
N-\frac{2d-2}{d-2}\log\log
N+\gamma}\right)^{\frac{1}{d-2}}\right\\}\right)\leq\frac{C}{\sqrt{N}}+Ce^{-k\gamma}\quad\forall\gamma>0.$
(4.2)

The constant $C$ depends on $\beta$, $\sup\Delta W$, and $k$.

###### Proof.

Let $r\in(0,1)$. Let $S$ be the event that there exists $i\in\\{1,\ldots,N\\}$
such that $B_{r}(x_{i})$ contains $3$ or more particles $x_{j}$. We have

$\mathbb{P}^{W,U}_{N,\beta}(S)\leq
CN\mathbb{P}^{W,U}_{N,\beta}(\\{X(B_{r}(x_{1}))\geq
3\\})\leq\begin{cases}CNr^{4+3\beta}\quad&\text{if}\ d=2,\\\
CNr^{2d}e^{-\frac{2\beta}{r^{d-2}}}\quad&\text{if}\ d\geq 3,\end{cases}$ (4.3)

by a union bound and Theorem 3.

Let $E_{k,r}$ be the event that $\eta_{k}\leq r$, and let $a_{j}<b_{j}$ be
random indices such that $\eta_{j}=|x_{a_{j}}-x_{b_{j}}|$ for $j=1,\ldots,k$,
which are well-defined almost surely. On the event $S^{c}\cap E_{k,r}$, we
have that $\\{a_{j},b_{j}\\}\cap\\{a_{\ell},b_{\ell}\\}=\varnothing$ for
$\ell\neq j$ almost surely. Let
$F_{j}=\bigcap_{\ell=1}^{j}\\{a_{\ell}=2\ell-1\\}\cap\\{b_{\ell}=2\ell\\}$. By
exchangeability, we have

$\mathbb{P}^{W,U}_{N,\beta}(E_{k,r}\cap
S^{c})\leq{\binom{N}{2}}^{k}\mathbb{P}^{W,U}_{N,\beta}(E_{k,r}\cap
F_{k})\leq{\binom{N}{2}}^{k}\mathbb{P}^{W,U}_{N,\beta}\left(\bigcap_{\ell=1}^{k}\\{|x_{2\ell}-x_{2\ell-1}|\leq
r\\}\right).$

Let $Y=X-\sum_{i=1}^{2k-2}\delta_{x_{i}}$, i.e. the point process with the
first $2k-2$ points deleted. We estimate

$\displaystyle\mathbb{P}^{W,U}_{N,\beta}\left(\bigcap_{\ell=1}^{k}\\{|x_{2\ell}-x_{2\ell-1}|\leq
r\\}\right)$ (4.4)
$\displaystyle=\mathbb{P}^{W,U}_{N,\beta}\left(\bigcap_{\ell=1}^{k-1}\\{|x_{2\ell}-x_{2\ell-1}|\leq
r\\}\right)\mathbb{P}^{W,U}_{N,\beta}\left(\\{|x_{2k}-x_{2k-1}|\leq r\\}\
\bigg{|}\ \bigcap_{\ell=1}^{k-1}\\{|x_{2\ell}-x_{2\ell-1}|\leq r\\}\right),$

and furthermore

$\mathbb{P}^{W,U}_{N,\beta}\left(\\{|x_{2k}-x_{2k-1}|\leq r\\}\ \bigg{|}\
\bigcap_{\ell=1}^{k-1}\\{|x_{2\ell}-x_{2\ell-1}|\leq
r\\}\right)=\mathbb{E}\left[\mathbb{P}^{W,U_{k-1}}_{N-2k+2,\beta}(\\{|x_{2k}-x_{2k-1}|\leq
r\\})\right].$ (4.5)

for the random potential
$U_{k-1}(x_{2k-1},x_{2k},\ldots,x_{N})=U(X_{N})+\sum_{j=2k-1}^{N}\sum_{\ell=1}^{2k-2}\mathsf{g}(x_{\ell}-x_{j}).$
The expectation $\mathbb{E}$ is over the law of the points $x_{\ell}$,
$\ell\in\\{1,\ldots,2k-2\\}$, conditioned to be pairwise close as above, and
$\mathbb{P}^{W,U_{k-1}}_{N-2k+2,\beta}$ is a measure on the particles
$(x_{2k-1},x_{2k},\ldots,x_{N})$.

For any positive integer $n$, we have by exchangeability that

$\displaystyle\mathbb{P}^{W,U_{k-1}}_{N-2k+2,\beta}(\\{|x_{2k}-x_{2k-1}|\leq
r\\})$
$\displaystyle\leq\frac{C_{n}}{N}\mathbb{P}^{W,U_{k-1}}_{N-2k+2,\beta}(\\{Y(B_{r}(x_{2k}))\geq
2\\})+\mathbb{P}^{W,U_{k-1}}_{N-2k+2,\beta}(\\{Y(B_{r}(x_{2k}))\geq n\\}).$

We apply Theorem 3 to bound each piece above. Collecting the estimates, (4.5),
and (4.4), we have proved

$\displaystyle\mathbb{P}^{W,U}_{N,\beta}\left(\bigcap_{\ell=1}^{k}\\{|x_{2\ell}-x_{2\ell-1}|\leq
r\\}\right)$
$\displaystyle\leq\begin{cases}C\left(\frac{1}{N}r^{2+\beta}+r^{2(n-1)+\beta{\binom{n}{2}}}\right)\mathbb{P}^{W,U}_{N,\beta}\left(\bigcap_{\ell=1}^{k-1}\\{|x_{2\ell}-x_{2\ell-1}|\leq
r\\}\right)\quad\text{if }d=2,\\\
C\left(\frac{1}{N}r^{2d-2}e^{-\frac{\beta}{r^{d-2}}}+r^{(n-1)d}e^{-\frac{c\beta
n^{2}}{r^{d-2}}}\right)\mathbb{P}^{W,U}_{N,\beta}\left(\bigcap_{\ell=1}^{k-1}\\{|x_{2\ell}-x_{2\ell-1}|\leq
r\\}\right)\quad\text{if }d\geq 3.\end{cases}$

for a dimensional constant $c>0$. We can iterate this estimate to see

$\mathbb{P}^{W,U}_{N,\beta}\left(\bigcap_{\ell=1}^{k}\\{|x_{2\ell}-x_{2\ell-1}|\leq
r\\}\right)\leq\begin{cases}C\left(\frac{1}{N}r^{2+\beta}+r^{2(n-1)+\beta{\binom{n}{2}}}\right)^{k}\quad&\text{if}\
d=2,\\\
C\left(\frac{1}{N}r^{2d-2}e^{-\frac{\beta}{r^{d-2}}}+r^{(n-1)d}e^{-\frac{c\beta
n^{2}}{r^{d-2}}}\right)^{k}\quad&\text{if}\ d\geq 3.\end{cases}$ (4.6)

We conclude the general argument by writing

$\mathbb{P}^{W,U}_{N,\beta}(E_{k,r})\leq\mathbb{P}^{W,U}_{N,\beta}(S)+\mathbb{P}^{W,U}_{N,\beta}(E_{k,r}\cap
S^{c})\leq\mathbb{P}^{W,U}_{N,\beta}(S)+\binom{N}{2}^{k}\mathbb{P}^{W,U}_{N,\beta}\left(\bigcap_{\ell=1}^{k}\\{|x_{2\ell}-x_{2\ell-1}|\leq
r\\}\right).$ (4.7)

The probability of $S$ is bounded in (4.3).

Next, we choose specific $r$ and $n$ to conclude the proposition. We must
consider $d=2$ and $d\geq 3$ separately. For $d=2$, we choose $r=\gamma
N^{-\frac{1}{2+\beta}}$ and $n=10$ to see

$\displaystyle\mathbb{P}^{W,U}_{N,\beta}(S)\leq
CN^{-\frac{2+2\beta}{2+\beta}}\gamma^{4+3\beta},$
$\displaystyle\mathbb{P}^{W,U}_{N,\beta}\left(\bigcap_{\ell=1}^{k}\\{|x_{2\ell}-x_{2\ell-1}|\leq
r\\}\right)\leq CN^{-2k}\gamma^{(2+\beta)k}+CN^{-9k}\gamma^{C_{0}(1+\beta)k}$

for some $C_{0}>0$, and (4.1) follows.

For $d\geq 3$, we will assume $r^{d-2}\leq\frac{4\beta}{3\log N}$, which with
our choice below will happen if $N\geq C$, so that the first summand in the
RHS of (4.6) dominates for $n$ large enough. It follows that

$\mathbb{P}^{W,U}_{N,\beta}\left(\bigcap_{\ell=1}^{k}\\{|x_{2\ell}-x_{2\ell-1}|\leq
r\\}\right)\leq C\frac{r^{(2d-2)k}}{N^{k}}e^{-\frac{\beta k}{r^{d-2}}},$

if $n$ is chosen large enough, and

$\mathbb{P}^{W,U}_{N,\beta}(S)\leq CN\exp\left(-2\beta\left(\frac{3\log
N}{4\beta}\right)\right)\leq CN^{-\frac{1}{2}}.$

We can then let $r=\left(\frac{\beta}{\log N-\frac{2d-2}{d-2}\log\log
N+\gamma}\right)^{\frac{1}{d-2}}$, and after a short computation using (4.7),
we find (4.2) holds. ∎

Our next goal is to prove that $\eta_{k}$, properly normalized, is tight as
$N\to\infty$. To do this, we must create close particle pairs using the model
computation, except with a new class of operators, and precisely estimate
energy and entropy costs.

For $i\in\\{1,\ldots,N\\}$, let $L_{i,R}$ be the event that $\min_{j\neq
i}|x_{j}-x_{i}|\geq R$, i.e. that the point $x_{i}$ is “lonely”. Clearly, if
$R$ is much larger than the interstitial distance, the event $L_{i,R}$ is
rare. We use [CHM18] to quantify this fact. The below result is their Theorem
1.5, written in our blown-up coordinates and corresponding to their inverse
temperature chosen as $N^{-1+2/d}\beta$ in terms of our $\beta$.

######  Theorem 7 ([CHM18], Theorem 1.5).

Let $W=V_{N}$ with $V$ satisfying (A1) in $d=2$ and (A2) in $d\geq 3$. Recall
the blown-up equilibrium measure $\mu_{\mathrm{eq}}^{N}$ from Section 1.7. We
have

$\mathbb{P}^{V_{N}}_{N,\beta}(\\{d_{\mathrm{BL},N}(X,\mu_{\mathrm{eq}}^{N})\geq
N\sqrt{\log N}r\\})\leq e^{-cN\log Nr^{2}}$ (4.8)

for any $r\geq c^{-1}$ for some $\beta$-dependent $c>0$, where, for
nonnegative measures $\nu_{1}$ and $\nu_{2}$ of mass $N$, we define

$d_{\mathrm{BL},N}(\nu_{1},\nu_{2}):=\sup_{\begin{subarray}{c}f\in
C_{N}^{0,1}(\mathbb{R}^{d})\\\ \|f\|_{C_{N}^{0,1}(\mathbb{R}^{d})}\leq
1\end{subarray}}\int_{\mathbb{R}^{d}}f(x)(\nu_{1}-\nu_{2})(dx),$

and $C_{N}^{0,1}(\mathbb{R}^{d})$ is the space of bounded Lipschitz functions
with norm

$\|f\|_{C^{0,1}(\mathbb{R}^{d})}=\sup_{x\in\mathbb{R}^{d}}N^{-1/d}|f(x)|+\sup_{x\neq
y\in\mathbb{R}^{d}}\frac{|f(x)-f(y)|}{|x-y|}.$

######  Lemma 4.2.

Let $W=V_{N}$ with $V$ satisfying (A1) in $d=2$ and (A2) in $d\geq 3$. We have

$\mathbb{P}^{V_{N}}_{N,\beta}(L_{1,R})\leq CR^{-d}+CN^{-\frac{1}{d}}\sqrt{\log
N}$

for all $R\geq 1$.

###### Proof.

We will mostly omit $V_{N},N,\beta$ from the notation.

Defining $\varphi(x)=\max(0,1-N^{-1/d}\mathrm{dist}(x,\mathrm{supp}\
\mu_{\mathrm{eq}}^{N}))$, one can easily check

$\|\varphi(x)\|_{C^{0,1}_{N}(\mathbb{R}^{d})}\leq CN^{-1/d}.$

Therefore

$\left|\int_{\mathbb{R}^{d}}\varphi(x)(X-\mu_{\mathrm{eq}}^{N})(dx)\right|\leq\min\left(CN^{-\frac{1}{d}}d_{\mathrm{BL},N}(X,\mu_{\mathrm{eq}}^{N}),N\right),$

and so for $r=c^{-1}$ large enough we have by Theorem 4.8

$\displaystyle\mathbb{E}\left[\left|\int_{\mathbb{R}^{d}}\varphi(x)(X-\mu_{\mathrm{eq}}^{N})(dx)\right|\right]$
$\displaystyle\leq
N\mathbb{P}(\\{d_{\mathrm{BL},N}(X,\mu_{\mathrm{eq}}^{N})\geq N\sqrt{\log
N}r\\})+CN^{1-\frac{1}{d}}\sqrt{\log N}r$ (4.9) $\displaystyle\leq Ne^{-cN\log
Nr^{2}}+CN^{1-\frac{1}{d}}\sqrt{\log N}r\leq CN^{1-\frac{1}{d}}\sqrt{\log N}.$

We will use (4.9) to show that most points are typically within
$\mathrm{supp}\ \varphi$. Indeed, we have

$\mathbb{E}\left[\int_{\mathbb{R}^{d}}\varphi(x)(X-\mu_{\mathrm{eq}}^{N})(dx)\right]\leq\mathbb{E}\left[-X((\mathrm{supp}\
\varphi)^{c})\right]=-\sum_{i=1}^{N}\mathbb{P}(\\{x_{i}\not\in\mathrm{supp}\
\varphi\\}),$

and so $\mathbb{P}(\\{x_{1}\not\in\mathrm{supp}\ \varphi\\})\leq
CN^{-\frac{1}{d}}\sqrt{\log N}$.

Let $\xi_{i}=\min(R,\min_{j\neq i}|x_{i}-x_{j}|)$, so the event $L_{i,R}$ is
equivalent to $\xi_{r}=R$. Since the balls $B_{\xi_{i}/2}(x_{i})$ are
disjoint, we must have

$|\\{i:x_{i}\in\mathrm{supp}\ \varphi,\xi_{i}=R\\}|\cdot R^{d}\leq
C|\\{x:\mathrm{dist}(x,\mathrm{supp}\ \varphi)\leq R\\}|\leq CN,$

where we assumed WLOG $R\leq N^{\frac{1}{d}}$ and used $\mathrm{supp}\
\mu_{\mathrm{eq}}$ compact in the last inequality. Thus

$\mathbb{P}(L_{1,R}\cap\\{x_{1}\in\mathrm{supp}\ \varphi\\})\leq CR^{-d},$

and the lemma follows. ∎

For a configuration $X_{N}$, let $\phi(i)=\phi_{X_{N}}(i)\in\\{1,\ldots,N\\}$
be the index of the closest particle to $i$, i.e.
$|x_{i}-x_{\phi(i)}|=\min_{i\neq j}|x_{i}-x_{j}|$. This is almost surely well-
defined. Also, let $T_{k,r}$ be the event that the cardinality of
$\\{\\{i,j\\}\ :\ |x_{i}-x_{j}|<r,i\neq j\\}$ is at most $k$. Define the
random indices $a_{\ell}<b_{\ell}$ such that
$\eta_{\ell}=|x_{a_{\ell}}-x_{b_{\ell}}|$. The next proposition uses a mimicry
operator to move $x_{i}$ and $x_{\phi(i)}$ closer together.

######  Proposition 4.3.

Suppose $U(X_{N})=\sum_{i=1}^{N}U_{1}(x_{i})$ for a superharmonic function
$U_{1}:\mathbb{R}^{d}\to\mathbb{R}$. Let $r\in(0,1)$ and let $\nu$ be a
rotationally symmetric probability measure supported in $\overline{B_{r}(0)}$
with a Lebesgue density, also denoted $\nu$. Abbreviate
$\mathbb{P}=\mathbb{P}^{W,U}_{N,\beta}$. For any $R\geq 1$, integer $n\geq 3$,
and $k\in\\{1,\ldots,N\\}$, we have that $\mathbb{P}(\\{\eta_{k}\geq r\\})$ is
bounded by

$\displaystyle{Ce^{C\beta
r^{2}+\beta\Delta_{\nu}}}\|\nu\|_{L^{\infty}(\mathbb{R}^{d})}M_{r,R}\left(\frac{k+n}{N}+\mathbb{P}(\\{X(B_{r}(x_{1}))\geq
3\\})+N\mathbb{P}(\\{X(B_{r}(x_{1}))\geq n\\})\right)$
$\displaystyle\quad+\frac{2k-2}{N}+\mathbb{P}(L_{1,R}),$

where $\Delta_{\nu}=\int_{\mathbb{R}^{d}}\mathsf{g}(y)\nu(dy)$ and
$M_{r,R}=\int_{\mathrm{Ann}_{[r,R]}(0)}e^{-\beta\mathsf{g}(y)}dy$. The
constant $C$ depends on $\sup\Delta W$ and $n$.

###### Proof.

We will abbreviate $\mathbb{P}=\mathbb{P}^{W,U}_{N,\beta}$ and
$\mathbb{E}=\mathbb{E}^{W,U}_{N,\beta}$ throughout the proof.

First note that $T_{k-1,r}=\\{\eta_{k}\geq r\\}$. Furthermore, on $T_{k-1,r}$
we have $|x_{1}-x_{\phi(1)}|\geq r$ unless $a_{\ell}=1$ or $b_{\ell}=1$ for
some $\ell\in\\{1,\ldots,k-1\\}$. We will furthermore want to fix the label
$\phi(1)$ and ensure $x_{1}$ is not $R$-lonely, which inspires the bound

$\displaystyle\mathbb{P}(T_{k-1,r})$
$\displaystyle\leq(N-1)\mathbb{P}(\\{|x_{1}-x_{2}|\geq
r\\}\cap\\{\phi(1)=2\\}\cap L_{1,R}^{c}\cap T_{k-1,r})+\mathbb{P}(L_{1,R})$
(4.10) $\displaystyle\quad+\mathbb{P}(\\{\exists\ell\in\\{1,\ldots,k-1\\}\
a_{\ell}=1\ \text{or}\ b_{\ell}=1\\}).$

The $N-1$ factor comes from the fact that $\phi(1)$ is equally likely to be
each of $\\{2,\ldots,N\\}$. The first probability on the RHS of (4.10) is
suitable to apply a transport procedure to move $x_{2}$ closer to $x_{1}$, but
first we bound the last term. By exchangeability, we have

$\mathbb{P}(\\{\exists\ell\in\\{1,\ldots,k-1\\}\ a_{\ell}=1\ \text{or}\
b_{\ell}=1\\})\leq\frac{2k-2}{N}$ (4.11)

since $\\{a_{\ell}:\ell=1,\ldots,k-1\\}\cup\\{a_{\ell}:\ell=1,\ldots,k-1\\}$
is random subset of $\\{1,\ldots,N\\}$ size at most $2k-2$.

We condition on $X_{N,3}=(x_{3},x_{4},\ldots,x_{N})$ to rewrite

$\displaystyle\mathbb{P}(\\{|x_{1}-x_{2}|\geq r\\}\cap\\{\phi(1)=2\\}\cap
L_{1,R}^{c}\cap T_{k-1,r})$ (4.12)
$\displaystyle\leq\mathbb{P}(T^{\prime}_{k-1,r}\cap\\{|x_{1}-x_{2}|\in[r,R]\\}\cap\\{\phi(1)=2\\})$
$\displaystyle=\mathbb{E}\left[\mathbf{1}_{T^{\prime}_{k-1,r}}\mathbb{P}(\\{|x_{1}-x_{2}|\in[r,R]\\}\cap\\{\phi(1)=2\\}\
|\ X_{N,3})\right],$

where $T^{\prime}_{k-1,r}$ is the event that the cardinality of
$\\{\\{i,j\\}\subset\\{3,\ldots,N\\}\ :\ |x_{i}-x_{j}|<r,i\neq j\\}$ is at
most $k-1$.

We next define a new type of isotropic averaging operator to apply to the
conditional probability above. For a rotationally symmetric probability
measure $\nu$ on $\mathbb{R}^{d}$, define the mimicry operators

$\displaystyle\mathrm{Mim}_{1,2,\nu}F(x_{1},x_{2})$
$\displaystyle=\int_{\mathbb{R}^{d}}F(x_{1},x_{1}+y)\nu(dy),$
$\displaystyle\mathrm{Mim}_{2,1,\nu}F(x_{1},x_{2})$
$\displaystyle=\int_{\mathbb{R}^{d}}F(x_{2}+y,x_{2})\nu(dy).$

Note that

$\mathrm{Mim}_{1,2,\nu}\sum_{j=3}^{N}\mathsf{g}(x_{2}-x_{j})\leq\sum_{j=3}^{N}\mathsf{g}(x_{1}-x_{j})$

and

$\mathrm{Mim}_{1,2,\nu}U_{1}(x_{2})\leq
U_{1}(x_{1}),\quad\mathrm{Mim}_{1,2,\nu}W(x_{2})\leq W(x_{1})+Cr^{2}$

since $U_{1}$ is superharmonic and $\mathrm{supp}\
\nu\subset\overline{B_{r}(0)}$. Define

$\Delta_{\nu}=\mathrm{Mim}_{1,2,\nu}\mathsf{g}(x_{1}-x_{2})=\mathrm{Mim}_{2,1,\nu}\mathsf{g}(x_{1}-x_{2})$

which depends only on $\nu$. Analogous results as above hold for
$\mathrm{Mim}_{2,1,\nu}$. It follows that

$e^{-\beta\mathrm{Mim}_{1,2,\nu}\mathcal{H}^{W,U}(X_{N})}+e^{-\beta\mathrm{Mim}_{2,1,\nu}\mathcal{H}^{W,U}(X_{N})}\geq
e^{-C\beta
r^{2}}e^{\beta(\mathsf{g}(x_{1}-x_{2})-\Delta_{\nu})}e^{-\beta\mathcal{H}^{W,U}(X_{N})}.$

Indeed, the first summand on the LHS dominates the RHS in the event that

$\sum_{j=3}^{N}\mathsf{g}(x_{1}-x_{j})+U_{1}(x_{1})+W(x_{1})\leq\sum_{j=3}^{N}\mathsf{g}(x_{2}-x_{j})+U_{1}(x_{2})+W(x_{2})$

since $x_{2}$ interacts with $x_{3},\ldots,x_{N}$ similar to how $x_{1}$ does
after applying the operator $\mathrm{Mim}_{1,2,\nu}$. When the reverse
inequality is true, the second summand on the LHS is dominating. The adjoints
are easily computed as

$\displaystyle\mathrm{Mim}_{1,2,\nu}^{\ast}F(x_{1},x_{2})$
$\displaystyle=\nu(x_{2}-x_{1})\int_{\mathbb{R}^{d}}F(x_{1},y)dy,$
$\displaystyle\mathrm{Mim}_{1,2,\nu}^{\ast}F(x_{1},x_{2})$
$\displaystyle=\nu(x_{1}-x_{2})\int_{\mathbb{R}^{d}}F(y,x_{2})dy.$

We will now apply the model computation to $\mathbb{P}(\cdot\ |\ X_{N,3})$.
For a normalizing factor $\mathcal{Z}(X_{N,3})$, we have a.s.

$\displaystyle\mathbb{P}(\\{|x_{1}-x_{2}|\in[r,R]\\}\cap\\{\phi(1)=2\\}\ |\
X_{N,3})$
$\displaystyle=\frac{1}{\mathcal{Z}(X_{N,3})}\iint_{\mathbb{R}^{d}\times\mathbb{R}^{d}}\mathbf{1}_{\\{|x_{1}-x_{2}|\in[r,R]\\}\cap\\{\phi(1)=2\\}}(x_{1},x_{2})e^{-\beta\mathcal{H}^{W,U}(X_{N})}dx_{1}dx_{2}$
$\displaystyle\leq\frac{e^{C\beta
r^{2}+\beta\Delta_{\nu}}}{\mathcal{Z}(X_{N,3})}(A_{1}+A_{2}),$

where (with a slight abuse of notation)

$A_{1}=\iint_{\mathbb{R}^{d}\times\mathbb{R}^{d}}\mathrm{Mim}_{1,2,\nu}^{\ast}\left(e^{-\beta\mathsf{g}(x_{1}-x_{2})}\mathbf{1}_{\\{|x_{1}-x_{2}|\in[r,R]\\}\cap\\{\phi(1)=2\\}}\right)(x_{1},x_{2})e^{-\beta\mathcal{H}^{W,U}(X_{N})}dx_{1}dx_{2}$

and $A_{2}$ identical but with $\mathrm{Mim}_{2,1,\nu}^{\ast}$ in the place of
$\mathrm{Mim}_{1,2,\nu}^{\ast}$. Note that $\mathrm{Mim}_{1,2,\nu}^{\ast}$ is
monotonic and

$\mathbf{1}_{\\{|x_{1}-x_{2}|\in[r,R]\\}\cap\\{\phi(1)=2\\}}\leq\mathbf{1}_{\\{|x_{1}-x_{2}|\in[r,R]\\}},$

whence

$\mathrm{Mim}_{1,2,\nu}^{\ast}\left(e^{-\beta\mathsf{g}(x_{1}-x_{2})}\mathbf{1}_{\\{|x_{1}-x_{2}|\in[r,R]\\}\cap\\{\phi(1)=2\\}}\right)(x_{1},x_{2})\leq\nu(x_{2}-x_{1})\int_{\mathrm{Ann}_{[r,R]}(x_{1})}e^{-\beta\mathsf{g}(x_{1}-y)}dy.$

Define $M_{r,R}=\int_{\mathrm{Ann}_{[r,R]}(0)}e^{-\beta\mathsf{g}(y)}dy$.
Since $\nu$ is supported in $\overline{B_{r}(0)}$, the RHS above is a.s.
bounded by

$\|\nu\|_{L^{\infty}(\mathbb{R}^{d})}M_{r,R}\mathbf{1}_{B_{r}(0)}(x_{1}-x_{2}),$

and we find

$\frac{A_{1}}{\mathcal{Z}(X_{N,3})}\leq
M_{r,R}\|\nu\|_{L^{\infty}(\mathbb{R}^{d})}\mathbb{P}(\\{|x_{1}-x_{2}|<r\\}\
|\ X_{N,3}).$

We can prove an identical bound for $A_{2}$ with the same argument, and using
the bounds in (4.12) shows

$\mathbb{P}(\\{|x_{1}-x_{2}|\geq r\\}\cap\\{\phi(1)=2\\}\cap L_{1,R}^{c}\cap
T_{k-1,r})\leq\kappa\mathbb{P}(\\{|x_{1}-x_{2}|<r\\}\cap{T^{\prime}_{k-1,r}}).$
(4.13)

where

$\kappa:=2e^{C\beta
r^{2}+\beta\Delta_{\nu}}M_{r,R}\|\nu\|_{L^{\infty}(\mathbb{R}^{d})}.$

We wish to partially recover the information $\phi(1)=2$ after the transport,
i.e. on the RHS of (4.13). To do so, we use the bound

$\mathbb{P}(\\{|x_{1}-x_{2}|<r\\}\cap\\{\phi(1)\neq
2\\})\leq\frac{C_{n}}{N}\mathbb{P}(\\{X(B_{r}(x_{1}))\geq
3\\})+\mathbb{P}(\\{X(B_{r}(x_{1}))\geq n\\}),$

finding (starting from (4.13))

$\displaystyle\mathbb{P}(\\{|x_{1}-x_{2}|\geq r\\}\cap\\{\phi(1)=2\\}\cap
L_{1,R}^{c}\cap T_{k-1,r})$ (4.14)
$\displaystyle\leq\kappa\bigg{(}\mathbb{P}(\\{|x_{1}-x_{2}|<r\\}\cap\\{\phi(1)=2\\}\cap{T^{\prime}_{k-1,r}})$
$\displaystyle\quad+\frac{C_{n}}{N}\mathbb{P}(\\{X(B_{r}(x_{1}))\geq
3\\})+\mathbb{P}(\\{X(B_{r}(x_{1}))\geq n\\})\bigg{)}.$

Let $T^{\prime}_{k-1,r}(j)$ be the event that
$\\{x_{i}:i\in\\{2,\ldots,N\\},i\neq j\\}$ contains at most $k-1$ pairs of
points within distance $r$. For example,
$T^{\prime}_{k-1,r}(2)=T^{\prime}_{k-1,r}$. Since the events
$\\{\phi(i)=j\\}$, $j=2,\ldots,N$ are disjoint up to probability $0$, we see

$\displaystyle\mathbb{P}(\\{|x_{1}-x_{2}|<r\\}\cap\\{\phi(1)=2\\}\cap{T^{\prime}_{k-1,r}})$
(4.15)
$\displaystyle=\frac{1}{N-1}\sum_{j=2}^{N}\mathbb{P}(\\{|x_{1}-x_{j}|<r\\}\cap\\{\phi(1)=j\\}\cap{T^{\prime}_{k-1,r}(j)})$
$\displaystyle\leq\frac{1}{N-1}\mathbb{P}(\\{|x_{1}-x_{\phi(1)}|<r\\}\cap
T^{\prime}_{k-1,r}(\phi(1))\\}).$

On the event $\\{|x_{1}-x_{\phi(1)}|<r\\}\cap T^{\prime}_{k-1,r}(\phi(1))\\}$,
the index $1$ is an exceptional index since there likely are very few
particles within distance $r$ of each other, so we expect the probability of
the event is of order $O(1/N)$. To be precise, if $X(B_{r}(x_{i}))\leq n$ for
all $i\in\\{1,\ldots,N\\}$, then $T^{\prime}_{k-1,j}(\phi(1))$ occurring
implies $T_{k-1+2n,r}$. Thus

$\displaystyle\mathbb{P}({\\{|x_{1}-x_{\phi(1)}|<r\\}}\cap{T^{\prime}_{k-1,j}(\phi(1))})$
(4.16) $\displaystyle\leq\mathbb{P}(\\{\exists i\ X(B_{r}(x_{i}))\geq
n\\})+\mathbb{P}({\\{|x_{1}-x_{\phi(1)}|<r\\}}\cap T_{k-1+2n,r})$
$\displaystyle\leq N\mathbb{P}(\\{X(B_{r}(x_{1}))\geq
n\\})+\mathbb{P}({\\{|x_{1}-x_{\phi(1)}|<r\\}}\cap T_{k-1+2n,r}).$

Since, by definition of $T_{k-1+2n,r}$, we have pointwise a.s.

$\sum_{i=1}^{N}\mathbf{1}_{\\{|x_{i}-x_{\phi(i)}|<r\\}}\mathbf{1}_{T_{k-1+2n,r}}\leq
2k-2+4n,$

we can apply exhchangeability to see

$\mathbb{P}({\\{|x_{1}-x_{\phi(1)}|<r\\}}\cap
T_{k-1+2n,r})\leq\frac{2k-2+4n}{N}.$

Collecting this estimate, (4.16), (4.15), and (4.14), we have

$\displaystyle\mathbb{P}(\\{|x_{1}-x_{2}|\geq r\\}\cap\\{\phi(1)=2\\}\cap
L_{1,R}^{c}\cap T_{k-1,r})$
$\displaystyle\leq\kappa\left(\frac{2k-2+4n}{N(N-1)}+\frac{C_{n}}{N}\mathbb{P}(\\{X(B_{r}(x_{1}))\geq
3\\})+2\mathbb{P}(\\{X(B_{r}(x_{1}))\geq n\\})\right).$

Finally, plugging this bound into (4.10) along with (4.11) proves the
proposition. ∎

######  Proposition 4.4.

Consider $W=V_{N}$ with $V$ satisfying (A1) in $d=2$ and (A2) in $d\geq 3$. In
$d=2$, the law of $N^{\frac{1}{2+\beta}}\eta_{k}$ is tight as $N\to\infty$ and
$\limsup_{N\to\infty}\mathbb{P}^{V_{N}}_{N,\beta}(\\{N^{\frac{1}{2+\beta}}\eta_{k}\geq\gamma\\})\leq
C\gamma^{-\frac{4+2\beta}{4+\beta}}$ for $\gamma>0$. For $d\geq 3$, let
$Z_{k}$ be defined by

$\eta_{k}=\left(\frac{\beta}{\log
N}\right)^{\frac{1}{d-2}}\left(1+\frac{2d-2}{(d-2)^{2}}\frac{\log\log N}{\log
N}+\frac{Z_{k}}{(d-2)\log N}\right).$

Then we have
$\limsup_{N\to\infty}\mathbb{P}^{V_{N}}_{N,\beta}(\\{Z_{k}\geq\gamma\\})\leq
Ce^{-\gamma/2}$.

###### Proof.

Both results are consequences of Proposition 4.3, Lemma 4.2, and our
clustering result Theorem 3. We adopt the notation from Proposition 4.3. For
the $d=2$ result, choose $r=\gamma N^{-\frac{1}{2+\beta}}$ for $\gamma\geq 1$,
$n=5$ (say), and let $\nu$ be the uniform probability measure on
$\mathrm{Ann}_{[r/2,r]}(0)$. Without loss of generality, we assume $r\leq 1$.
We compute

$\displaystyle M_{r,R}\leq CR^{2+\beta},$ $\displaystyle\Delta_{\nu}\leq-\log
r+C,$ $\displaystyle\|\nu\|_{L^{\infty}(\mathbb{R}^{2})}\leq Cr^{-2}.$

Applying Proposition 4.3, Lemma 4.2, and Theorem 3, we see

$\mathbb{P}^{V_{N}}_{N,\beta}(\\{\eta_{k}\geq r\\})\leq
CN^{-\frac{1}{d}}\sqrt{\log N}+CR^{-2}+CR^{2+\beta}\gamma^{-2-\beta}e^{C\beta
N^{-\frac{2}{2+\beta}}\gamma^{2}}\left(1+N^{1-\frac{4+3\beta}{2+2\beta}}\gamma^{c_{\beta}}\right)$
(4.17)

for a constant $C$ depending on $k$ and some constant $c_{\beta}>0$. Taking
$\limsup_{N\to\infty}$ of both sides and optimizing in $R$ proves that

$\limsup_{N\to\infty}\mathbb{P}^{V_{N}}_{N,\beta}(\\{N^{\frac{1}{2+\beta}}\eta_{k}\geq\gamma\\})\leq
CR^{-d}+CR^{2+\beta}\gamma^{-2-\beta}\leq
C\gamma^{-\frac{4+2\beta}{4+\beta}}.$

For the $d\geq 3$ result, choose $r=\left(\frac{\beta}{\log
N-\frac{2d-2}{d-2}\log\log N-\gamma}\right)^{\frac{1}{d-2}}$ for $\gamma>0$
and WLOG assume $r\leq 1$. Let $\nu$ be the uniform measure on the annulus
$\mathrm{Ann}_{[(1-(\log N)^{-1})r,r]}(0)$, and compute

$\displaystyle M_{r,R}\leq CR^{d},$
$\displaystyle\Delta_{\nu}\leq\frac{1}{(1-(\log
N)^{-1})^{d-2}r^{d-2}}\leq\frac{1}{r^{d-2}}\left(1+\frac{d-2}{\log N}+C(\log
N)^{-2}\right),$
$\displaystyle\|\nu\|_{L^{\infty}(\mathbb{R}^{d})}\leq\frac{C\log N}{r^{d}}.$

We can then estimate

$\displaystyle\frac{e^{C\beta
r^{2}+\beta\Delta_{\nu}}}{N}M_{r,R}\|\nu\|_{L^{\infty}(\mathbb{R}^{d})}$
$\displaystyle\leq
CR^{d}\exp\left(\frac{\beta}{r^{d-2}}\left(1+\frac{d-2}{\log
N}\right)+\log\log N-d\log r-\log N\right)$ $\displaystyle\leq
CR^{d}\exp\left(-\gamma+d-2-\frac{2d-2}{d-2}\frac{\log\log N}{\log
N}\right)\leq CR^{d}e^{-\gamma}.$

Thus we have

$\displaystyle\mathbb{P}^{V_{N}}_{N,\beta}(\\{\eta_{k}\geq r\\})$
$\displaystyle\leq CN^{-\frac{1}{d}}\sqrt{\log
N}+CR^{-d}+CR^{d}e^{-\gamma}(1+N\mathbb{P}^{V_{N}}_{N,\beta}(\\{X(B_{r}(x_{1}))\geq
3\\})$
$\displaystyle\quad+N^{2}\mathbb{P}^{V_{N}}_{N,\beta}(\\{X(B_{r}(x_{1}))\geq
n\\}))$ $\displaystyle\leq CN^{-\frac{1}{d}}\sqrt{\log
N}+CR^{-d}+CR^{d}e^{-\gamma}(1+Ne^{-\frac{2\beta}{r^{d-2}}}+N^{2}e^{-\frac{(n-1)\beta}{r^{d-2}}})$
$\displaystyle\leq CN^{-\frac{1}{d}}\sqrt{\log
N}+CR^{-d}+CR^{d}e^{-\gamma}(1+e^{C\gamma-\log N+C\log\log N}),$

where we chose $n=4$. We conclude

$\limsup_{N\to\infty}\mathbb{P}^{V_{N}}_{N,\beta}(\\{\eta_{k}\geq r\\})\leq
CR^{-d}+CR^{d}e^{-\gamma}\leq Ce^{-\gamma/2}.$

To conclude the result on $Z_{k}$, note that

$r=\left(\frac{\beta}{\log
N}\right)^{\frac{1}{d-2}}\left(1+\frac{2d-2}{(d-2)^{2}}\frac{\log\log N}{\log
N}+\frac{\gamma}{(d-2)\log N}\right)+O((\log
N)^{-\frac{1}{d-2}-2+\varepsilon})$

for any $\varepsilon>0$. ∎

##  5\. Discrepancy Bounds

In this section, we prove Theorem 5 and Theorem 6. All implicit constants $C$
may depend on $\beta$ and on various characteristics of $W$ or $V$. We let
$\mu(dx)=\frac{1}{c_{d}}\Delta W(x)dx$ throughout.

It will be necessary to consider more general domains $\Omega$ for our
discrepancy bounds. We will work with a domain $\Omega\subset\mathbb{R}^{d}$
that is an $\alpha$-thin annulus for fixed $\alpha\in(0,1]$. To be precise, we
will take either $\Omega$ to be a ball of radius $R$ if $\alpha=1$ or
$\Omega=\mathrm{Ann}_{[R-\alpha R,R]}(z)$ for $\alpha\in(0,1)$. This means
there exists some $C>0$ such that

$C^{-1}\alpha R^{d}\leq|\Omega|\leq C\alpha R^{d}\quad\text{where}\
2R=\mathrm{diam}(\Omega).$ (5.1)

We also define thickened and thinned versions of $\Omega$. For $s\geq 0$,
define
$\Omega_{s}=\Omega\cup\\{x\in\mathbb{R}^{d}:\mathrm{dist}(x,\partial\Omega)\leq
s\\}$, and for $s<0$, define
$\Omega_{s}=\\{x\in\Omega:\mathrm{dist}(x,\partial\Omega)\geq|s|\\}$. We will
use $2R$ for $\mathrm{diam}(\Omega)$ throughout the first two subsections. We
assume $\alpha R\geq C$ dependent on $\beta$, $\sup\Delta W$, and
$\inf_{\Omega_{R}}\Delta W$.

We will consider the event $E_{\rho,r,M}$ for parameters $\rho,r,M>0$. It is
defined by

$E_{\rho,r,M}=\\{X(\Omega)\geq\mu(\Omega)+\rho|\Omega|\\}\cap\\{X(\Omega_{5r}\setminus\Omega_{-5r})\leq
MR^{d-1}r\\}.$ (5.2)

Let $\phi:\mathbb{R}^{d}\to\mathbb{R}$ be a smooth, nonnegative, radial
function with $\int_{\mathbb{R}^{d}}\phi(x)dx=1$ and support within
$B_{1}(0)$. For $s>0$, define $\phi_{s}=s^{-d}\phi(x/s)$, and
$\psi_{s}=\phi_{s}\ast\phi_{1}$. For a measure (or function) $\lambda$, we
will write $\Phi_{s}\lambda$ for $\phi_{s}\ast\lambda$ when it makes sense. We
will apply the isotropic averaging argument with operator
$\mathrm{Iso}_{\mathbb{X}(\Omega),\psi_{r}}$ (identifying $\psi_{r}$ with the
measure $\psi_{r}dx$). The analysis mainly hinges on a precise lower bound for

$\Delta_{\rho,r,M}:=\inf_{X_{N}\in
E_{\rho,r,M}}\mathcal{H}^{W,U}(X_{N})-\mathrm{Iso}_{\mathbb{X}(\Omega),\psi_{r}}\mathcal{H}^{W,U}(X_{N}).$
(5.3)

We want to be able to take $\rho$ to be very small and still achieve
$\Delta_{\rho,r,M}\gg 1$ for appropriate choices of $r$ and $M$.

###  5.1. Proof idea.

The main idea of the proof is to use a continuum approximation for the energy
change upon isotropic averaging to find a lower bound for $\Delta_{\rho,r,M}$.
To a continuous charge density, we associate an isotropic averaging procedure
adapted to $\phi_{r}$ in which every “infinitesimal point charge” constituting
the continuum is replaced by a infinitesimal charge shaped like $\phi_{r}$.

Crucially, the Coulomb energy of a continuous charge distribution contains the
self-energy or “the diagonal”. That is, to a bounded charge distribution
$\nu$, we associate the Coulomb energy
$\frac{1}{2}\iint\mathsf{g}(x-y)\nu(dx)\nu(dy)$. After isotropic averaging,
i.e. replacing $\nu$ by $\Phi_{r}\nu$, the Coulomb energy is

$\frac{1}{2}\iint_{\mathbb{R}^{d}\times\mathbb{R}^{d}}\mathsf{g}(x-y)\Phi_{r}\nu(dx)\Phi_{r}\nu(dy)=\frac{1}{2}\iint_{\mathbb{R}^{d}\times\mathbb{R}^{d}}(\Phi_{r}^{2}\mathsf{g})(x-y)\nu(dx)\nu(dy).$

The continuum Coulomb energy change is therefore

$-\mathcal{E}_{r}(\nu)\quad\text{where}\quad\mathcal{E}_{r}(\nu):=\frac{1}{2}\iint_{\mathbb{R}^{d}\times\mathbb{R}^{d}}(\mathsf{g}-\Phi_{r}^{2}\mathsf{g})(x-y)\nu(dx)\nu(dy).$
(5.4)

Two facts form the crux of the method: (1) $\mathcal{E}_{r}(\nu)$ is convex
and (2) $\mathsf{g}-\Phi_{r}^{2}\mathsf{g}$ is compactly supported (on scale
$r$).

Considering the energy term $\int W(x)\nu(dx)$ associated to the potential
$W$, which one can think of as the Coulomb interaction between $\nu$ and a
signed background charge density of $-\mu$, we can precisely compare (5.4) to
the change in this term upon smearing $\nu$ by $\phi_{r}$. The energy change
is
$\iint_{\mathbb{R}^{d}\times\mathbb{R}^{d}}(\mathsf{g}-\Phi_{r}\mathsf{g})(x-y)\nu(dx)\mu(dy)$.
Note that the kernel is different from that of (5.4), but also the
$\frac{1}{2}$ factor is absent in comparison. In the case that
$\nu(\Omega)\geq\mu(\Omega)+\rho|\Omega|$ and $\nu$ is supported on $\Omega$,
the net energy change is favorable for the choice $r=(\alpha R)^{1/3}$ under
some conditions on $\rho>0$, provided we apply some appropriate modifications
near the boundary of $\Omega$ to overcome any boundary layer effects and
approximate $\mu$ by a constant measure near $\Omega$. The modifications
create only boundary errors since $\mathsf{g}-\Phi_{r}^{2}\mathsf{g}$ is
supported on a scale $r$ much smaller than $R$. Our argument is capturing the
fact that overcrowding is unfavorable for the interaction on length scale $r$
and below.

The remainder of the proof involves relating our continuum approximation above
to the true change in energy of a point configuration upon isotropic
averaging. The first step, and the reason for isotropic averaging using
$\psi_{r}$ instead of $\phi_{r}$, is that we must renormalize by replacing our
point charges by microscopic continuous charges shaped like $\phi_{1}$. This
allows us to make sense of the self energy of the charges (whereas the self-
energy of a point charge is infinite) and so directly relate to the continuum
problem. As alluded to above, we must also deal with boundary effects, which
are well controlled by the parameter $M$ in the event $E_{\rho,r,M}$ and our
local law Theorem 1. Finally, there are some lower order entropy factors to
consider, after which our model computation applies to conclude Theorem 5.

Regarding Theorem 6, we use rigidity for the fluctuation of smooth linear
statistics from [Ser20] to find a screening region whenever the absolute
discrepancy in a ball $B_{R}(z)$ is large. A screening region takes the form
of an $\alpha$-thin annulus $\Omega$ just inside or outside $\partial
B_{R}(z)$ that has an excess of positive charge. We can then apply our
incompressibility estimate to conclude.

###  5.2. Discrepancy upper bound.

In this subsection, we prove Theorem 5. We will actually generalize the result
to $\alpha$-thin annuli $\Omega$ under some conditions.

We will first control $\Delta_{\rho,r,M}$ from (5.3) in terms of an “energy”
functional $\mathcal{E}_{r}(\nu)$ defined in (5.4). We will only need to
consider measures $\nu$ with bounded densities, so for simplicity we restrict
to this case from the start.

We let $q\in\mathbb{R}$ denote a positive constant to be chosen later and
introduce $\mathrm{Leb}(dx)$ to denote Lebesgue measure on $\mathbb{R}^{d}$.
We let $\nu_{|A}$ denote the restriction of $\nu$ to a set $A$ for any measure
$\nu$. Define

$\mathcal{E}_{r}(\nu;q)=\mathcal{E}_{r}(\nu+q\mathrm{Leb}_{|\Omega_{2r}\setminus\Omega})-q\iint_{\mathbb{R}^{d}\times\mathbb{R}^{d}}(\mathsf{g}-\Phi_{r}\mathsf{g})(x-y)dx(\nu(dy)+q\mathrm{Leb}_{|\Omega_{r}\setminus\Omega}(dy)).$
(5.5)

The above definition models the full energy change for the continuum
approximation discussed in Section 5.1. Note that we extend the measure $\nu$
slightly past the boundary $\Omega$ by $qdx$ to eliminate boundary layer
effects. In Proposition 5.3, we will see that $\nu=q\mathrm{Leb}_{|\Omega}$ is
a minimizer of $\mathcal{E}_{r}(\cdot;q)$ among bounded measures supported on
$\Omega$ with mass $q|\Omega|$.

######  Lemma 5.1.

We have

$\displaystyle\mathrm{Iso}_{\\{1,2\\},\psi_{r}}\mathsf{g}(x_{1}-x_{2})$
$\displaystyle=(\phi_{1}\ast\phi_{1}\ast\phi_{r}\ast\phi_{r}\ast\mathsf{g})(x_{1}-x_{2}),$
$\displaystyle\mathrm{Iso}_{\\{1,2\\},\psi_{r}}\mathsf{g}(x_{1}-x_{3})$
$\displaystyle\leq\mathsf{g}(x_{1}-x_{3}).$

Furthermore,

$\displaystyle\mathrm{Iso}_{\\{1\\},\psi_{r}}W(x_{1})=W(x_{1})$
$\displaystyle+\int_{\mathbb{R}^{d}}(\mathsf{g}-\Phi_{1}\mathsf{g})(y-x_{1})\mu(dy)$
(5.6)
$\displaystyle+\iint_{\mathbb{R}^{d}\times\mathbb{R}^{d}}(\mathsf{g}-\Phi_{r}\mathsf{g})(y-x)\phi_{1}(x-x_{1})dx\mu(dy).$

###### Proof.

The first two equations follow immediately from the definition of
$\mathrm{Iso}_{\\{1,2\\},\psi_{r}}$ and superharmonicity of $\mathsf{g}$; see
the proof of Proposition 2.1 for a similar calculation.

Considering now isotropic averaging of the potential term and letting $\sigma$
be surface measure on the unit sphere, we have

$\frac{1}{\sigma(B_{1}(0))}\int_{\partial
B_{1}(0)}W(z+r\theta)\sigma(d\theta)=W(z)+\int_{B_{r}(z)}(\mathsf{g}(x-z)-\mathsf{g}(r))\mu(dx).$
(5.7)

Defining $\mathsf{g}_{s}=(\mathsf{g}(x)-\mathsf{g}(s))_{+}$, we compute using
radial symmetry of $\phi_{r}$:

$\displaystyle\phi_{r}\ast W(z)=\int_{\mathbb{R}^{d}}\phi_{r}(y)W(z+y)dy$
$\displaystyle=\int_{\mathbb{R}^{d}}\phi_{r}(y)\frac{1}{\sigma(B_{1}(0))}\int_{\partial
B_{1}(0)}W(z+|y|{\theta})d\theta dy$
$\displaystyle=W(z)+\left(\int_{\mathbb{R}^{d}}\phi_{r}(y)\mathsf{g}_{|y|}dy\right)\ast\mu(x).$

Note that $\mathsf{g}_{s}=\mathsf{g}-\delta^{(s)}\ast\mathsf{g}$, where
$\delta^{(s)}$ is Dirac delta “smeared” evenly on the sphere $\partial
B_{s}(0)$, and so

$\int_{\mathbb{R}^{d}}\phi_{r}(y)\mathsf{g}_{|y|}(x)dy=\mathsf{g}(x)-\int_{\mathbb{R}^{d}}\int_{\mathbb{R}^{d}}\mathsf{g}(x-z)\phi_{r}(y)\delta^{(|y|)}(dz)dy=\mathsf{g}(x)-\int_{\mathbb{R}^{d}}\mathsf{g}(x-z)\phi_{r}(z)dz.$

This is $(\mathsf{g}-\Phi_{r}\mathsf{g})(x)$. In the last equality, we used
that $\int(\delta^{(|y|)}(dz)\phi_{r}(y))dy=\phi_{r}(z)dz$, formally, since
$\phi$ is radial. By applying the above computation twice, we have

$\displaystyle\psi_{r}\ast W(x_{i})$ $\displaystyle=\phi_{r}\ast(\phi_{1}\ast
W)(x_{i})=(\phi_{1}\ast
W)(x_{i})+((\mathsf{g}-\Phi_{r}\mathsf{g})\ast\phi_{1}\ast\mu)(x_{i})$
$\displaystyle=W(x_{i})+(\mathsf{g}-\Phi_{1}\mathsf{g})\ast\mu(x_{i})+((\mathsf{g}-\Phi_{r}\mathsf{g})\ast\phi_{1}\ast\mu)(x_{i}).$

The last line with $i=1$ gives (5.6). ∎

The next lemma directly relates the energy change of a point charge system
after isotropic averaging to the functional $\mathcal{E}_{r}$. There are error
terms corresponding to self-energy terms ($\mathrm{Error}_{\mathrm{vol}}$),
boundary layer effects ($\mathrm{Error}_{\mathrm{bl}}$), and approximation of
$\mu$ by $q$.

######  Lemma 5.2.

Let $r>1$ and $q\in\mathbb{R}$. We have

$\displaystyle\mathcal{H}^{W,U}(X_{N})-\mathrm{Iso}_{\mathbb{X}(\Omega),\psi_{r}}\mathcal{H}^{W,U}(X_{N})$
(5.8)
$\displaystyle\geq\mathcal{E}_{r}((\Phi_{1}X_{|\Omega})_{|\Omega};q)+\iint_{\mathbb{R}^{d}\times\Omega}(\mathsf{g}-\Phi_{r}\mathsf{g})(y-x)(q\mathrm{Leb}-\mu)(dx)(\Phi_{1}X_{|\Omega})(dy)$
$\displaystyle\quad-\mathrm{Error}_{\mathrm{bl}}-\mathrm{Error}_{\mathrm{vol}},$

where

$\mathrm{Error}_{\mathrm{bl}}\leq
Cr^{2}|q|X(\Omega\setminus\Omega_{-3r})+Cr^{3}q^{2}R^{d-1}+Cr^{2}(1+|q|)X(\Omega\setminus\Omega_{-1}),$
(5.9)

and

$\mathrm{Error}_{\mathrm{vol}}\leq-X(\Omega)\mathsf{g}(4r)+CX(\Omega).$ (5.10)

###### Proof.

By Lemma 5.1, we have

$\mathrm{Iso}_{\mathbb{X}(\Omega),\psi_{r}}\mathcal{H}^{0}(X_{N})\leq\frac{1}{2}\sum_{\begin{subarray}{c}i\neq
j\\\
\\{i,j\\}\not\subset\mathbb{X}(\Omega)\end{subarray}}\mathsf{g}(x_{i}-x_{j})+\frac{1}{2}\sum_{i,j\in\mathbb{X}(\Omega)}(\Phi_{1}^{2}\Phi_{r}^{2}\mathsf{g})(x_{i}-x_{j})-\frac{1}{2}X(\Omega)\Phi_{1}^{2}\Phi_{r}^{2}\mathsf{g}(0).$

Note that we have added and subtracted the diagonal terms in the second sum on
the RHS. We use that $\Phi_{1}$ is self-dual to write

$\frac{1}{2}\sum_{i,j\in\mathbb{X}(\Omega)}(\Phi_{1}^{2}\Phi_{r}^{2}\mathsf{g})(x_{i}-x_{j})=\frac{1}{2}\iint_{\mathbb{R}^{d}\times\mathbb{R}^{d}}\Phi_{r}^{2}\mathsf{g}(x-y)(\Phi_{1}X_{|\Omega})(dx)(\Phi_{1}X_{|\Omega})(dy).$

Considering the interaction within $\mathcal{H}^{0}(X_{N})$, we estimate

$\frac{1}{2}\sum_{\begin{subarray}{c}i,j\in\mathbb{X}(\Omega)\\\ i\neq
j\end{subarray}}\mathsf{g}(x_{i}-x_{j})\geq\frac{1}{2}\iint_{\mathbb{R}^{d}\times\mathbb{R}^{d}}\mathsf{g}(x-y)(\Phi_{1}X_{|\Omega})(dx)(\Phi_{1}X_{|\Omega})(dy)-\frac{1}{2}X(\Omega)\Phi_{1}^{2}\mathsf{g}(0)$

using $\Phi_{1}^{2}\mathsf{g}\leq\mathsf{g}$ pointwise and duality like above.
It follows that

$\displaystyle\mathcal{H}^{0}(X_{N})-\mathrm{Iso}_{\mathbb{X}(\Omega),\psi_{r}}\mathcal{H}^{0}(X_{N})$
$\displaystyle\geq\frac{1}{2}\iint_{\mathbb{R}^{d}\times\mathbb{R}^{d}}(\mathsf{g}-\Phi_{r}^{2}\mathsf{g})(x-y)(\Phi_{1}X_{|\Omega})(dx)(\Phi_{1}X_{|\Omega})(dy)$
(5.11)
$\displaystyle\quad+\frac{1}{2}X(\Omega)(\Phi_{1}^{2}\Phi_{r}^{2}\mathsf{g}(0)-\Phi_{1}^{2}\mathsf{g}(0)).$

We can compare the double integral above to
$\mathcal{E}_{r}((\Phi_{1}X_{|\Omega})_{|\Omega}+q\mathrm{Leb}_{|\Omega_{2r}\setminus\Omega})$.
They will only differ by boundary layer terms. We start by simply restricting
the integral to $\Omega\times\Omega$ using
$\mathsf{g}-\Phi_{r}^{2}\mathsf{g}\geq 0$. After doing so, we find

$\frac{1}{2}\iint_{\mathbb{R}^{d}\times\mathbb{R}^{d}}(\mathsf{g}-\Phi_{r}^{2}\mathsf{g})(x-y)(\Phi_{1}X_{|\Omega})^{\otimes
2}(dx,dy)\geq\mathcal{E}_{r}((\Phi_{1}X_{|\Omega})_{|\Omega}+q\mathrm{Leb}_{|\Omega_{2r}\setminus\Omega})-T_{1}-T_{2}$

where

$\displaystyle T_{1}$
$\displaystyle=q\iint_{\Omega\times\mathbb{R}^{d}}(\mathsf{g}-\Phi_{r}^{2}\mathsf{g})(x-y)\Phi_{1}X_{|\Omega}(dx)\mathrm{Leb}_{|\Omega_{2r}\setminus\Omega}(dy)\leq
Cr^{2}|q|X(\Omega\setminus\Omega_{-2r-1}),$ $\displaystyle T_{2}$
$\displaystyle=q^{2}\iint_{\mathbb{R}^{d}\times\mathbb{R}^{d}}(\mathsf{g}-\Phi_{r}^{2}\mathsf{g})(x-y)\mathrm{Leb}_{|\Omega_{2r}\setminus\Omega}(dy)\mathrm{Leb}_{|\Omega_{2r}\setminus\Omega}(dy)\leq
Cr^{3}q^{2}R^{d-1}.$

We also bound

$\frac{1}{2}X(\Omega)(\Phi_{1}^{2}\Phi_{r}^{2}\mathsf{g}(0)-\Phi_{1}^{2}\mathsf{g}(0))\geq
X(\Omega)(\mathsf{g}(4r)-C).$

This term, as well as $T_{1}$ and $T_{2}$, are absorbed into
$\mathrm{Error}_{\mathrm{vol}}$ and $\mathrm{Error}_{\mathrm{bl}}$.

We now handle the potential terms. Like usual, we can handle the $U$ term
using superharmonicity. For the $W$ term, we use Lemma 5.1 to see

$\displaystyle\mathrm{Iso}_{\mathbb{X}(\Omega),\psi_{r}}\sum_{i\in\mathbb{X}(\Omega)}W(x_{i})=\sum_{i\in\mathbb{X}(\Omega)}W(x_{i})$
(5.12)
$\displaystyle\quad+\iint_{\mathbb{R}^{d}\times\mathbb{R}^{d}}(\mathsf{g}-\Phi_{1}\mathsf{g})(x-y)X_{|\Omega}(dx)\mu(dy)+\iint_{\mathbb{R}^{d}\times\mathbb{R}^{d}}(\mathsf{g}-\Phi_{r}\mathsf{g})(x-y)(\Phi_{1}X_{|\Omega})(dx)\mu(dy).$

The middle term on the RHS in (5.12) contributes to the volume error
$\mathrm{Error}_{\mathrm{vol}}$. It is bounded by

$\iint_{\mathbb{R}^{d}\times\mathbb{R}^{d}}(\mathsf{g}-\Phi_{1}\mathsf{g})(x-y)X_{|\Omega}(dx)\mu(dy)\leq
CX(\Omega).$

For the last term in (5.12), we replace $\mu$ by $q$ to generate the term

$\iint_{\mathbb{R}^{d}\times\mathbb{R}^{d}}(\mathsf{g}-\Phi_{r}\mathsf{g})(x-y)(\Phi_{1}X_{|\Omega})(dx)(q\mathrm{Leb}-\mu)(dy)$

in (5.8). We then estimate

$\displaystyle
q\iint_{\mathbb{R}^{d}\times\mathbb{R}^{d}}(\mathsf{g}-\Phi_{r}\mathsf{g})(x-y)(\Phi_{1}X_{|\Omega})(dx)dy$
$\displaystyle\leq
q\iint_{\Omega\times\mathbb{R}^{d}}(\mathsf{g}-\Phi_{r}\mathsf{g})(x-y)(\Phi_{1}X_{|\Omega})(dx)dy+Cr^{2}|q|X(\Omega\setminus\Omega_{-1})$
$\displaystyle\leq
q\iint_{\mathbb{R}^{d}\times\mathbb{R}^{d}}(\mathsf{g}-\Phi_{r}\mathsf{g})(x-y)((\Phi_{1}X_{|\Omega})_{|\Omega}+q\mathrm{Leb}_{|\Omega_{r}\setminus\Omega})(dx)dy+Cr^{2}|q|X(\Omega\setminus\Omega_{-1}),$

where we used $\|\mathsf{g}-\Phi_{r}\mathsf{g}\|_{L^{1}(\mathbb{R}^{d})}\leq
Cr^{2}$ and the fact that the mass of $\Phi_{1}X_{|\Omega}$ lying outside of
$\Omega$ is bounded by $X(\Omega\setminus\Omega_{-1})$. Assembling the above
estimates proves the lemma. ∎

We now turn to studying minimizers of $\mathcal{E}_{r}(\cdot;q)$ conditioned
on the weight given to $\Omega$. The following proposition is the key
technical result of Section 5, and it is the reason for considering the
precise form of the energy $\mathcal{E}_{r}(\cdot;q)$.

######  Proposition 5.3.

Let $\Omega$ an $\alpha$-thin annulus and $q\in\mathbb{R}$. We have

$\inf_{\nu:\nu(\Omega)=q|\Omega|}\mathcal{E}_{r}(\nu;q)\geq 0$ (5.13)

where the infimum is over measures $\nu$ supported on $\overline{\Omega}$ with
a bounded Lebesgue density.

###### Proof.

First, note that $\mathcal{E}_{r}(\cdot)$ as defined in (5.4) is non-negative
when applied to measures with bounded density and compact support. This is
because the Fourier transform of $\delta_{0}-\phi_{r}\ast\phi_{r}$ is non-
negative. Indeed, the Fourier transform on $\mathbb{R}^{d}$ of
$\phi_{r}\ast\phi_{r}$ is real since $\phi_{r}$ is radial, and it is bounded
above by $1$ (in the normalization for which $\hat{\delta}_{0}=1$) since it is
a probability density. Since $\mathsf{g}$ has positive Fourier transform, the
Fourier transform of
$\mathsf{g}-\Phi_{r}^{2}\mathsf{g}=(\delta_{0}-\phi_{r}\ast\phi_{r})\ast\mathsf{g}$
is non-negative, and non-negativity of $\mathcal{E}_{r}(\cdot)$ follows from
Plancherel’s theorem.

Let $\nu$ be a measure supported on $\overline{\Omega}$ with a bounded
Lebesgue density and $\nu(\Omega)=q|\Omega|$. We use that
$\mathcal{E}_{r}(\cdot;q)$ is quadratic to expand

$\displaystyle\mathcal{E}_{r}(\nu;q)$
$\displaystyle=\mathcal{E}_{r}(q\mathrm{Leb}_{|\Omega};q)+\mathcal{E}_{r}(q\mathrm{Leb}_{|\Omega}-\nu)$
(5.14)
$\displaystyle\quad+\iint_{\mathbb{R}^{d}\times\mathbb{R}^{d}}(\mathsf{g}-\Phi_{r}^{2}\mathsf{g})(x-y)(\nu-q\mathrm{Leb}_{|\Omega})(dx)(q\mathrm{Leb}_{|\Omega_{2r}})(dy)$
$\displaystyle\quad-q\iint_{\mathbb{R}^{d}\times\mathbb{R}^{d}}(\mathsf{g}-\Phi_{r}\mathsf{g})(x-y)dx(\nu-q\mathrm{Leb}_{|\Omega})(dy).$

We claim that both terms on the last line are $0$. Indeed, we have

$\int_{\mathbb{R}^{d}}(\mathsf{g}-\Phi_{r}\mathsf{g})(x-y)dy=c_{1,\phi,r},\quad\int_{\mathbb{R}^{d}}(\mathsf{g}-\Phi_{r}^{2}\mathsf{g})(x-y)dy=c_{2,\phi,r}$
(5.15)

for constants $c_{1,\phi,r},c_{2,\phi,r}$ independent of $x$, and the same
holds for $\mathrm{Leb}_{|\Omega_{2r}}(dy)$ in place of $dy$ as long as
$x\in\Omega$. The claim follows from $\nu(\Omega)=q$. Since
$\mathcal{E}_{r}(q\mathrm{Leb}_{|\Omega}-\nu)\geq 0$, we have proved that the
infimum in (5.13) is attained at $\nu=q\mathrm{Leb}_{|\Omega}$.

It remains to compute $\mathcal{E}_{r}(q\mathrm{Leb}_{|\Omega};q)$. We write
$\mathcal{E}_{r}(q\mathrm{Leb}_{|\Omega};q)=T_{1}+T_{2}-T_{3}$ for

$\displaystyle T_{1}$
$\displaystyle=\frac{q^{2}}{2}\iint_{\mathbb{R}^{d}\times\mathbb{R}^{d}}(\mathsf{g}-\Phi_{r}\mathsf{g})(x-y)\mathrm{Leb}_{|\Omega_{2r}}(dx)\mathrm{Leb}_{|\Omega_{2r}}(dy),$
$\displaystyle T_{2}$
$\displaystyle=\frac{q^{2}}{2}\iint_{\mathbb{R}^{d}\times\mathbb{R}^{d}}\Phi_{r}(\mathsf{g}-\Phi_{r}\mathsf{g})(x-y)\mathrm{Leb}_{|\Omega_{2r}}(dx)\mathrm{Leb}_{|\Omega_{2r}}(dy),$
$\displaystyle T_{3}$
$\displaystyle=q^{2}\iint_{\mathbb{R}^{d}\times\mathbb{R}^{d}}(\mathsf{g}-\Phi_{r}\mathsf{g})(x-y)dx\mathrm{Leb}_{|\Omega_{r}}(dy).$

Note that $\mathsf{g}-\Phi_{r}\mathsf{g}\geq 0$, and so

$T_{1}\geq\frac{q^{2}}{2}|\Omega_{r}|\inf_{y\in\Omega_{r}}\int_{\Omega_{2r}}(\mathsf{g}-\Phi_{r}\mathsf{g})(x-y)=\frac{1}{2}q^{2}c_{1,\phi,r}|\Omega_{r}|.$

We use $\Phi_{r}\mathrm{Leb}_{|\Omega_{2r}}\geq\mathrm{Leb}_{|\Omega_{r}}$ to
see

$\displaystyle T_{2}$
$\displaystyle=\frac{q^{2}}{2}\iint_{\mathbb{R}^{d}\times\mathbb{R}^{d}}(\mathsf{g}-\Phi_{r}\mathsf{g})(x-y)\Phi_{r}\mathrm{Leb}_{|\Omega_{2r}}(dx)\mathrm{Leb}_{|\Omega_{2r}}(dy)$
$\displaystyle\geq\frac{q^{2}}{2}\iint_{\Omega_{r}\times\Omega_{2r}}(\mathsf{g}-\Phi_{r}\mathsf{g})(x-y)dxdy\geq\frac{c_{1,\phi,r}q^{2}}{2}|\Omega_{r}|.$

Similarly, we have $T_{3}\leq q^{2}c_{1,\phi,r}|\Omega_{r}|$, and combining
the bounds on $T_{1},T_{2},T_{3}$ finishes the proof. ∎

######  Proposition 5.4.

Let $\Omega$ be an $\alpha$-thin annulus with diameter $2R$. Assume that the
parameters $\rho,\alpha,q>0$ and $M\geq 1$ satisfy the following for fixed
constants $C_{i}$, $i=1,2,3,4,5$.

  1. (1)

There is bounded excess:

$\rho\leq C_{1}.$ (5.16)

  2. (2)

The constant $q$ approximates $\mu$:

$\|\mu-q\|_{L^{\infty}(\Omega_{R})}\leq C_{2}^{-1}\rho.$ (5.17)

  3. (3)

The annulus is not too thin:

$\alpha R\geq C_{3}.$ (5.18)

  4. (4)

There is significant charge excess:

$\frac{(\alpha R)^{2/3}\rho}{1+\mathbf{1}_{d=2}\log(\alpha R)}\geq C_{4}.$
(5.19)

  5. (5)

The boundary layer density is not too high:

$M\leq C_{5}^{-1}(\alpha R)^{2/3}\rho.$ (5.20)

Assume that, dependent on $(\inf_{\Omega}\mu)^{-1}$, $\sup_{\Omega_{R}}\mu$,
$\beta$, we have $C_{i}\gg 1$ for $i=2,3,4,5$ and $C_{2}\geq C_{1}$. Then we
have

$\Delta_{\rho,r,M}\geq C^{-1}(\alpha R)^{2/3}\rho(\mu(\Omega)+\rho|\Omega|)$
(5.21)

for $r=(\alpha R)^{1/3}$ and the quantity $\Delta_{\rho,r,M}$ defined in
(5.3).

###### Proof.

Let $r=(\alpha R)^{1/3}$ and let $X_{N}\in E_{\rho,r,M}$ be arbitrary.

Step 1: We begin by estimating the energy change for the continuum
approximation discussed in Section 5.1. First, note that $C_{2}\geq C_{1}$
means that

$q\leq\|\mu-q\|_{L^{\infty}(\Omega_{R})}+\|\mu\|_{L^{\infty}(\Omega_{R})}\leq
C_{2}^{-1}\rho+C\leq C_{2}^{-1}C_{1}+C\leq C.$

Define $q_{X}=q+m_{X}$ for

$m_{X}:=\frac{1}{|\Omega|}(\Phi_{1}X_{|\Omega}(\Omega)-q|\Omega|).$

The parameter $m_{X}$ acts as an excess charge density beyond $q$ accounting
for boundary layer effects. We estimate it by

$\displaystyle\Phi_{1}X_{|\Omega}(\Omega)\geq
X(\Omega)-X(\Omega\setminus\Omega_{-1})$
$\displaystyle\geq\mu(\Omega)+\rho|\Omega|-MR^{d-1}r$
$\displaystyle\geq(q+\rho)|\Omega|-MR^{d-1}r-\|\mu-q\|_{L^{\infty}(\Omega)}|\Omega|.$

Applying our assumptions, we see

$\displaystyle
m_{X}\geq\rho-\frac{MR^{d-1}r}{|\Omega|}-\|\mu-q\|_{L^{\infty}(\Omega)}$
$\displaystyle\geq\rho-CM(\alpha R)^{-2/3}-\|\mu-q\|_{L^{\infty}(\Omega)}$
(5.22) $\displaystyle\geq\rho-CC_{5}^{-1}\rho-
C_{2}^{-1}\rho\geq\frac{1}{2}\rho.$

Since $q_{X}|\Omega|=\Phi_{1}X_{|\Omega}(\Omega)$, by Proposition 5.3 we have

$\mathcal{E}_{r}((\Phi_{1}X_{|\Omega})_{|\Omega};q_{X})\geq 0.$

We now lower bound $\mathcal{E}_{r}((\Phi_{1}X_{|\Omega})_{|\Omega};q)$ by
comparison to $\mathcal{E}_{r}((\Phi_{1}X_{|\Omega})_{|\Omega};q_{X})$. First,
for any measure $\nu$ with bounded density, we use that
$\mathcal{E}_{r}(\cdot)$ is quadratic to compute

$\displaystyle\mathcal{E}_{r}(\nu+q\mathrm{Leb}_{|\Omega_{2r}\setminus\Omega})-\mathcal{E}_{r}(\nu+q_{X}\mathrm{Leb}_{|\Omega_{2r}\setminus\Omega})$
$\displaystyle=\frac{q-q_{X}}{2}\iint_{\mathbb{R}^{d}\times(\Omega_{2r}\setminus\Omega)}(\mathsf{g}-\Phi_{r}^{2}\mathsf{g})(x-y)(2\nu+(q_{X}+q)\mathrm{Leb}_{|\Omega_{2r}\setminus\Omega})(dx)dy$
$\displaystyle\geq-\frac{|q-q_{X}|}{2}\|\mathsf{g}-\Phi_{r}^{2}\mathsf{g}\|_{L^{1}(\mathbb{R}^{d})}(2|\nu|+(q_{X}+q)\mathrm{Leb}_{|\Omega_{2r}\setminus\Omega})(\Omega_{5r}\setminus\Omega_{-3r})$
$\displaystyle\geq-C|q-q_{X}|r^{2}(|\nu|(\Omega_{5r}\setminus\Omega_{-2r})+CR^{d-1}r).$

The bound in the second to last line follows from the fact that
$\mathsf{g}-\Phi_{r}^{2}\mathsf{g}$ has support of diameter at most $4r$ and
so we only need to integrate over $x\in\Omega_{5r}\setminus\Omega_{-3r}$. For
$\nu=(\Phi_{1}X_{|\Omega})_{|\Omega}$, we see

$\mathcal{E}_{r}((\Phi_{1}X_{|\Omega})_{|\Omega}+q\mathrm{Leb}_{|\Omega_{2r}\setminus\Omega})-\mathcal{E}_{r}((\Phi_{1}X_{|\Omega})_{|\Omega}+q_{X}\mathrm{Leb}_{|\Omega_{2r}\setminus\Omega})\geq-
Cm_{X}(M+C)R^{d-1}r^{3}.$

Plugging this computation into (5.5), we find

$\displaystyle\mathcal{E}_{r}((\Phi_{1}X_{|\Omega})_{|\Omega};q)-\mathcal{E}_{r}((\Phi_{1}X_{|\Omega})_{|\Omega};q_{X})$
$\displaystyle\geq-
Cm_{X}(M+C)R^{d-1}r^{3}+q_{X}\iint_{\mathbb{R}^{d}\times\mathbb{R}^{d}}(\mathsf{g}-\Phi_{r}\mathsf{g})(x-y)dx(\Phi_{1}X_{|\Omega}+q_{X}\mathrm{Leb}_{|\Omega_{r}\setminus\Omega})(dy)$
$\displaystyle\quad-q\iint_{\mathbb{R}^{d}\times\mathbb{R}^{d}}(\mathsf{g}-\Phi_{r}\mathsf{g})(x-y)dx(\Phi_{1}X_{|\Omega}+q\mathrm{Leb}_{|\Omega_{r}\setminus\Omega})(dy)$
$\displaystyle\geq-
Cm_{X}(M+C)R^{d-1}r^{3}+(q_{X}-q)\iint_{\mathbb{R}^{d}\times\mathbb{R}^{d}}(\mathsf{g}-\Phi_{r}\mathsf{g})(x-y)dx(\Phi_{1}X_{|\Omega}+q\mathrm{Leb}_{|\Omega_{r}\setminus\Omega})(dy)$
$\displaystyle\geq-Cm_{X}(M+C)\alpha
R^{d}+m_{X}\|\mathsf{g}-\Phi_{r}\mathsf{g}\|_{L^{1}(\mathbb{R}^{d})}\Phi_{1}X_{|\Omega}(\Omega).$

We can compute
$\|\mathsf{g}-\Phi_{r}\mathsf{g}\|_{L^{1}(\mathbb{R}^{d})}=c_{\phi}r^{2}$ for
some constant $c_{\phi}>0$. Furthermore by (5.16), (5.18), and (5.20), we have
$(M+C)\alpha R^{d}\ll r^{2}\alpha R^{d}\leq
Cc_{\phi}r^{2}\Phi_{1}X_{|\Omega}(\Omega)$. It follows

$\mathcal{E}_{r}((\Phi_{1}X_{|\Omega})_{|\Omega};q)-\mathcal{E}_{r}((\Phi_{1}X_{|\Omega})_{|\Omega};q_{X})\geq\frac{1}{2}c_{\phi}m_{X}r^{2}\Phi_{1}X_{|\Omega}(\Omega).$

We apply our estimate (5.22) on $m_{X}$ and the similar estimate
$\Phi_{1}X_{|\Omega}(\Omega)\geq\frac{1}{2}X(\Omega)$ to see

$\mathcal{E}_{r}((\Phi_{1}X_{|\Omega})_{|\Omega};q)\geq\frac{1}{4}c_{\phi}r^{2}\rho
X(\Omega).$ (5.23)

This term will be the main energy benefit of isotropic averaging. Using
$\inf_{\Omega}\mu\geq C^{-1}$, we can estimate

$\mathcal{E}_{r}((\Phi_{1}X_{|\Omega})_{|\Omega};q)\geq C^{-1}\rho r^{2}\alpha
R^{d}\geq C^{-1}C_{4}\alpha R^{d}.$ (5.24)

Step 2: We now relate the isotropic averaging energy change associated to $X$
to the quantity $\mathcal{E}_{r}((\Phi_{1}X_{|\Omega})_{|\Omega};q)$ using
Lemma 5.1. We claim that

$\mathcal{H}^{W,U}(X_{N})-\mathrm{Iso}_{\mathbb{X}(\Omega),\psi_{r}}\mathcal{H}^{W,U}(X_{N})\geq\frac{1}{2}\mathcal{E}_{r}((\Phi_{1}X_{|\Omega})_{|\Omega};q).$
(5.25)

This claim, combined with (5.23), will finish the proof. By step 1, it is
sufficient to bound the error terms within Lemma 5.2 by
$\frac{1}{8}c_{\phi}r^{2}\rho X(\Omega)$. Using the notation from that lemma,
we compute

$\displaystyle\mathrm{Error}_{\mathrm{bl}}$ $\displaystyle\leq
Cr^{3}(1+q)MR^{d-1}+Cr^{3}q^{2}R^{d-1}\leq C\alpha R^{d}(M+1)$
$\displaystyle\leq C\alpha R^{d}(C_{5}^{-1}(\alpha R)^{2/3}\rho+1)\leq C\alpha
R^{d}+CC_{5}^{-1}\rho r^{2}X(\Omega).$

Applying (5.23) and (5.24), we see that

$\mathrm{Error}_{\mathrm{bl}}\leq\frac{1}{100}\mathcal{E}_{r}((\Phi_{1}X_{|\Omega})_{|\Omega};q).$

Next, we estimate the volume error term. In $d=2$, we can compute using (5.19)

$\mathrm{Error}_{\mathrm{vol}}\leq X(\Omega)(\log(\alpha R)+C)\leq
C_{4}^{-1}X(\Omega)(\alpha
R)^{2/3}\rho+CX(\Omega)\leq\frac{1}{400}c_{\phi}r^{2}\rho X(\Omega).$ (5.26)

In the last inequality, we used that $r^{2}\rho\geq C_{4}\gg 1$. In $d\geq 3$,
we can delete $\log(\alpha R)$ above and have the same final result. By
(5.23), we have dominated the volume error term by
$\frac{1}{100}\mathcal{E}_{r}((\Phi_{1}X_{|\Omega})_{|\Omega};q)$. The last
remaining error term is related to the approximation of $\mu$ by $q$:

$\iint_{\mathbb{R}^{d}\times\Omega}(\mathsf{g}-\Phi_{r}\mathsf{g})(q-\mu)(dx)(\Phi_{1}X_{|\Omega})(dy)\geq-
c_{\phi}r^{2}\|q-\mu\|_{L^{\infty}(\Omega_{r})}X(\Omega)\geq-
C_{2}^{-1}c_{\phi}r^{2}\rho X(\Omega).$ (5.27)

Using (5.23) and $C_{2}\gg 1$ allows us to dominate this term as well.
Assembling the above allows us to conclude the claim (5.25) and the proof. ∎

We will apply Proposition 5.4 to prove Theorem 5, but first we take care of
the case in which $\rho$ is very large using our high density law Theorem 1.

######  Proposition 5.5.

Let $\Omega$ be an $\alpha$-thin annulus of diameter $2R$ such that $\alpha
R\geq 1-\log\alpha$. Let $\rho\geq C_{1}$ for a fixed constant $C_{1}$ taken
large enough. Then we have

$\mathbb{P}^{W,U}_{N,\beta}(\\{X(\Omega)\geq\mu(\Omega)+\rho|\Omega|\\})\leq
e^{-\alpha^{d+2}R^{d+2}}.$

###### Proof.

Let $(A_{\lambda})_{\lambda\in\Lambda}$ be a covering of $\Omega$ by balls
$A_{\lambda}$ of radius $\alpha R$ of cardinality at most $C\alpha^{-d+1}$. We
have

$\displaystyle\mathbb{P}^{W,U}_{N,\beta}(\\{X(\Omega)\geq\mu(\Omega)+\rho|\Omega|\\})$
$\displaystyle\leq\mathbb{P}^{W,U}_{N,\beta}(\\{X(\Omega)\geq\rho|\Omega|\\})$
$\displaystyle\leq|\Lambda|\sup_{\lambda\in\Lambda}\mathbb{P}^{W,U}_{N,\beta}(\\{X(A_{\lambda})\geq
C^{-1}\alpha^{d-1}\rho|\Omega|\\}).$

If $C_{1}$ is large enough, we have that
$\alpha^{d-1}\rho|\Omega|\gg|A_{\lambda}|$, so we may apply Theorem 1 to see

$\displaystyle\mathbb{P}^{W,U}_{N,\beta}(\\{X(\Omega)\geq\mu(\Omega)+\rho|\Omega|\\})$
$\displaystyle\leq C\alpha^{-d+1}e^{-C^{-1}(\alpha\rho|\Omega|)^{d+2}}$
$\displaystyle\leq C\alpha^{-1}e^{-C^{-1}C_{1}^{2}\alpha^{d+2}R^{d+2}}\leq
e^{-\alpha^{d+2}R^{d+2}}.$

∎

We now consider the more general case.

######  Proposition 5.6.

Let $\Omega$ be an $\alpha$-thin annulus of diameter $2R$. Let $C_{1}$ be the
constant fixed in Proposition 5.5. Suppose we have parameters $q>0$ and
$\rho>0$ such that conditions (5.17), (5.18), (5.19) hold for large enough
constants $C_{2},C_{3},C_{4}$ with $C_{i}\gg C_{1}$ for $i=2,3,4$. Then we
have

$\mathbb{P}^{W,U}_{N,\beta}(\\{X(\Omega)\geq\mu(\Omega)+\rho|\Omega|\\})\leq
e^{-c(\alpha R)^{2/3}\rho(\mu(\Omega)+\rho|\Omega|)}+e^{-c(\alpha
R)^{d/3+2}\rho^{2}}+e^{-\alpha^{d+2}R^{d+2}},$ (5.28)

for some $c>0$.

###### Proof.

Let $r=(\alpha R)^{1/3}$ and $M=C_{5}^{-1}\rho r^{2}$ and for a large enough
constant $C_{5}$ ($C_{5}=\sqrt{C_{4}}$ will work if $C_{4}$ is chosen large
enough).

Step 1: We first bound the probability of $E_{\rho,r,M}$ by the isotropic
averaging argument. Let $n_{\rho}=\lceil\rho|\Omega|+\mu(\Omega)\rceil$ and
$M_{\rho}=\lceil MR^{d-1}r\rceil$ and $N_{1}=\lfloor
C_{1}|\Omega|+\mu(\Omega)\rfloor$. We write

$E_{\rho,r,M}\subset\\{X(\Omega)\geq\mu(\Omega)+C_{1}|\Omega|\\}\cup\bigcup_{m=0}^{M_{\rho}}\bigcup_{n=n_{\rho}}^{N_{1}}\bigcup_{\begin{subarray}{c}\mathcal{M}\subset\\{1,\ldots,N\\}\\\
|\mathcal{M}|=m\end{subarray}}\bigcup_{\begin{subarray}{c}\mathcal{N}\subset\\{1,\ldots,N\\}\\\
|\mathcal{N}|=n\end{subarray}}E_{\mathcal{M},\mathcal{N},r,M}$ (5.29)

for

$E_{\mathcal{M},\mathcal{N},r,M}=\\{\mathbb{X}(\Omega)=\mathcal{N}\\}\cap\\{\mathbb{X}(\Omega_{5r}\setminus\Omega)=\mathcal{M}\\}\cap\\{X(\Omega_{5r}\setminus\Omega_{-5r})\leq
MR^{d-1}r\\}.$

Note that

$\mathrm{Iso}_{\mathcal{N},\psi_{r}}^{\ast}\mathbf{1}_{E_{\mathcal{M},\mathcal{N},r,M}}\leq\mathbf{1}_{\\{\mathbb{X}(\Omega_{5r})=\mathcal{M}\cup\mathcal{N}\\}}.$

Indeed, since $\mathrm{Iso}^{\ast}_{\mathcal{N},\psi_{r}}$ is convolution by a
probability measure, it is bounded by $1$ as an operator $L^{\infty}\to
L^{\infty}$. By a similar argument as in the proof of Proposition 2.1, we have
$\mathrm{Iso}_{\mathcal{N},\psi_{r}}^{\ast}\mathbf{1}_{E_{\mathcal{M},\mathcal{N},r,M}}(X_{N})=0$
if $\mathbb{X}(\Omega_{5r})\neq\mathcal{M}\cup\mathcal{N}$.

By isotropic averaging, we conclude, for $|\mathcal{N}|=n\leq N_{1}$, that

$\mathbb{P}^{W,U}_{N,\beta}(E_{\mathcal{M},\mathcal{N},r,M})\leq
e^{-\beta\Delta_{\rho_{n},r,M}}\mathbb{P}^{W,U}_{N,\beta}(\\{\mathbb{X}(\Omega_{5r})=\mathcal{M}\cup\mathcal{N}\\})$

where $\rho_{n}:=\frac{n-\mu(\Omega)}{|\Omega|}$ and $\Delta_{\rho_{n},r,M}$
is as in (5.3). By Proposition 5.5, a union bound, and exchangeability, we
have

$\displaystyle\mathbb{P}^{W,U}_{N,\beta}(E_{\rho,r,M})$ $\displaystyle\leq
e^{-\alpha^{d+2}R^{d+2}}$
$\displaystyle\quad+\sum_{m=0}^{M_{\rho}}\sum_{n=n_{\rho}}^{N_{1}}{\binom{N}{m}}{\binom{N-m}{n}}e^{-\beta\Delta_{\rho_{n},r,M}}\mathbb{P}^{W,U}_{N,\beta}(\\{\mathbb{X}(\Omega_{5r})=\\{1,\ldots,m+n\\}\\}).$

Above, we only summed over $\mathcal{M}$ and $\mathcal{N}$ disjoint. We have

$\mathbb{P}(\\{\mathbb{X}(\Omega_{5r})=\\{1,\ldots,m+n\\}\\})=\frac{1}{{\binom{N}{m+n}}}\mathbb{P}(\\{X(\Omega_{5r})=m+n\\}\\})$

and

$\frac{{\binom{N}{m}}{\binom{N-m}{n}}}{{\binom{N}{m+n}}}=\frac{(m+n)!}{m!n!}\leq\frac{(n+m)^{m}}{m!}\leq
2n^{m}$

whenever $m\leq n$, which for us is always the case if $C_{5}$ is large enough
(independently of $C_{i}$, $i=2,3,4$). Letting $j=m+n$, we have

$\displaystyle\mathbb{P}^{W,U}_{N,\beta}(E_{\rho,r,M})$ $\displaystyle\leq
e^{-\alpha^{d+2}R^{d+2}}+\sum_{n=n_{\rho}}^{N_{1}}\sum_{j=n}^{n+M_{\rho}}2n^{j-n}e^{-\beta\Delta_{\rho_{n},r,M}}\mathbb{P}^{W,U}_{N,\beta}(\\{X(\Omega_{5r})=j\\})$
(5.30) $\displaystyle\leq
e^{-\alpha^{d+2}R^{d+2}}+\sum_{n=n_{\rho}}^{N_{1}}2n^{M_{\rho}}e^{-\beta\Delta_{\rho_{n},r,M}}.$

Note that $M_{\rho}\leq 2MR^{d-1}r=2C_{5}^{-1}R^{d-1}r^{3}\rho\leq
C_{5}^{-1}\alpha R^{d}r^{2}\rho/\log(N_{1})$. Thus for $n\leq N_{1}$ we have

$n^{M_{\rho}}\leq e^{M_{\rho}\log N_{1}}\leq
e^{CC_{5}^{-1}r^{2}\rho|\Omega|}.$

Since $\beta\Delta_{\rho_{n},r,M}\geq C^{-1}r^{2}\rho|\Omega|$ by Proposition
5.4, we can bound

$n^{M_{\rho}}e^{-\beta\Delta_{\rho_{n},r,M}}\leq
e^{-\frac{\beta}{2}cr^{2}\rho_{n}(\mu(\Omega)+\rho_{n}|\Omega|)}$

for some $c>0$ for $C_{5}$ large enough. The series in the RHS of (5.30) can
therefore be dominated by a geometric series with rate $1/2$ with the same
first term, and so

$\mathbb{P}^{W,U}_{N,\beta}(E_{\rho,r,M})\leq
e^{-\alpha^{d+2}R^{d+2}}+4e^{-\frac{\beta}{2}cr^{2}\rho(\mu(\Omega)+\rho|\Omega|)}.$
(5.31)

Step 2: We now consider the event that $X(\Omega)\geq\mu(\Omega)+\rho|\Omega|$
on the complement of $E_{\rho,r,M}$. It suffices to bound the probability that
$X(\Omega_{5r}\setminus\Omega_{-5r})\geq MR^{d-1}r.$ Let
$\\{A_{\lambda}\\}_{\lambda\in\Lambda}$ be a covering of
$\Omega_{5r}\setminus\Omega_{-5r}$ by balls of radius $r$ of cardinality at
most $C(R/r)^{d-1}$. We have

$\mathbb{P}^{W,U}_{N,\beta}(\\{X(\Omega_{5r}\setminus\Omega_{-5r})\geq
MR^{d-1}r\\})\leq\left(\frac{CR}{r}\right)^{d-1}\sup_{\lambda\in\Lambda}\mathbb{P}^{W,U}_{N,\beta}(\\{X(A_{\lambda})\geq
C^{-1}Mr^{d}\\}).$

Note that $(\alpha R)^{2/3}\rho\geq C_{4}$ by (5.19), so we have $M\geq
C_{5}^{-1}C_{4}.$ If we take $C_{4}\gg C_{5}$, we have $M\gg 1$ and we may
apply Theorem 1 to see

$\mathbb{P}^{W,U}_{N,\beta}(\\{X(\Omega_{5r}\setminus\Omega_{-5r})\geq
MRr\\})\leq\left(\frac{CR}{r}\right)^{d-1}e^{-C^{-1}r^{d+6}\rho^{2}}\leq
e^{-cr^{d+6}\rho^{2}}$

for some $c>0$. Assembling (5.31) and the above finishes the proof. ∎

It is now a simple matter to prove Theorem 5.

###### Proof of Theorem 5.

We will apply Proposition 5.6 to the $1$-thin annulus $\Omega=B_{R}(z)$ with a
certain parameter $\rho$. For the constant $q$ approximation to
$\mu=\frac{1}{c_{d}}\Delta W$, we take $q=\mu(z)$. If $W$ is quadratic near
$B_{R}(z)$, this approximation is exact, but we focus on the general case. One
finds that

$\|\mu-q\|_{L^{\infty}(B_{2R}(z))}\leq CN^{-1/d}R,$

whence we have the restriction $\rho\gg N^{-1/d}R$ in applying Proposition
5.6. We also have the restriction $\rho\gg R^{-2/3}(1+\mathbf{1}_{d=2}\log R)$
from (5.19). If $R\leq N^{\frac{3}{5d}},$ then the latter restriction is the
only relevant one, and we achieve

$\mathbb{P}^{W,U}_{N,\beta}(\\{X(B_{R}(z))\geq\mu(B_{R}(z))+\rho|B_{R}(z)|\\})\leq
e^{-cR^{d}T}+e^{-cR^{(d+2)/3}T^{2}}+e^{-R^{d+2}},$

as desired, where we set $\rho=TR^{-2/3}(1+\mathbf{1}_{d=2}\log R)$ for a
large $T>0$. ∎

###  5.3. Upgrading the discrepancy bound.

In this subsection, we upgrade Theorem 5 using rigidity results for smooth
linear statistics. We will assume conditions stated in Theorem 6 throughout.
We do not spend undue effort trying to optimize our bounds in $\beta$ or $V$
since our results are generally weaker than those of [AS21]. Instead, our
purpose is to show how our overcrowding estimates can be upgraded using known
rigidity bounds for smooth linear statistics. In particular, we demonstrate
that overcrowding bounds are sufficient to bound the absolute discrepancy,
rather than just the positive part of the discrepancy. The mechanism for this
is consists of finding a screening region of excess positive charge near the
boundary of a ball with large absolute discrepancy. We note that the idea of
obtaining a screening region is already present and features prominently in
[Leb21], and we do not add anything fundamentally new to this procedure, but
rather adapt it for our overcrowding estimate.

For $\alpha\in(0,1]$ and $R\in(0,\infty)$, let
$\xi_{R,\alpha}:\mathbb{R}\to\mathbb{R}$ be a function satisfying

  * •

$0\leq\xi_{R,\alpha}\leq 1$

  * •

$\xi_{R,\alpha}(x)=1\quad\forall x\in[-R,R]$ and
$\xi_{R,\alpha}(x)=0\quad\forall x\not\in(-R-\alpha R,R+\alpha R)$

  * •

$\xi^{\prime}_{R,\alpha}(x)\leq 0$ for $x\geq 0$.

  * •

$\sup_{x\in\mathbb{R}}|\xi^{(k)}_{R,\alpha}(x)|\leq C_{k}(\alpha R)^{-k}$ for
$k=1,2,3,4$.

In what follows, we will consider the map $x\mapsto\xi_{R,\alpha}(|x|)$ from
$\mathbb{R}^{d}\to\mathbb{R}$, and by abuse of notation we will write this map
as $\xi_{R,\alpha}$. We will also write
$\mathrm{Fluct}(\phi):=\int\phi(x)\mathrm{fluct}(dx)$ for the ($N$-dependent)
fluctuation measure $\mathrm{fluct}$ defined in (1.4).

######  Theorem 8 (Corollary of [Ser20], Theorem 1).

Under the conditions on $W=V_{N}$ stated above Theorem 6, for a large enough
constant $C>0$ and $\alpha R\geq C$, we have