MICROSCOPIC DESCRIPTION OF LOG AND COULOMB GASESSYLVIA SERFATY
Abstract. These are the lecture notes of a course taught at the Park City Mathematics
Institute in June 2017. They are intended to review some recent results, obtained in largepart with Thomas Leblé, on the statistical mechanics of systems of points with logarithmicor Coulomb interactions. After listing some motivations, we describe the “electric approach"which allows to get concentration results, Central Limit Theorems for fluctuations, and a
Large Deviations Principle expressed in terms of the microscopic state of the system.

1. Introduction and motivations
We are interested in the following class of energies(1.1)

HN (x1 , . . . , xN ) :=X

g(xi − xj ) +NX

N V (xi ).

i=1

1≤i6=j≤N

where x1 , . . . , xN are N points (or particles) in the Euclidean space Rd (d ≥ 1), and N islarge. The potential V is a confining potential, growing fast enough at infinity, on which weshall make assumptions later. The pair interaction potential g is given by either(1.2)(Log1 case)g(x) = − log |x|,in dimension d = 1,

(1.3)

(Log2 case)g(x) = − log |x|,in dimension d = 2,

(1.4)

(Coul case) g(x) = |x|2−d in dimension d ≥ 3.

We will also say a few things about the more general case
(1.5)

(Riesz case) g(x) = |x|−s ,with max(0, d − 2) ≤ s < d, in dimension d ≥ 1.

The interaction as in (1.2) (resp. (1.3)) corresponds to a one-dimensional (resp. twodimensional) logarithmic interaction, we will call Log1, Log2 the logarithmic cases. The
Log2 interaction is also the Coulomb interaction in dimension 2. For d ≥ 3, Coul corresponds
to the Coulomb interaction in higher dimension. In both instances Log2 and Coul we have− ∆g = cd δ0

(1.6)
where
(1.7)

cd = 2π

if d = 2

cd = (d − 2)|Sd−1 | for d ≥ 3.

Finally, the Riesz cases max(d − 2, 0) < s < d in (1.4) correspond to more general Riesz
interactions, in what is called the potential case. The situation where s > d, i.e. that for
which g is not integrable near 0 is called the hypersingular case. Some generalizations of2010 Mathematics Subject Classification. 60F05, 60K35, 60B20, 82B05, 60G15, 82B21, 82B26, 15B52.
Key words and phrases. Coulomb gases, log gases, random matrices, jellium, large deviations, point
processes.

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SYLVIA SERFATY

what we discuss here to that case are provided in [HLSS], see references therein for further
background and results.
Whenever the parameter s appears, it will be with the convention that s = 0 in thelogarithmic cases.
~ N for (x1 , . . . , xN ) and dX~ N for dx1 . . . dxN .
We will use the notational shortcut X
For any β > 0, we consider the canonical Gibbs measure at inverse temperature β, givenby the following density
(1.8)

~N) =
dPN,β (X

where
(1.9)

ˆ
ZN,β =β~ N ) dX~N
exp − HN (X
ZN,β
2
1





β
~ N ) dX~N.
exp − HN (X2The term ZN,β is a normalizing constant, called the partition function, which plays an important role in understanding the physics of the system.
We now review various motivations for studying such systems.
1.1. Fekete points and approximation theory. Fekete points arise in interpolation theory
as the points minimizing interpolation errors for numerical integration [SaTo]. Indeed, if one
is looking for N interpolation points {x1 , . . . , xN } in K such that the relation
ˆNX
wj f (xj )f (x)dx =K

j=1

is exact for the polynomials≤ N − 1. One sees that one needs to compute the´ kof degree
PN
coefficients wj such that K x = j=1 wj xkj for 0 ≤ k ≤ N − 1, and this computation is easy
if one knows to invert the Vandermonde matrix of the {xj }j=1...N . The numerical stabilityof this operation is as large as the condition number of the matrix, i.e. as the Vandermondedeterminant of the (x1 , . . . , xN ). The points that minimize the maximal interpolation error
for general functions are easily shown to be the Fekete points, defined as those that maximizeY|xi − xj |i6=jor equivalently minimize−X

log |xi − xj |.

i6=jThey are often studied on manifolds, such as the d-dimensional sphere. In Euclidean space,
one also considers “weighted Fekete points" which maximizeY|xi − xj |e−N

Pi

V (xi )

i6=j

or equivalently minimize
−

X
i6=j

log |xi − xj | + N

NX

V (xi )

i=1

which in dimension 2 corresponds exactly to the minimization of our Hamiltonian HN in theparticular case Log2. They also happen to be zeroes of orthogonal polynomials, see [Sim].

MICROSCOPIC DESCRIPTION OF LOG AND COULOMB GASES

3

Since − log |x| can be obtained as lims→0 1s (|x|−s − 1), there is also interest in studying“Riesz s-energies", i.e. the minimization of(1.10)

1|xi − xj |s
i6=jX

for all possible s, hence a motivation for (1.4). Varying s from 0 to ∞ connects Fekete points tooptimal packing problems. The optimal packing problem is solved in 1, 2 and 3 dimensions.
The solution in dimension 2 is the triangular lattice, in dimension 3 it is the FCC (facecentered cubic) lattice. In higher dimension, the solution is in general not known except indimensions 8 and 24. In high dimension, where the problem is important for error-correcting
codes, it is expected that it is not a lattice.
Note that the triangular lattice is conjectured to have to universally minimizing property[CoKu] i.e. to be the minimizer for a broad class of interactions. An analogous role is played indimensions 8 and 24 by the E8 and Leech lattices, respectively, and remarkably the universallyminimizing property of these two lattices has just been recently proven [Via, CKMRV].
For these aspects, we refer to the the review papers [SK, BHS] and references therein (inthat context such systems are mostly studied on the d-dimensional sphere or torus).
1.2. Statistical mechanics. The ensemble given by (1.8) in the Log2 case is called in physics
a two-dimensional Coulomb gas or one-component plasma (see e.g. [AJ, JLM, Ja, SM] for aphysical treatment). Two-dimensional Coulomb interactions arise in quantum mechanics: thedensity of the many-body wave function of free fermions in a harmonic trap is the same as theGibbs measure of the Log1 gas with β = 2 [DDMS], and the fractional quantum Hall effect isalso described via a two-dimensional Coulomb gas [Gir, Sto]. Ginzburg-Landau vortices [SS1]and vortices in superfluids and Bose-Einstein condensates also interact like two-dimensional
Coulomb particles, cf. below. The Coul case with d = 3 can be seen as a toy (classical) model
for matter (see e.g. [PenSm, JLM, LiLe, LN]).
The general Riesz case can be seen as a generalization of the Coulomb case, motivations forstudying Riesz gases are numerous in the physics literature (in solid state physics, ferrofluids,
elasticity), see for instance [Maz, BBDR05, CDR, To], they can also correspond to systems
with Coulomb interaction constrained to a lower-dimensional subspace.
In all cases of interactions, the systems governed by the Gibbs measure PN,β are consideredas difficult systems in statistical mechanics because the interactions are truly long-range,
singular, and the points are not constrained to live on a lattice.
As always in statistical mechanics [Huan], one would like to understand if there are phasetransitions for particular values of the (inverse) temperature β. For the systems studied here,
one may expect what physicists call a liquid for small β, and a crystal for large β. The
meaning of crystal in this instance is not to be taken literally as a lattice, but rather as asystem of points whose 2-point correlation function
ρ2 (x, y) = T2 (x − y)(assuming translation invariance) does not decay too fast as x − y → ∞.
This phase-transition at finite β has been conjectured in the physics literature for the Log2
case (see e.g. [BST66, CLWH82, CC83]) but its precise nature is still unclear (see e.g. [Sti98]
for a discussion). In view of the recent progress in computational physics concerning suchphenomenon in two-dimensional systems (see e.g. [KK15]), which suggest a possibly very

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SYLVIA SERFATY

subtle transition between the liquid and solid phase, the question seems yet out of reach fora rigorous treatment.
1.3. Two component plasma case. The two-dimensional “one component plasma", consisting of positively charged particles, has a “two-component" counterpart which consists in
N particles x1 , . . . , xN of charge +1 and N particles y1 , . . . , yN of charge −1 interacting
logarithmically, with energy
X
X
X
~N, Y
~N ) = −HN (Xlog |xi − xj | −log |yi − yj | +log |xi − yj |i6=jand the Gibbs measure

i6=j

1

i,j

~ ~
~ N dY~N .
e−βHN (XN ,YN ) dX

ZN,β
Although the energy is unbounded below (positive and negative points attract), the Gibbs
measure is well defined for β small enough, more precisely the partition function converges forβ < 2. The system is then seen to form dipoles which do not collapse, thanks to the thermal
agitation. The two-component plasma is interesting due to its close relation to two important
theoretical physics models: the XY model and the sine-Gordon model (cf. the review [Spe]),which exhibit a Kosterlitz-Thouless phase transition (made famous by the 2016 physics Nobel
prize, cf. [BHG]) consisting in the binding of these “vortex-antivortex" dipoles.
1.4. Random matrix theory. The study of (1.8) has attracted a lot of attention due to its
connection with random matrix theory (we refer to [Fo] for a comprehensive treatment). Random matrix theory (RMT) is a relatively old theory, pionereed by statisticians and physicists
such as Wishart, Wigner and Dyson, and originally motivated by the understanding of thespectrum of heavy atoms, see [Me]. For more recent mathematical reference see [AGZ, D, Fo].
The main question asked by RMT is : what is the law of the spectrum of a large randommatrix ? As first noticed in the foundational papers of [Wi,Dy], in the particular cases (1.2)–(1.3) the Gibbs measure (1.8) corresponds in some particular instances to the joint law of theeigenvalues (which can be computed algebraically) of some famous random matrix ensembles:• for Log2, β = 2 and V (x) = |x|2 , (1.8) is the law of the (complex) eigenvalues of an
N × N matrix where the entries are chosen to be normal Gaussian i.i.d. This is called
the Ginibre ensemble.
• for Log1, β = 2 and V (x) = x2 /2, (1.8) is the law of the (real) eigenvalues of an
N × N Hermitian matrix with complex normal Gaussian iid entries. This is called theGaussian Unitary Ensemble.
• for Log1, β = 1 and V (x) = x2 /2, (1.8) is the law of the (real) eigenvalues of an N ×N
real symmetric matrix with normal Gaussian iid entries. This is called the Gaussian
Orthogonal Ensemble.
• for Log1, β = 4 and V (x) = x2 /2, (1.8) is the law of the eigenvalues of an N × N
quaternionic symmetric matrix with normal Gaussian iid entries. This is called theGaussian Symplectic Ensemble.
One thus observes in these ensembles the phenomenon of “repulsion of eigenvalues": they
repel each other logarithmically, i.e. like two-dimensional Coulomb particles.
For the Log1 and Log2 cases, at the specific temperature β = 2, the law (1.8) acquires
a special algebraic feature : it becomes a determinantal process, part of a wider class ofprocesses (see [HKPV,Bor]) for which the correlation functions are explicitly given by certain
determinants. This allows for many explicit algebraic computations, on which there is a large

MICROSCOPIC DESCRIPTION OF LOG AND COULOMB GASES

5

literature. In particular, one can compute an expansion of log ZN,β as N → ∞ (see [Me]) andthe limiting processes at the microscopic scale [BSi]. However, many relevant quantities thatcan be computed explicitly for β = 2 are not exactly known for the β 6= 2 case, even in thecase of the potential V (x) = |x|2 . The particular case of (1.2) for all β and general V , also
called β-ensembles, has however been well-understood. In particular, thanks to the worksof [Jo, Sh, BEY1, BEY2, BFG, BL], one has expansions of log ZN,β , Central Limit Theorems
for linear statistics, and universality (after suitable rescaling) of the microscopic behavior andlocal statistics of the points, i.e. the fact that they are essentially independent of V .
Considering the coincidence between a statistical mechanics model and the law of thespectrum of a random matrix model for several values of the inverse temperature, it is also
natural to ask whether such a correspondence exists for any value of β, i.e. whether (1.8)
can be seen as a law of eigenvalues for some random matrix model. The answer is positivein dimension 1 for any β: a somehow complicated model of tridiagonal matrices can beassociated to the Gibbs measure of the one-dimensional log gas at inverse temperature β,see [KN1, DE]. This and other methods allow again to compute a lot explicitly, and to derivethat the microscopic laws of the eigenvalues are those of a so called sine-β process [VV].
1.5. Complex geometry and theoretical physics. Two-dimensional Coulomb systems(in the determinantal case β = 2) are of interest to geometers because they serve to construct Kähler-Einstein metrics with positive Ricci curvature on complex manifolds, cf. [Ber].
Another important motivation is the construction of Laughlin states for the Fractional Quantum Hall effect, which effectively reduces to the study of a two-dimensional Coulomb gas(cf. [Gir, Sto, RY]). When studying the Fractional Quantum Hall effect on a complex manifold, the coefficients in the expansion of the (logarithm of the) partition function have interpretations as geometric invariants, and it is thus of interest to be able to compute them,cf [Kl].
1.6. Vortices in condensed matter physics. In superconductors with applied magnetic
fields, and in rotating superfluids and Bose-Einstein condensates, one observes the occurrenceof quantized “vortices" (which are local point defects of superconductivity or superfluidity,
surrounded by a current loop). The vortices repel each other, while being confined together
by the effect of the magnetic field or rotation, and the result of the competition between
these two effects is that, as predicted by Abrikosov [Abri], they arrange themselves in aparticular triangular lattice pattern, called Abrikosov lattice, cf. Fig. 1 (for more pictures, seewww.fys.uio.no/super/vortex/). When restricting to a two-dimensional situation, it canbe shown formally, cf. [Se, Chap. 1] for a formal derivation, but also rigorously [SS1, SS4],
that the minimization problem governing the behavior of such systems can be reduced, in
terms of the vortices, to the minimization of an energy of the form (1.1) in the case Log2.
This naturally leads to the question of understanding the connection between minimizers of(1.1) + (1.3) and the Abrikosov triangular lattice.
2. Equilibrium measure and leading order behavior
2.1. The macroscopic behavior: empirical measure. It is well-known since [Cho58] (seee.g. [SaTo] for the logarithmic cases), that under suitable assumptions on V , the leading order
behavior of HN is governed by the minimization of the functional
¨
ˆ
(2.1)
IV (µ) :=g(x − y)dµ(x)dµ(y) +V (x)dµ(x)Rd ×RdRd

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SYLVIA SERFATY

Figure 1. Abrikosov lattice, H. F. Hess et al. Bell Labs Phys. Rev. Lett. 62,
214 (1989)
defined over the space P(Rd ) of probability measures on Rd , which may also take the value+∞.
Note that IV (µ) is simply a continuum version of the discrete Hamiltonian HN . From thepoint of view of statistical mechanics, IV is the “mean-field" limit energy of HN , while wewill see that from the point of view of probability, IV plays the role of a rate function.
Under suitable assumptions, there is a unique minimizer of IV on the space P(Rd ) of
probability measures on Rd , it is called the equilibrium measure and we denote it by µV . Its
uniqueness follows from the strict convexity of IV . For existence, one assumes(A1): V is finite l.s.c. and bounded below(A2): (growth assumption)
V (x)+ g(x) = +∞.
2|x|→+∞
One then proceeds in a standard fashion, taking a minimizing sequence for IV and using that
IV is coercive and lower semi-continuous. Finally, one has




lim

Theorem 1 (Frostman [Fro], existence and characterization of the equilibrium measure).
Under the assumptions (A1)-(A2), the minimum of IV over P(Rd ) exists, is finite and isachieved by a unique µV , which has a compact support of positive capacity. In addition µV is
uniquely characterized by the fact that

V

µV


q.e. in Rd h + 2 ≥c
(2.2)

V


 hµV +
= c q.e. in the support of µV
2
where
ˆ
µV
(2.3)
h (x) :=
g(x − y)dµV (y)Rdis the electrostatic potential generated by µV ; and then the constant c must beˆ
1
V (x)dµV (x).
(2.4)
c = IV (µV ) −2 Rd

MICROSCOPIC DESCRIPTION OF LOG AND COULOMB GASES

7

The proof of this theorem can easily be adapted from [SaTo, Chap. 1] or [Se, Chap. 2]. The
relations (2.2) can be seen as the Euler-Lagrange equations associated to the minimizationof IV , they are obtained by making variations of the form (1 − t)µV + tν where µV is theminimizer, ν is an arbitrary probability measure of finite energy, and t ∈ [0, 1].
Here, the capacity of a set (see [SaTo, AdHed] or [LiLo, Sec. 11.15]) is an appropriate
notion of size, suffice it to say that q.e. means except on a set of capacity zero, and that a setof null capacity has zero Lebesgue measure (but the converse is not true).
Remark 2.1. Note that by (1.6), in dimension d ≥ 2, the function hµV solves−∆hµV = cd µV
where cd is the constant defined in (1.7).
Example 2.2 (C 1,1 potentials and RMT examples). In general, the relations (2.2) say thatthe total potential hµV + V2 is constant on the support of the charges. Moreover, in dimensiond ≥ 2, applying the Laplacian on both sides of (2.2) and using Remark 2.1 gives that, on theinterior of the support of the equilibrium measure, if V ∈ C 1,1 ,∆V(2.5)
cd µV =
2i.e. the density of the measure on the interior of its support is given by ∆V
2cd . For example if V
∆Vis quadratic, then the density 2cd is constant on the interior of its support. This corresponds
to the important examples of the Hamiltonians which arise in random matrix theory, more
precisely :• in dimension d = 2, for V (x) = |x|2 , one may check that µV = π1 1B1 where 1 denotesa characteristic function and B1 is the unit ball, i.e. the equilibrium measure is thenormalized Lebesgue measure on the unit disk (by uniqueness, µV should be radiallysymmetric, and the combination of (2.5) with the constraint of being a probability
measure imposes the support to be B1 ). This is known as the circle law for the Ginibre
ensemble in the context of Random Matrix Theory (RMT). Its derivation is attributed
to Ginibre [Gin], Mehta [Me], an unpublished paper of Silverstein and Girko [Gi].
• in dimension d ≥ 3, the same holds, i.e. for V (x) = |x|2 we have µV = cdd 1Bby(d−2)1/dthe same reasoning.
• in dimensiond = 1, with g = − log | · | and V (x) = x2 , the equilibrium measure is√1µV (x) = 2π
4 − x2 1|x|≤2 , which corresponds in the context of RMT (GUE and GOE
ensembles) to Wigner’s semi-circle law, cf. [Wi, Me].
The equilibrium measure µV can also be interpreted in terms of the solution to a classicalobstacle problem, which is essentially dual to the minimization of IV , and better studied fromthe PDE point of view (in particular the regularity of µV and of the boundary of its support).
For this aspect, see [Se, Chap. 2].
Definition 2.3. From now on, we denote by ζ the functionV(2.6)
ζ = hµV + − c.
2
We note that in view of (2.2), ζ ≥ 0 a.e. and ζ = 0 µV -a.e.
In the rest of the course, we will always assume that the support of µV , denoted Σ, is aset with a nice boundary (say C 1,α ), and that the density µV (x) of µV is a Hölder continuousfunction in Σ.

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SYLVIA SERFATY

2.2. Large Deviations Principle at leading order. Let us start by recalling the basic
definitions associated to Large Deviations Principles (cf. for instance [DZ]).
Definition 2.4 (Rate function). Let X be a metric space (or a topological space). A ratefunction is a l.s.c. function I : X → [0, +∞], it is called a good rate function if its sub-levelsets {x, I(x) ≤ α} are compact.
Definition 2.5 (Large deviations). Let {PN }N be a sequence of Borel probability measures
on X and {aN }N a sequence of positive real numbers diverging to +∞. Let also I be a (good)rate function on X. The sequence {PN }N is said to satisfy a large deviation principle (LDP)
at speed aN with (good) rate function I if for every Borel set E ⊂ X the following inequalities
hold :
1
1
log PN (E) ≤ lim suplog PN (E) ≤ − inf I(2.7)− inf I ≤ lim inf◦
N →+∞ aNĒN →+∞ aN
E◦

where E (resp. Ē) denotes the interior (resp. the closure) of E for the topology of X.
Formally, it means that PN (E) should behave roughly like e−aN inf E I . The rate function I
is the rate of exponential decay of the probability of rare events, and the events with larger
probability are the ones on which I is smaller.
For us, a convenient macroscopic observable is given by the empirical measure of the par~ N ∈ (Rd )N we formticles: if X
(2.8)

~
µemp
N [XN ] :=

N
1 X
δx ,
N i=1 iwhich is a probability measure on Rd . The minimisation of IV determines the macroscopic(or global) behavior of the system in the following sense:~• Minimizers of HN are such that µemp
N [XN ] converges to µV as N → ∞ (cf. [Se, Chap.
2])~• In fact µemp
N [XN ] converges weakly to µV as N → ∞ almost surely under the canonical
Gibbs measure PN,β , see below.
In other words, not only the minimisers of the energy, but almost every (under the Gibbs
measure) sequence of particles is such that the empirical measure converges to the equilibrium
measure. Since µV does not depend on the temperature, in particular the asymptotic macroscopic behavior of the system is independent of β. This is formally stated as the following
Large Deviations Principle, due to [PeHi] (in dimension 2), [BG] (in dimension 1) and [BZ]
(in dimension 2) for the particular case of a quadratic potential (and β = 2), see also [Ber] forresults in a more general (still determinantal) setting of multidimensional complex manifolds,
or [CGZ] which recently treated more general singular g’s and V ’s. We present here the result
for the Coulomb gas in any dimension and general potential, which is not more difficult, forthe proof see [Se].
We need an additional assumption:(A3) Given β, for N large enough, we have
ˆ



V (x)
(Log1, Log2) exp −βN− log |x|dx < ∞2ˆβ(Coul) exp − N V (x) dx < +∞.
2

MICROSCOPIC DESCRIPTION OF LOG AND COULOMB GASES

9

Note in particular that (A3) ensures that the integral in (1.9) is convergent, hence ZN,βwell-defined.
Theorem 2 (Large deviations principle for the Coulomb gas at speed N 2 ). Let β > 0 be
given and assume that V is continuous, satisfies (A3) and that (1 − α0 )V satisfies (A2) forsome α0 > 0. Then the sequence {PN,β }N of probability measures on P(Rd ) satisfies a large
deviations principle at speed N 2 with good rate function β2 ÎV where ÎV = IV − minP(Rd ) IV =IV − IV (µV ). Moreover
1
β
β
(2.9)
lim
log ZN,β = − IV (µV ) = − min IV .
2
N →+∞ N
22 P(Rd )Here the underlying topology is that of weak convergence on P(Rd ).
The heuristic reading of the LDP is that
(2.10)β2

PN,β (E) ≈ e− 2 N (minE IV −min IV ) ,which in view of the uniqueness of the minimizer of IV implies as stated above that configurations whose empirical measure does not converge to µV as N → ∞ have exponentiallydecaying probability.
2.3. Further questions. In contrast to the macroscopic result, several observations (e.g. bynumerical simulation, see Figure 2.3) suggest that the behavior of the system at microscopic
scale1 depends heavily on β.

Figure 2. Case Log2 with N = 100 and V (x) = |x|2 , for β = 400 (left) and
β = 5 (right).
The questions that one would like to answer are then to describe the system beyond themacroscopic scale, at the mesoscopic (i.e. N −α for 0 < α < 1/d) scale, and at the microscopic(N −1/d ) scale.
P
Since one already knows that N
i=1 δxi − N µV is small (more precisely  N in some appropriate norm), one knows that the so-called discrepancy
ˆNX
D(x, r) :=δxi − N dµV
Br (x) i=1

is o(N ) as long as r > 0 is fixed. Is this still true at the mesoscopic scale for r of the orderN −α with α < 1/d? Is it true down to the microscopic scale, i.e. for r = RN −1/d with R  1?
Does it hold regardless of the temperature? This would correspond to a rigidity result. Note
1Since the N particles are typically confined in a set of order O(1), the microscopic, inter-particle scale isO(N −1/d ).

10

SYLVIA SERFATY

that point processes with discrepancies growing like the perimeter of the ball have been called
hyperuniform and are of interest to physicists for a variety of applications (cf. [To]).
Once one proves rigidity down to the microscopic scale, one may also like to characterize
the fluctuations of linear statistics of the formˆNX
f (xi ) − N f dµV ,i=1

where f is a regular enough test-function. In the logarithmic cases, they are proven toconverge to a Gaussian distribution whose variance depends on the temperature, as will beseen below.
Another point of view is that of large deviations theory. Then, one wishes to study amicroscopic observable, the microscopic point process obtained after blow-up by N 1/d , andcharacterize it as minimizing a certain energy (or rate function, in the case with temperature),thus connecting to a crystallization question.
In all the cases, one wants to understand precisely how the behavior depends on β, butalso on V . It is believed that at the macroscopic and microscopic levels, in the logarithmic
cases the behavior is independent on V , a universality feature.

3. Splitting of the Hamiltonian and electric approach
We now start to present the approach to these problems initiated in [SS5] and continued
in [SS6, RouSe, PetSer, LS1]. It relies on a splitting of the energy into a fixed leading orderterm and a next order term expressed in terms of the charge fluctuations, and on a rewritingof this next order term via the “electric potential" generated by the points.
3.1. The splitting formula. The splitting consists in an exact formula that separates theleading (N 2 ) order term in HN from next order terms.
~
Since we expect µemp
N [XN ] to converge to µV , we may try to “expand" around µV . In all
the sequel, we denote for any probability measure µ,(3.1)~ N ] − µ) =~ N ] = N (µemp [XfluctµN [XNNX

δxi − N µ.

i=1

~ N dependence.
Unless ambiguous, we will drop the X
Lemma 3.1 (Splitting formula). Assume µV is absolutely continuous with respect to the~ N ∈ (Rd )N we haveLebesgue measure. For any N and any X(3.2)~ N ) = N 2 IV (µV ) + 2N
HN ( X

N
X~N)
ζ(xi ) + FNµV (X

i=1

where 4 denotes the diagonal of Rd × Rd and we define for any probability measure µ
¨
µ ~
(3.3)
FN (XN ) =g(x − y) dfluctµNV (x)dfluctµNV (y).
4c

MICROSCOPIC DESCRIPTION OF LOG AND COULOMB GASES

11

Proof. We may write~N) =HN ( XX
i6=jg(xi − xj ) + N

NX

V (xi )

i=1

ˆ
emp
2
V dµempg(x − y)dµemp
(x)dµ
(y)
+
N
N (x)
N
N
d
c
R
4
ˆ
¨
2
2
V dµV
g(x − y)dµV (x)dµV (y) + N
= N
Rd
4c
¨
ˆ
+ 2N
g(x − y)dµV (x)dfluctN (y) + N
V dfluctN4c
Rd¨
g(x − y)dfluctN (x)dfluctN (y).
+
¨

= N2

(3.4)

4c

We now recall that ζ was defined in (2.6) by
ˆ
V
V
µV
(3.5)
ζ =h + −c=g(· − y) dµV (y) + − c22
Rd
and that ζ = 0 in Σ (with the assumptions we made, one can check that ζ is continuous, so
the q.e. relation can be upgraded to everywhere).
With the help of this we may rewrite the medium line in the right-hand side of (3.4) as¨ˆ
2Ng(x − y)dµV (x)dfluctN (y) + N
V dfluctN
4c
Rd
ˆ
ˆ
V
µV
= 2N
(h + )dfluctN = 2N
(ζ + c)dfluctN
2
Rd
Rd
ˆ
ˆ
ˆ
ˆ
2
2
−
2N
ζdµ
+
2N
c
dfluct=2N
ζdµemp= 2N 2
ζdµemp
V
N
N .
N
Rd

Rd

Rd

Rd

and µVThe last equality is due to the facts that ζ ≡ 0 on the support of µV and that µemp
N
are both probability measures. We also have to notice that since µV is absolutely continuouswith respect to the Lebesgue measure, we may include the diagonal back into the domain of
integration. By that same argument, one may recognize in the first line of the right-hand sideof (3.4), the quantity N 2 IV (µV ), cf. (2.1).
The function ζ can be seen as an effective potential, whose sole role is to confine the points
to the set Σ. We have thus reduced to studying FNµV . It is a priori not clear of which orderthis term is, and whether it is bounded below!
3.2. Electric interpretation. To go further, we use an electric interpretation of the energy
FNµV , as first used in [SS5], and the rewriting of the energy via truncation as in [RouSe,PetSer].
Such a computation allows to replace the sum of pairwise interactions of all the charges
and “background" by an integral (extensive) quantity, which is easier to handle. This will be
the first time that the Coulomb nature of the interaction is really used.
~ N of points in
3.2.1. Electric potential and truncation. Electric potential. For any N -tuple Xdthe space R , and any probability density µ, we define the (electric) potential generated by

12

SYLVIA SERFATY~ N and µ as
X
(3.6)

ˆ
µ
HN
(x) :=−g(x − y)
Rd

N
X

!

δxi − N dµ (y).

i=1

µ ~
~ N dependence forNote that in principle it should be denoted HN[XN ](x), but we omit the Xµthe sake of lightness of notation. The potential HN satisfies

(3.7)

µ
− ∆HN
= cd

N
X

!

δxi − N dµ

in Rd .

i=1µNote that HN decays at infinity, because the charge distribution fluctN is compactly supported

and has zero total charge, hence, when seen from infinity behaves like a dipole. More precisely,µµ
decays like the secondHNdecays like ∇g at infinity, that is O( |x|1d−1 ) and its gradient ∇HN
derivative D2 g, that is O( |x|1 d ).
~ N ∈ (Rd )N be fixed. For any ~η = (η1 , . . . , ηN ) we define theTruncated potential. Let X
truncated potential
(3.8)

µ
µ
HN,~
η (x) = HN (x) −NX(g(x − xi ) − g(ηi ))+

i=1

where (·)+ denotes the positive part of a number. Note that in view of the singular behaviorµof g at the origin, HN
diverges at each xi , and here we “chop off" these infinite peaks at
distance ηi from xi . We will also denote(3.9)

fη (x) = (g(x) − g(η))+ ,and point out that fη is supported in B(0, η). We note that
!
ˆ
N
X
µ
(ηi )
(3.10)
HN,~η (x) :=g(x − y)δxi − N dµ (y),
R2

i=1

(η )
where δxi i denotes the uniform measure of mass 1 on ∂B(xi , ηi ). By HN,η we simply denote

HN,~η when all the ηi are chosen equal to η.
3.2.2. Re-expressing the interaction term. Formally, using Green’s formula (or Stokes’ theorem) and the definitions, one would like to write that in the Coulomb cases
ˆ
ˆ
ˆ
1
1
µ
µ
µ
µ 2
µ ~
|∇HN
|
(3.11)FN (XN ) = HN dfluctN = HN (− ∆HN ) ≈cd
cd
This is the place where we really use for the first time in a crucial manner the Coulombic
nature of the interaction kernel g. Such a computation allows to replace the sum of pairwiseinteractions of all the charges and “background" by an integral (extensive) quantity, which isµeasier to handle in some sense. However, (3.11) does not make sense because ∇HNfails to beµ2
in L due to the presence of Dirac´masses. Indeed, near each atom xi , the vector-field ∇HNbehaves like ∇g and the integrals B(0,η) |∇g|2 are divergent in all dimensions. Another way
to see this is that the Dirac masses charge the diagonal 4 and so 4c cannot be reduced to
the full space.´ The point of the truncation above is precisely to remedy this and give a wayµ 2
of computing |∇HN
| in a “renormalized" fashion.
We now give a proper meaning to the statement.

MICROSCOPIC DESCRIPTION OF LOG AND COULOMB GASES

13~ N , for any ~η with ηi ≤ 1 such that the B(xi , ηi ) are disjoint, for any
Lemma 3.2. Given X
2absolutely continuous probability measure µ we have
!
!
ˆ
N
N
X
X
1
µ 2
µ ~
2
|∇HN,~η | − cdg(ηi ) + O N kµkL∞
ηi .
(3.12)FN (XN ) =
cd
Rd
i=1
i=1
If the balls B(xi , ηi ) are not assumed to be disjoint, then we still have that the left-hand sideis larger than the right-hand side.
The last term in the right-hand side of (3.12) should be thought of as a small error. In
practice, we will take ηi ≤ N −1/d η and let η → 0. The error is then O(η 2 N 2−2/d ).
´Proof. For the proof, we drop the superscripts µ. First we notice that Rd |∇HN,~η |2 is aconvergent integral and that
!
!
ˆ
¨
N
N
X
X
2
(ηi )
(ηi )
(3.13)|∇HN,~η | = cdg(x − y)dδxi − N µ (x)dδxi − N µ (y).
Rd

i=1

i=1

~ N are contained in the ballIndeed, we may choose R large enough so that all the points of X
BR = B(0, R). By Green’s formula and (3.10), we have
!
ˆ
ˆ
ˆ
N
X
∂H
N
HN,~η
δx(ηi i ) − N µ .
|∇HN,~η |2 =HN,~η
(3.14)
− cd
∂ν
∂BR
BR
BR
i=1
In view of the decay of HN and ∇HN mentioned above, the boundary integral tends to 0 as
R → ∞, and so we may write
!
ˆ
ˆ
N
X
(ηi )
2
δ xi − N µ|∇HN,~η | = cd
HN,~η
Rd

Rd

i=1

and thus (3.13) holds. We may next write
!
!
¨
N
N
X
X
(ηi )
(ηi )
(3.15)
g(x − y)dδxi − N µ (x)dδxi − N µ (y)
i=1

i=1

¨
−

g(x − y) dfluctN (x) dfluctN (y)
4c

=

N
X
i=1

g(ηi ) +

¨
X



(η )

g(x − y) δx(ηi i ) δxjj − δxi δxj

i6=j

+ 2N
´

N
X

¨







g(x − y) δxi − δx(ηi i ) µ.

i=1

´
(η )
(η )
Let us now observe that g(x − y)δxi i (y), the potential generated by δxi i is equal to g(x −(η )y)δxi outside of B(xi , ηi ), and smaller otherwise. Since its Laplacian is −cd δxi i , a negative
measure, this is also a superharmonic function, so by the maximum principle, its value ata point xj is larger or equal to its average on a sphere centered at xj . Moreover, outside
B(xi , ηi ) it is a harmonic function, so its values are equal to its averages. We deduce fromthese considerations, and reversing the roles of i and j, that for each i 6= j,
ˆ
ˆ
ˆ
(ηj )
(ηi ) (ηj )g(x − y)δxi δxj ≤ g(x − y)δxi δxj ≤ g(x − y)δxi δxj

14

SYLVIA SERFATY

with equality if B(xi , ηi ) ∩ B(xj , ηj ) = ∅. We conclude that the second term in the right-handside of (3.15) is nonnegative and equal to 0 if all the balls are disjoint. Finally, by the above´´(η )considerations, since g(x − y)δxi i coincides with g(x − y)δxi outside B(xi , ηi ), we may
rewrite the last term in the right-hand side of (3.15) as
N ˆX
2N(g(x − xi ) − g(ηi ))dµ.
i=1

B(xi ,ηi )On the other hand, recalling (3.9), if µ ∈ L∞ then we have
N
N ˆ
X
X
ηi2 .
fηi dµ ≤ Cd kµkL∞(3.16)
i=1

Rd

i=1

Indeed, it suffices to observe that
ˆ r
ˆ η
ˆ
d−1
g0 (r)rd dr,(g(r) − g(η))rdr = −fη =(3.17)
B(0,η)

00

with an integration by parts (and an abuse of notation, i.e. viewing g as a function on R)
and using the explicit form of g it follows thatˆ
(3.18)|fη | ≤ Cd η 2 .
B(0,η)



By the definition (3.9), we thus have obtained the result.

3.3. The case d = 1. In the case g(x) = − log |x| in dimension 1 (1.2), or in the Riesz cases(1.5), g is no longer the Coulomb kernel, so the formal computation (3.11) does not work.
However g is in the case (1.2) the kernel of the half-Laplacian, and it is known that the halfLaplacian can be made to correspond to the Laplacian by adding one extra space dimension.
In the same way, in the case (1.5), g is the kernel of a second order local operator, after adding
one extra space dimension. In other words, in the case (1.2) we should imbed the space RµV
into the two-dimensional space R2 and consider the harmonic extension of HN, defined in(3.6), to the whole plane. That extension will solve an appropriate Laplace equation, andwe will reduce dimension 1 to a special case of dimension 2. An analogue procedure, due toCaffarelli-Silvestre [CaffSi] applies to the case (1.5). This is the approach that was proposedin [SS6] for the Log1 case and in [PetSer] for the Riesz case.
Let us now get more specific about the extension procedure in the case Log1. We view R
as identified with R × {0} ⊂ R2 = {(x, y), x ∈ R, y ∈ R}. Let us denote by δR the uniform
measure on R × {0}, i.e. such that for any smooth ϕ(x, y) (with x ∈ R, y ∈ R) we have
ˆ
ˆ
ϕδR =
ϕ(x, 0) dx.
R2R

Let us still consider µ a measure on R (such as µV the equilibrium measure on R associatedto IV as in Theorem 1).
Given x1 , . . . , xN ∈ R, as explained above we identify them with the points (x1 , 0), . . . , (xN , 0)µµ2
in R2 , and we may then define the potentials HNand truncated potentials HN,~
η in R by
µ
HN
=g∗

N
X
i=1

!

δ(xi ,0) − N µδR

µ
HN,~
η =g∗

N
X
(η)

!

δ(xi ,0) − N µδR .

i=1

MICROSCOPIC DESCRIPTION OF LOG AND COULOMB GASES

15

Since g is naturally extended to a function in R2 , these potentials make sense as functions inµR2 and HNsolves
µ
− ∆HN
= 2π

(3.19)NX!

δ(xi ,0) − N µδR .

i=1
µ
HN

is nothing else than the harmonic extension to R2 , away from the real axis, of thepotential defined in dimension 1 by the analogue of (3.6). This is closely related to theStieltjes transform, a commonly used object in Random Matrix Theory (more precisely theµgradient of HN
is like the Stieltjes transform).
´µ 2
The proof of Lemma 3.2 then goes through without change, if one replaces Rd |∇HN,~η|´µ 2
with R2 |∇HN,~η | .
3.4. The electric energy controls the fluctuations. Since (3.7) (resp. (3.19)) holds,µ
we can immediately relate the fluctuations to HN, and via the Cauchy-Schwarz inequality,
control them by the electric energy.
Proposition 3.3. Let ϕ be a compactly supported Lipschitz function in Rd supported in U ,and µ be a bounded probability density on Rd . Let ~η be a N -tuple of distances such that
~ N , we haveηi ≤ N −1/d , for each i = 1, . . . , N . For each configuration X
ˆ


1
µ
1− 1d
k
(3.20)
ϕ fluctµN ≤ Ck∇ϕkL∞ |U | 2 k∇HN,~
2 (U ) + N
Lη
Rd

where C depends only on d.
Proof. In the 1D logarithmic case, we first extend ϕ to a smooth compactly supported test
function in R2 coinciding with ϕ(x) for any (x, y) such that |y| ≤ 1 and equal to 0 for |y| ≥ 2.
In view of (3.7) (resp. (3.19)) and applying Cauchy-Schwarz, we have
ˆ X
ˆ
N

1
1
µV
µV
(ηi )
δxi − N µV ϕ =· ∇ϕ ≤ C|U | 2 k∇ϕkL∞ k∇HN,~(3.21)∇HN,~ηη kL2 (U ) ,
cd Rd
i=1
(resp. with an integral over U × R in the 1D log case). Moreover, since ηi ≤ N −1/d , we have
!
ˆ
ˆ X
N
N

X

1
µ
(ηi )
δxi − N µ ϕ =(δxi − δx(ηi i ) ) ϕ ≤ N 1− d k∇ϕkL∞ .
(3.22)
fluctN −
i=1

i=1



The result follows.
Corollary 3.4. Applying this with ηi = N −1/d , we deduce in view of Lemma 3.2 thatˆ
(3.23)
ϕ fluctµN
Rd




1

1
1

2
µ ~
1−
N

2
d
+N
Ck∇ϕkL∞ |Supp ϕ| FN (XN ) + d log N + CN kµkL∞


≤
1


1
2
1

µ ~

Ck∇ϕkL∞ |Supp ϕ| 2 FN
(XN ) + N 2− d kµkL∞ 2 + N 1− dwhere C depends only on d and kµkL∞ .

for Log1, Log2
for Coul,

16

SYLVIA SERFATY

3.5. Consequences for the energy and partition function. Thanks to the splitting, we
can expand the energy to next order. For instance, combining (3.2) and Lemma 3.2, choosing
again ηi = N −1/d , we easily obtain~N
Corollary 3.5 (First energy lower bound). If µV ∈ L∞ , then for any X(3.24)~ N ) ≥ N 2 IV (µV ) + 2N
HN (X

N
X



ζ(xi ) −i=1

N
log N 1Log1,Log2 − CkµV kL∞ N 2−2/d ,dwhere C depends only on d.
Let us show how this lower bound easily translates into an upper bound for the partition
function ZN,β in the case with temperature. In the cases Coul it is better to normalize theenergy differently and define(3.25)

1

PN,β =ZN,ββe− 2 Nmin( 2 −1,0)
~N)
d
HN (X

~N
dX

and define ZN,β accordingly. Note that in the case Log1, Log2, this does not change anything.
Corollary 3.6 (An easy upper bound for the partition function). Assume that V is continuous, such that µV exists, satisfies (A3) and has an L∞ density. Then for all β > 0, and for
N large enough, we haveβ2βN log N 1Log1,Log2 + C(1 + β)Nlog ZN,β ≤ − N min(2, d +1) IV (µV ) +2
4

where C depends only on µV and the dimension.
To prove this, let us state a lemma that we will use repeatedly and that exploits assumption(A3).
Lemma 3.7. Assume that V is continuous, such that µV exists, and satisfies (A3). We haveˆ
(3.26)

e

lim

N →+∞

−βN

!1

PN

ζ(xi )
i=1

N

~N
dX

= |ω|

(Rd )N

where ω = {ζ = 0}.
Proof. First, by separation of variables, we have
ˆ
e

−βN

PN

ζ(xi )
i=1

ˆ

!1

~N
dX

(Rd )NN

e−βN ζ(x) dx.

=
Rd

Second, we recall that since µV is a compactly supported probability measure, hµV must
asymptotically behave like g(x) as |x| → ∞, thus ζ = hµV + V2 − c grows like g(x) + 12 V − c.
´The assumption (A3) thus ensures that for N large enough, e−βN ζ(x) dx < +∞. Moreover,by definition of ω,
e−βN ζ → 1ωas N → +∞pointwise and monotonically, and ω has finite measure in view of the growth of hµV and thusof ζ. The monotone convergence theorem allows to conclude.
MICROSCOPIC DESCRIPTION OF LOG AND COULOMB GASES

17

Proof of the corollary. Inserting (3.24) into the definition of ZN,β we are led to
β

2
2
β
log ZN,β ≤ N min( d −1,0) − N 2 IV (µV ) +N log N 1Log1,Log2 + CβN 2− d
2
2d
ˆ

PN
~N .
+ log
e−N β i=1 ζ(xi ) dX


(3.27)



Using Lemma 3.7 to handle the last term, we deduce thatlog ZN,β ≤ Nmin( 2d −1,0)



β

2
β
− N 2 IN (µV ) +N log N 1Log1,Log2 + CβN 2− d22d
+ N (log |ω| + oN (1))
which gives the conclusion.
The converse inequality to Corollary 3.6 actually holds and we have

Proposition 3.8. Assume that V is continuous, such that µV exists, satisfies (A3) and has
an L∞ density. Then for all β > 0, and for N large enough, we have
(3.28)

β
2β
N log N 1Log1,Log2 + O(N ).
log ZN,β = − N min(2, d +1) IV (µV ) +22dwhere O(N ) depends only on β, µV and the dimension.
One of the final goals on this course will be to present a more precise result with theconstant in the order N term identified and characterized variationally. This will also provide
information on the behavior of the configurations at the microscale. Anticipating the result,
we will be able to show that in the logarithmic cases, log ZN,β has an expansion where the V
dependence can be decoupled:(3.29)

β N log Nβ− N C(β, d)log ZN,β = − N 2 IV (µV ) +
2
2
d
ˆ
β
−N 1−
µV (x) log µV (x) dx + o((β + 1)N ),2dΣ

where C(β, d) is a constant depending only on β and the dimension (1 or 2). Obtaining (3.28)
alone without an explicit constant is easier, and we will assume it for now.
In the sequel, it is convenient to use the splitting to rewrite(3.30)~N) =dPN,β (XPNµmin( 2 −1,0)1~ N )+2Nd− β2 NFNV (Xζ(xi )~N
i=1edX
KN,β (µV , ζ)

where for any function ξ growing sufficiently fast at infinity, KN,β (µ, ξ) is defined asˆPNmin( 2 −1,0)
µ ~
β
d
FN
(XN )+2N
ξ(xi )~N.
i=1(3.31)
KN,β (µ, ξ) = e− 2 NdX
In view of what precedes, we have(3.32)β

KN,β (µV , ζ) = ZN,β e 2 N

min( 2 +1,2)d
IV (µV ).

18

SYLVIA SERFATY

3.6. Consequence: concentration bounds. With Proposition 3.8, we deduce an upperbound on the exponential moments of the electric energy FN :Corollary 3.9. We have, for some constant C depending on β and V ,log EPN,β exp(3.33)



β
4



2

~N) + (N min( d −1,0) FNµV (XNlog N )1Log1,Log2
d



≤ CN.

Proof. We may write
β min( 2 −1,0) µV ~d
EPN,β expN
FN (XN )
4
!!
ˆ
N
X
1
β
µV ~
min( 2d −1,0)
~N
=
exp −
N
(FN (XN ) + 2N
2ζ(xi ))dX
KN,β (µV , ζ)
4
i=1






=

KN, β (µV , 2ζ)2KN,β (µV , ζ).

Taking the log and using (3.32), (3.28) to expand both terms up to order N yields theresult.

As proved in [Leb2] in the case Log2(and as should be true in higher dimensions as well)this control can also be improved into a local control at all mesoscales: by this we mean thecontrol


 ˆ
β
2
|∇HN,~η |≤ C|U |N,
(3.34)
log EPN,β exp
4 U
for suitable ~η and cubes U of sidelength N −α , α < 12 .
Combining (3.33) with (3.23), we may then easily obtain concentration bounds for thefluctuations:
Corollary 3.10 (Concentration of fluctuations). For any Lipschitz function ϕ, we have

(3.35)

log EPN,β e´2µt( ϕfluctNV )≤ CN

where the constant C depends on t, β and ϕ.
This already gives us a good control on the´fluctuations: by√ Markov’s inequality we may
for instance deduce that the probability that fluctµNV ϕ  N is exponentially small (thecontrol on the exponential moment is a stronger information though), or other moderate deviations bounds (compare with the total mass of fluctµNV which is order N ). This also improveson the a-priori concentration bound in N log N generally obtained in the one-dimensional
logarithmic case by the non-electric approach [BG1, MMS]. We also refer to [CHM] for a
recent improved result in the same spirit valid in all Coulomb cases.
This is however not the best control one can obtain. The rest of these notes will be devoted
to two things: obtaining a better control on the fluctuations when ϕ is assumed to be moreregular, and obtaining a full LDP at next order for empirical fields, providing the exact
constant in the order N term of (3.28).

MICROSCOPIC DESCRIPTION OF LOG AND COULOMB GASES

19

4. CLT for fluctuations in the logarithmic cases
In this section, we restrict our attention to the logarithmic cases Log1, Log2.
Let us define the fluctuations of the linear statistics associated to ξN as the random variableˆξ dfluctµNV ,(4.1)
FluctN (ξ) :=R2

where fluctµNV

is as in (3.1). The goal of this section is to present a result that FluctN (ξ)
converges in law to a Gaussian random variable with explicit mean and variance. This isachieved by proving that the Laplace transform of FluctN (ξ) converges to that of the Gaussian.
This approach was pioneered in [Jo] and immediately leads to dealing with a Coulomb gas
with potential V replaced by Vt = V + tξ with t small. We then follow here the approachof [LS2,BLS] where we use a simple change of variables which is a transport map between theequilibrium measure µV and the equilibrium measure µVt for the perturbed potential. Notethat the use of changes of variables in this context is not new, cf. [Jo, BG1, Sh, BFG]. In our
approach, it essentially replaces the use of the so-called “loop" or Dyson-Schwinger equations.
4.1. Reexpressing the fluctuations as a ratio of partition functions. We start byintroducing the notation related to the perturbed potential and equilibrium measure.
Definition 4.1. For any t ∈ R, we define• The perturbed potential Vt as Vt := V + tξ and the perturbed equilibrium measureµ Vt .
• The next-order confinement term ζt := ζVt , as in (2.6).
µ~ N ) as in (3.3).
• The next-order energy FNVt (X• The next-order partition function KN,β (µVt , ζt ) as in (3.31).
Lemma 4.2 (Reexpressing the Laplace transform of fluctuations). For any t ∈ R we haveβ(4.2) EPN,β exp − N tFluctN (ξ)
2
ˆ



KN,β (µVt , ζt )β 2
exp − N IVt (µVt ) − IV (µV ) − t ξdµV.
=
KN,β (µV , ζ)
2






Proof. First, we notice that, for any t in R


Vt
ZN,β
β
β 2
exp(− N tFluctN (ξ)) =
exp
N t
2
ZN,β
2



(4.3)

EPN,β

ˆ



ξ dµV

Using the splitting formula (3.2) and the definition of KN,β as in (3.31) we see that for any t
(4.4)

Vt
KN,β (µVt , ζt ) = ZN,βexp

thus combining (4.3) and (4.4), we obtain (4.2).



β 2
N IVt (µVt ) ,2To compute the limit of the Laplace transform of FluctN (ξ) we will just need to apply the
2sformula (4.2) with t = − βN
for some arbitrary number s, hence t will indeed be very small.
We will see below that the term in the exponential in (4.2) is computable in terms of ξ andK(µVt ,ζt )then there will remain to study the ratio of partition functions KN,β.
N,β (µV ,ζ)20

SYLVIA SERFATY

4.2. Transport and change of variables. Our method introduced in [LS2, BLS] is basedon the construction of a transport φt such that φt #µV = µVt , with φt = Id + tψ. In fact thetransport chosen will not satisfy exactly φt #µV = µVt but will satisfy it approximately as
t → 0, and in these notes will neglect the error for simplicity.
In order to construct the correct transport map, one first needs to understand the perturbed
equilibrium measure µVt , or in other words, to understand how µV varies when V varies. In
the case Log1, the dependence of µV on V is indirect, but well understood. At least in the“one-cut case" (i.e. when the support of µV is a single interval), the right transport map (ast → 0) is found by inverting with respect to ξ a so-called “master operator" Ξ, which alreadyarose in the Dyson-Schwinger approach [BG1, BG2, Sh, BFG].
In dimension 2 (and in fact in all Coulomb cases), the perturbed equilibrium measure ist∆ξ, andeasy to compute when ξ is supported in Σ: one can check that it is µVt = µV − 2πβa correct transport map is Id + tψ with ψ = − cd∇ξµV . The case where ξ is not supported in Σ
i.e. has a support intersecting ∂Σ is much more delicate: one needs to understand how ∂Σ
is displaced under the perturbation. This is described precisely (in all dimensions) in [SeSe].
The PDE approach used there replaces Sakai’s theory used in [AHM], which is restricted tothe two-dimensional analytic setting.
In the interior (two-dimensional) Coulomb case, it follows from a direct computation basedon the exact formula for µVt that
ˆ
ˆ


s2
β
|∇ξ|2 .
− N 2 IVt (µVt ) − IV (µV ) − t ξdµV =
(4.5)
lim
2s
2
2πβ
2
N →∞,t=− βNR´
case, thanks to the analysis of [SeSe], we find that |∇ξ|2 gets replaced by´In the generalΣ
2ΣR2 |∇ξ | , where ξ denotes the harmonic extension of ξ outside Σ.
In the one-dimensional case, a more delicate computation based on the above facts reveals
that
ˆ
ˆ


β
s2
(4.6)
lim
− N 2 IVt (µVt ) − IV (µV ) − t ξdµV =ξ 0 ψdµV ,2s22
N →∞,t=− βN
where ψ is the transport map defined above. We have thus identified the limit of the exponential term in (4.2) and now turn to the ratio of partition functions.
Once the (approximate) transport has been constructed and proven to be regular enough(it is typically once less differentiable than ξ) and such that we approximately have ζt ◦φt = ζ,using the change of variables yi = φt (xi ) we find
ˆ



exp −

(4.7) KN,β (µVt , ζt ) =NXβ  µVt~ N )) + 2N
FN (Φt (Xζt ◦ φt (xi )
2
i=1

+
ˆ
=

N
X



~N
log | det Dφt |(xi ) dX

i=1



exp −

N
N
 X

X
β  µVt~ N )) + 2N
~N.
FN (Φt (Xζ(xi ) +log | det Dφt |(xi ) dX
2
i=1
i=1

Thus
PN
µV
µV ~
β
KN,β (µVt , ζt )t~
= EPN,β e− 2 (FN (Φt (XN ))−FN (XN ))+ i=1 log | det DΦt |(xi )KN,β (µV , ζ)


(4.8)



MICROSCOPIC DESCRIPTION OF LOG AND COULOMB GASES21µ~ N )) − F µV (X~ N ).
hence we need to evaluate the difference of energies FNVt (Φt (XN

4.3. Energy comparison. Let us present that computation in the Log1 case for simplicity,
knowing that it has a more complex analogue in the Log2 case.
~ N ∈ RN , and any ψ ∈ C 2 (R), defining
Lemma 4.3. For any probability density µ, any Xc¨ψ(x) − ψ(y)~ N , ψ, µ] =
(4.9)
A[X
dfluctµN (x) dfluctµN (y)x−y~ N ) = (φt (x1 ), · · · , φt (xN )), we haveand letting Φt (X(4.10)~ N )) − F µ (X~N) −FNφt #µ (Φt (X
N

N
X
i=1

log φ0t (xi ) +t~ N , ψ, µ]
A[X
2




~ N ) + N log N .
≤ Ct2 FNµ (XThis is the point that essentially replaces the loop equations.
Proof. We may write~ N )) − F µ (X~N)
FNφt #µ (Φt (X
N
ˆ
N
N

 X
X
δφt (xi ) − N φt #µ (y)δφt (xi ) − N φt #µ (x)=−log |x − y|
R2 \4

+
ˆ

i=1

i=1

ˆ

R2 \4

=−

log
R2 \4

log |x − y|dfluctµN (x)dfluctµN (y)|φt (x) − φt (y)|dfluctµN (x)dfluctµN (y)|x − y|

ˆ

=−

log
R2

N
X
|φt (x) − φt (y)|log |φ0t (xi )|.
dfluctµN (x)dfluctµN (y) +|x − y|i=1

Using that by definition φt = Id + tψ where ψ is in Cc2 (R), and the fact thatlog(1 + tx) − tx
≤ CKt2
C 2 (K)
for all compact sets K, some constant CK and t small enough, we get by the chain rulelog

|φt (x) − φt (y)|ψ(x) − ψ(y)=t
+ t2 εt (x, y),|x − y|x−ywith kεt kC 2 (R2 ) uniformly bounded in t. Applying Proposition 3.3 twice, we get that¨~ N ) + N log N .
εt (x, y)dfluctµN (x)dfluctµN (y) ≤ Ct2 FNµ (XThis yields the result in the Log1 case.

22

SYLVIA SERFATY

4.4. Computing the ratio of partition functions. Inserting the result of Lemma 4.3 (orits two-dimensional analogue) into (4.8) we obtain(4.11)

KN,β (µVt , ζt )KN,β (µV , ζ)

= EPN,β exp (1 −

N
β X
βt ~
)
log | det Dφt |(xi ) + A[XN , ψ, µV ]2d i=14~ N ) + N log N ))+ O(t2 (FNµV (X




β
βt ~
)FluctN [log | det Dφt |] + A[XN , ψ, µV ]2d4~ N ) + N log N ))+ O(t2 (F µV (X= EPN,β exp (1 −Nβ× e(1− 2d )N´

log | det Dφt |dµV.

Let us now examine the terms in this right-hand side.
First, remembering that φt #µV = µVt (approximately), we have det Dφt = µVµV◦φt andtthus, by definition of the push-forward,ˆˆˆ
(4.12)
log | det Dφt |dµV = log µV dµV − log µVt dµVt ,which is of order t i.e. O( Ns ), as N → ∞. More precisely, we may compute that in the Log1
case
ˆ
ˆ

 ˆ



β
β 2s
(4.13)lim
1−NµV log µV − µVt log µVt = − 1 −ψ 0 dµV2s22 β
N →∞,t=− βN

´
s
1
1
and in the Log2 interior case the analogue is 2π−∆ξ log ∆V .
β
4There now remains to evaluate all the terms in the expectation. We note that by the
Cauchy-Schwarz inequality, we have for instance(4.14)

log E(ea+b+c ) ≤ log E(e4a ) + log E(e4b ) + log E(e4c ).

First, by (3.33) and Hölder’s inequality we have the control
22~ N ) + N log N ))) = O( s N ) = O( s )log EPN,β exp(O(t2 (FNµV (XN2N(up to changing β in the formula), with the choice t = O(s/N ).
Since φt = Id + tψ with ψ regular enough, we find by combining (3.23) and (3.33) that√slog EPN,β (FluctN [log | det Dφt |]) ≤ O(t N ) = O( √ ).
N
Next, we turn to the term A (which we call anisotropy), which is the most delicate one.
There are two ways to handle this term and conclude. The first is direct and works in the
one-dimensional one-cut regular case, i.e. when the operator Ξ that finds the right transport
map is always invertible. This is described in the appendix of [BLS] and ressembles themethod of [BG1]. The second way works in all logarithmic (one-dimensional possibly multicut or critical, and two-dimensional) and calls instead the information on log ZN,β obtained in(3.29). It is described in [BLS] in the one-dimensional case and [LS2] in the two-dimensionalcase. Let us now present each.

(4.15)

MICROSCOPIC DESCRIPTION OF LOG AND COULOMB GASES

23

4.5. Conclusion in the one-dimensional one-cut regular case. We note that the term
A can be seen as being essentially also a fluctuation term withˆˆ
~ N , ψ, µV ] = f (x)dfluctN (x)(4.16)A[Xf (x) :=
ψ̂(x, y)dfluctN (y)where

ψ(x) − ψ(y)x−yis as regular as the regularity of ξ allows (note that this fact is only true in dimension 1 andso the argument breaks in dimension 2!). Using Proposition 3.3 twice, we obtain(4.17)
ˆψ̂(x, y) :=k∇f kL∞ ≤~ N ) + N log N + CN
∇x ψ̂(x, y)dfluctN (y) ≤ Ck∇x ∇y ψ̂kL∞ FNµV (Xand~ N , ψ, µV ]| =
|A[X

ˆ



~ N ) + N log N + CN
f (x)dfluctN (x) ≤ Ck∇f kL∞ FNµV (X

1

2

1
2





~ N ) + N log N + CN .
≤ Ckψ̂kC 2 FNµV (X
In view of (3.33), we deduce thatβ
~ N , ψ, µV ] ≤ O( s N ) = O(s).
EPN,β − tA[X
2
N
K

(µ

,ζ )

Vt t
Combining all these elements and inserting them into (4.11), we deduce that KN,βN,β (µV ,ζ)is of order 1 as N → ∞, and inserting into (4.2) together with (4.5)-(4.6) we obtain that
EPN,β (esFluctN (ξ) ) is bounded (in terms of s and ξ) as N → ∞. We may then bootstrap thisinformation by applying it to f , first improving (4.17) intoˆ
kDk f kL∞ ≤Dxk ψ̂(x, y)dfluctN (y)~ N ) + N log N + CN
≤ CkDxk ∇y ψ̂kL∞ FNµV (X12

obtaining (with Hölder’s inequality) if k is large enough, that√slog EPN,β etFluctN (f ) ≤ Ct N = O( √ ).
N
We then conclude that in fact
β~(4.18)
log EPN,β − tA[XN , ψ, µV ] ≤ oN (1).
2
Combining all the previous relations and (4.14), (4.5)–(4.6), and inserting into (4.2), we
conclude that
lim log EPN,β esFluctN (ξ) = mξ s + vξ s2
N →∞where mξ is the mean and vξ is a variance given by the explicit formulae above.
We have thus obtained that the Laplace transform of FluctN (ξ) converges to that of a
Gaussian of mean mξ and variance vξ , concluding the proof in the one-dimensional one-cut
regular case. Note that reinserting into (4.11) we have in fact obtained a precise expansion
K(µVt ,ζt )d, which also allows to directly evaluate dt
of KN,βlog KN,β (µVt , ζt ) up to order o(N ).
t=0N,β (µV ,ζ)24

SYLVIA SERFATY

But this is valid for all variations of the type V + tξ of the potential, hence by interpolating
between two potentials V0 and V1 which are such that their equilibrium measures are both
one-cut regular, we can compute log KN,β (µV1 , ζ1 ) − log KN,β (µV0 , ζ0 ) up to an error o(N ) andobtain effectively the formula (3.29). Moreover, since all the terms arising in the exponent in(4.11) are essentially fluctuations, the reasoning can be bootstrapped by inserting the CLTdresult and obtaining an expansion to higher order of dt
log KN,β (µVt , ζt ). This is possible
t=0
because the operator Ξ is always invertible in the one-cut case, allowing to build the transport
map associated to the fluctuation of any function, provided that function is regular enough.
This way, one may obtain a relative expansion of log ZN,β with an error of arbitrary order,
provided the potentials are regular enough, which is what is found in [BG1].
4.6. Conclusion in the two-dimensional case or in the general one-cut case. In thesecases, we control the term A differently: instead we use the important relation (3.29) which
we have assumed so far, and which provides another way of computing the left-hand side of(4.11). Comparing these two ways applied to a small, fixed t, we find on the one-hand
ˆ

 ˆ

KN,β (µVt , ζt )βNµV log µV − µVt log µVt + o(N )log= 1−KN,β (µV , ζ)2dand on the other hand combining (4.11), (4.12), (4.15),
ˆ

 ˆ

KN,β (µVt , ζt )
β
log
= 1−
N
µV log µV − µVt log µVt
KN,β (µV , ζ)
2d


βt ~
+ log EPN,β exp A[XN , ψ, µV ] + o(N )4This implies thatβt ~log EPN,β exp A[XN , ψ, µV ] = o(N ),4and this is true for any t small enough. Applying then Hölder’s inequality leads to−s ~log EPN,β exp
A[XN , ψ, µV ] = o(1).
2NThis replaces (4.18) and allows to conclude. The variance is given by the right-hand sideof (4.5) and (4.6) respectively, the mean is given by the right-hand side of (4.13) or its
two-dimensional analogue.
In both cases, we have obtained the following
Theorem 3. Assume ξ is regular enough. In the one-dimensional multi-cut or critical cases,
assume in addition that ξ is in the range of Ξ, and an analogous condition in the twodimension case with several connected components. Then FluctN (ξ) converges in law to a
Gaussian random variable of explicit mean and variance.
Let us make several additional comments:• In dimension 1, this theorem was first proven in [Jo] for polynomial V and ξ analytic.
It was later generalized in [Sh, BG1, BG2, BL, LLW], still with strong assumptions on
ξ, and to the critical case in [BLS].
• If the extra conditions do not hold, then the CLT is not expected to hold. Rather,
the limit should be a Gaussian convoled with a discrete Gaussian variable, as shownin the Log1 case in [BG2].

MICROSCOPIC DESCRIPTION OF LOG AND COULOMB GASES

25´• In ´dimension 2, the precise form of the variance as C |∇ξ|2 (or more generallyC |∇ξ Σ |2 ), means that fluctN converges to a so-called Gaussian Free Field. This
result was proven for the determinantal case β = 2 in [RV] (for V quadratic) and[AHM] under analyticity assumptions. It was then proven for all β simultaneously
in [LS2] and [BBNY], with ξ roughly assumed to be C 4 . Note that the result also
applies in the case where ξ is supported at a mesoscale, i.e. ξN (x) = ξ(xN α ) withα < 21 (the analogous result in dimension 1 is proven in [?]). The results of [AHM,LS2]
are the only ones that also apply to the case where ξ is not supported inside Σ.
• This theorem is to be compared to Corollary 3.10. It shows that if ξ is smooth enough,the fluctuations FluctN (ξ) are in fact much smaller than could be expected. They are
typically of order 1, √to be compared with the sum of N iid random variables which
is typically or order N . This a manifestation of rigidity, which even holds down to
the mesoscales.
• Moderate deviation bounds similar to those of [BBNY1] can also easily be obtained
as a by-product of the method presented above.
5. The renormalized energy
The goal of this section is to define a “renormalized energy" or jellium energy, that willarise as a rate function for the next order Large Deviation Principle that will be presented
below.
It allows to compute a total Coulomb interaction for an infinite system of discrete point“charges" in a constant neutralizing background of fixed density m > 0. Such a system isoften called a (classical) jellium in physics.
Starting again from the equations defining the electric potentials, (3.7) and (3.19), we
first perform a blow-up at scale N 1/d (the typical lengthscale of the systems). Denoting
x0i = N 1/d xi the blown-up points of the configuration, and the blown-up equilibrium measureµV 0 (x) = µV (xN −1/d ), we then define for shortcut the “electric fields"
(5.1)

µV 0
EN = ∇HN

µV 0
=g∗
∇HN

N
X

!
0

δx0i − µV δRd ,i=1

where the measure δRd is needed only in the one-dimensional case, cf (3.19). This way, EN
solves the linear equation(5.2)− div EN = cd

N
X



δx0i − µV 0 δRd ,i=1

in which it is easy, at least formally, to pass to the limit N → ∞. Note that this is written
when centering the blow-up at the origin. When µV is continuous, the blown-up measure µV 0
then converges to the constant µV (0). If another origin x was chosen, the constant would beµV (x), the local density of the neutralizing background charge.
On the other hand, as N → +∞, the number of points becomes infinite and they fillup the whole space, their local density remaining bounded (by control on the fluctuations).
PdThis way N
i=1 δxi is expected to converge to a distribution C ∈ Config(R ), where for A a
Borel set of Rd we denote by Config(A) the set of locally finite point configurations in A orequivalently the set of non-negative, purely atomic Radon measures on A giving an integer
mass
to singletons. We mostly use C for denoting a point configuration and we will write CP
for p∈C δp .

26

SYLVIA SERFATY

We thus wish to define the energy associated to an electric field E solving an equation of
the form(5.3)− div E = cd (C − mδRd )in Rd+kwhere C ∈ Config(Rd ) and m is a nonnegative constant (representing the density of points),
and k = 1 in the case Log1 where we need to add one space dimension, and 0 otherwise. We
say that an electric field E is compatible with (C, m) if it satisfies (5.3). By formulating interms of electric fields, we have not retained the information that EN was a gradient.
5.1. Definitions. Let C be a point configuration, m ≥ 0 and let E be compatible with (C, m).
For any η ∈ (0, 1) we define the truncation of E as
(5.4)

Eη (x) := E(x) −X∇fη (x − p),p∈C

where fη is as in (3.9). Let us observe that


(5.5)

− div Eη = cd X

δp(η) − mδRd  .

p∈CThis procedure is exactly the same, at the level of the electric fields, as the truncation procedure described in the previous sections.
In the sequel, R still denotes the d-dimensional cubes [−R/2, R/2]d .
Definition 5.1. The (Coulomb) renormalized energy of E with background m is
(5.6)

W(E, m) := lim Wη (E, m)
η→0where we let(5.7)

1Wη (E, m) := lim sup d
R→∞ Rˆ|Eη |2 − mcd g(η).
R ×RkDefinition 5.2. Let C be a point configuration and m ≥ 0. We define the renormalizedenergy of C with background m as
(5.8)

W(C, m) := inf{W(E, m) | E compatible with (C, m)}with the convention inf(∅) = +∞.
The name renormalized energy (originating in Bethuel-Brezis-Hélein [BBH] in the context
of two-dimensional Ginzburg-Landau vortices) reflects the fact that the integral of |E|2 isinfinite, and is computed in a renormalized way by first applying a truncation and thenremoving the appropriate divergent part mcd g(η).
It is not a priori clear how to define a total Coulomb interaction of such a jellium system,
first because of the infinite size of the system as we just saw, second because of the lack of
local charge neutrality of the system. The definitions we presented avoid having to go throughcomputing the sum of pairwise interactions between particles (it would not even be clear howtothem), but instead replace it with (renormalized variants of) the extensive quantity´ sum|E|2 (see (3.11) and the comments following it).

MICROSCOPIC DESCRIPTION OF LOG AND COULOMB GASES

27
b

b

b
b

b

b
b
b
b

b
b

b

b

b

b
b b

bb

b

b
b

b

r

b

b

b
b b

b

b
b

b b

b b

b

b

b b
b

b

b
b

b
b
b

b

b

∼ n−1/d
b

b

b

0
b

b
b

b

b

r

b
b

b

b

x0

b

b

b

×n1/d
b

b
b

b
b

b
b

b

bb

b

b
b

b

b

1

b
b

b

b
b

b
b
b b

b
b

b
b
b

b
b

Σ

b

b

b

b

b

b b
b
b

b
b

b
b

b

b b
b
b
b
b

b

∼1

b
b

b
b
b
b
b

Figure 3. An arbitrary blown-up configuration
5.2. Scaling properties. The dependence in m can be scaled out as follows: If E ∈ Elecm ,we define σm E by


·
1/d−1
σm E := mE.
m1/d
We have σm E ∈ Elec1 and in the case Coul,(

W(E, m) = m2−2/d W(σm E, 1)
Wη (E, m) = m2−2/d Wηm1/d (σm E, 1)(5.9)
and in the cases Log1, Log2(

(5.10)

W(E, m) = mW(σm E, 1) − 2π
d m log mWη (E, m) = mWηm1/d (σm E, 1) − 2π
d m log m.

It also follows that(

(5.11)

W(C, m) = mW(σm C, 1) − 2π
d m log m2−2/dW(C, m) = m
W(σm C, 1)

for Log1, Log2
for Coul.

In view of the above scaling relations, it suffices to study W(·, 1). It is proven in [RouSe]that min W(·, 1) is finite and achieved for any d ≥ 2. Moreover, the minimum coincides withthe limit as R → +∞ of the minima of W(·, 1) on configurations that are (RZ)d -periodic (i.e.
that live on the torus TR = Rd /(RZ)d ) (with R ∈ N).

28

SYLVIA SERFATY

5.3. Partial results on the minimization of W, crystallization conjecture. We haveseen that the minima of W can be achieved as limits of the minima over periodic configurations(with respect to larger and larger tori). In the Log1 case, a convexity argument (for which werefer to [SS6, Prop. 2.3]) shows that the minimum is achieved when the points are equally
spaced, in other words for the lattice or crystalline distribution Z.
In higher dimension, determining the value of min W is an open question. The only questionthat we can answer so far is that of the minimization over the restricted class of pure latticeconfigurations, in dimension d = 2 only, i.e. vector fields which are gradient of functions
that are periodic with respect to a lattice Z~u + Z~v with det(~u, ~v ) = 1, corresponding toconfigurations of points that can be identified with Z~u + Z~v . In this case, we have :Theorem 4 (The triangular lattice is the minimizer over lattices in 2D). The minimum of W
over this class of vector fields is achieved uniquely by the one corresponding to the triangular“Abrikosov" lattice.
Here the triangular lattice means Z + Zeiπ/3 , properly scaled, i.e. what is called the
Abrikosov lattice in the context of superconductivity. This result is essentially a result of
number theory, proven in the 50’s.
One may ask whether this triangular lattice does achieve the global minimum of W. The
fact that the Abrikosov lattice is observed in superconductors, combined with the fact that
W can be derived as the limiting minimization problem of Ginzburg-Landau [SS4], justify
conjecturing this. It was recently proven in [BS] that this conjecture is equivalent to a
conjecture of Brauchart-Hardin-Saff [BHS] on the next order term in the asymptotic expansionof the minimal logarithmic energy on the sphere (an important problem in approximation
theory, also related to Smale’s “7th problem for the 21st century"), which is obtained byformal analytic continuation, hence by very different arguments.
In dimension d ≥ 3 the minimization of W even restricted to the class of lattices is an
open question, except in dimensions 8 and 24. Similarly, one may conjecture that in low
dimensions, the minimum of W is achieved by some particular lattice, see [BLe]. In large
dimensions, lattices are not expected to be minimizing.
~ N = (x1 , . . . , xN ) in (Rd )N , we
5.4. Renormalized energy for point processes. Given X~ 0 as the finite configuration rescaled by a factor N 1/drecall that we define X
N(5.12)~ 0 :=XNNX

δN 1/d xi .

i=1

It is a point configuration (an element of Config), which represents the N -tuple of particles~ N seen at the microscopic scale.
X~ N ] defined
The object we wish to describe is the limit of the tagged empirical field EmpN [Xasˆ
~ N ] := 1
(5.13)
EmpN [Xδ~ 0 dx,|Σ| Σ (x, θN 1/d x ·XN )where θx denotes the translation by −x. It is a probability measure on Σ × Config. For any~ 0 is an element of Config which represents the N -tuple of particles
x ∈ Σ, the term θN 1/d x · X
N~ N centered at x and seen at microscopic scale (or, equivalently, seen at microscopic scaleXand then centered at N 1/d x). In particular any information about this point configuration inMICROSCOPIC DESCRIPTION OF LOG AND COULOMB GASES

29~ 0 around x. We may
a given ball (around the origin) translates to an information about XN00~ as encoding the behavior of X
~ around x.
thus think of θN 1/d x · X
NN~ N ] will be a probability measure on Σ × Config, typically denoted
The limit of EmpN [XP̄ , whose first marginal is the (normalized) Lebesgue measure on Σ, and which is stationary
with respect to the second variable. The space of such probability measures is denoted as
Ps (Σ × Config). We may then define the renormalized energy for P̄ ∈ Ps (Σ × Config) as
ˆEP̄ x [W(·, µV (x))] dx,
W(P̄ , µ) =(5.14)
Σx
where P̄ is the disintegration with respect to the variable x of the measure P̄ (informally it

is P̄ (x, ·)).
5.5. Lower bound for the energy in terms of the empirical field. We may now relate
rigorously the energy HN to W by the following
Proposition 5.3. Assume ∂Σ is C 1 and µ has a continuous density on its support. Let~ N }N be a sequence of N -tuples of points in Rd . If the left-hand side below is bounded{X~ N ] converges to some P̄ in
independently of N , then up to extraction the sequence EmpN [X
Ps (Σ × Config), and we have~ N ) + N log N ≥ W(P̄ , µV ) in the cases Log1, Log2
 lim inf F µV (XNN →∞d(5.15)
2µV ~ lim inf N 1− d FN
(XN ) ≥ W(P̄ , µV ) in the cases Coul.
N →∞~ N let us define the tagged electric field process PN [X~ N ] (thenProof. For each configuration X~ N ) bywe will drop the Xˆ1~ N ](x, E) :=(5.16)PN [Xδ
dx,|Σ| Σ (x,θN 1/d x ·EN )where EN is as in (5.1). The result follows from relatively standard considerations, afternoting that by Fubini’s theorem we may write, for any R > 1,ˆˆ
ˆ1
µV 0 2(5.17)|∇HN,η | ≥ N
1y∈R|EN,η (N 1/d x + y)|2 dx dy
Rd |R |RdΣˆ= N |Σ| fR,η (E) dPN (x, E)
where fR,η is defined byˆ
1fR,η (E) =|Eη |2|R | Rµ
if E is of the form ∇HN
for some N -point configuration, and +∞ otherwise. It then suffices
to show that {PN }N is tight, and that up to extraction PN converges to some stationary
tagged electric field process which can be “projected down" to some P̄ ∈ Ps (Σ × Config), andtake the limit N → ∞, R → ∞, then η → 0 in (5.17).

6. Large Deviations Principle for empirical fields
We now wish to state a large deviations result at the level of the point processes for the
Gibbs measure (3.25), proven in [LS1].

30SYLVIA SERFATY

6.1. Specific relative entropy. We first need to define the analogue of the entropy at thelevel of point processes: the specific relative entropy with respect to the Poisson point process.
The Poisson point process with intensity m is the point process characterized by the fact thatfor any bounded Borel set B in Rd(m|B|)n −m|B|en!
where N (B) denotes the number of points in B. The expectation of the number of points in
B can then be computed to be m|B|, and one also observes that the number of points in two
disjoint sets are independent, thus the points “don’t interact".
For any m ≥ 0, we denote by Πm the (law of the) Poisson point process of intensity m in
Rd (it is an element of Ps (Config)). Let P be in Ps (Config). We define the specific relative
entropy of P with respect to Π1 as
1(6.1)
ent[P |Π1 ] := lim d ent[PR |Π1R ],R→∞ R
where PR , Π1R denote the restrictions of the processes to the hypercube R . Here, ent[·|·]
denotes the usual relative entropy of two probability measures defined on the same probability
space, namely
ˆdµ
ent[µ|ν] := logdµdνif µ is absolutely continuous with respect to ν, and +∞ otherwise.
P (N (B) = n) =Lemma 6.1. The following properties are known (cf. [RAS]) :(1) The limit in (6.1) exists for P stationary.
(2) The map P 7→ ent[P |Π1 ] is affine and lower semi-continuous on Ps (Config).
(3) The sub-level sets of ent[·|Π1 ] are compact in Ps (Config) (it is a good rate function).
(4) We have ent[P |Π1 ] ≥ 0 and it vanishes only for P = Π1 .
Next, if P̄ is in Ps (Σ × Config), we defined the tagged specific relative entropy asˆent[P̄ |Π1 ] :=ent[P̄ x |Π1 ]dx.
(6.2)
Σ

For any N ≥ 1 we define QN,β as
(6.3)



2
N
exp −N d βζ(xi ) dxiY ,dQN,β (x1 , . . . , xN ) :=´
2
i=1

Rd exp−N d βζ(x) dx

which exists by assumption, and we let Q̄N,β be the push-forward of QN,β by the map EmpN ,as defined in (5.13).
We also introduce the following constant:(6.4)

cω,Σ := log |ω| − |Σ| + 1,

where ω is the zero-set of ζ (see (2.6)).
The following result expresses how the specific relative entropy evaluates the volume of~ N near a reference tagged point process P̄ . It is a Large Deviations Principle“microstates" X
for the tagged empirical field, when the points are distributed according to a reference measureon (Rd )N where there is no interaction. This is a microscopic or so-called “type III" analogueof Sanov’s theorem for the empirical measures.

MICROSCOPIC DESCRIPTION OF LOG AND COULOMB GASES

31

Proposition 6.2. For any A ⊂ Ps (Σ × Config), we have
(6.5)

1
log Q̄N,β (A)N →∞ N1≤ lim sup log Q̄N,β (A) ≤ − inf ent[P̄ |Π1 ] − cω,Σ .
P̄ ∈Ā
N →∞ N− inf ent[P̄ |Π1 ] − cω,Σ ≤ lim inf
Å∩Ps,1

With this result at hand we may expect the LDP result we are searching for (or at leastan LDP upper bound): indeed, the splitting (3.2) and the rewriting (3.30) may be combined
to Proposition 5.3, which bounds from above the exponent, while Proposition 6.2 allows toevaluate the volume. The hard part is to obtain the LDP lower bound, which involves instead
an upper bound for the energy, together with a lower bound for the volume of configurations.
6.2. Statement of the main result. For any β > 0, we define a free energy functional F β
as
β
(6.6)
F β (P̄ ) := W(P̄ , µV ) + ent[P̄ |Π1 ].
2
For any N, β we let PN,β be push-forward of the canonical Gibbs measure PN,β by thetagged empirical field map EmpN as in (5.13) (in other words, PN,β is the law of the tagged
empirical field when the particles are distributed according to PN,β ).
Theorem 5 (Large Deviation Principle for the tagged empirical fields [LS1]). Under suitable
assumptions on the regularity of ∂Σ and µV , for any β > 0 the sequence {PN,β }N satisfies alarge deviation principle at speed N with good rate function F β − inf F β .
In particular, in the limit N → ∞, the law PN,β concentrates on minimizers of F β .
• One readily sees the effect of the temperature: in the minimization there is a competition between the term W based on the renormalized energy, which is expected
to favor very ordered configurations, and the entropy term which in contrast favors
disorder (the entropy is minimal for a Poisson point process). As β → ∞, we expect
configurations to concentrate on minimizers of W hence to crystallize on lattices (indimension 1, there is a complete proof [Leb1]), in low dimensions. As β → 0, thelimiting point processes (if they exist) converge to the Poisson point process.
• This result is valid in the Log1, Log2, Coul and also Riesz cases.
• In the case Log2, it is proven in [Leb2] that the similar result holds down to themesoscales.
• The corresponding, simpler, result for minimizers of HN (which are of interest on theirown, as seen in Section 1) can be proven as well. More precisely, the lower bound of
Proposition 5.3 is sharp, and the scaling property of W allow to obtain(6.7)~ N ) = N 2 IV (µV )min HN (X
+



ˆ






− Nd log N + N





1
N 1+ d cd,smin W(·, 1)

s

− 12

ˆΣ

µV log µV +

Σ

1+ s

1
min W(·, 1)
cd

in the cases Log1, Log2

µV d in the cases Coul, Riesz.

~ N minimizes HN then, up to extraction EmpN [X~ N ] converges to a
Moreover, if X
minimizer of W(·, µV ). These are results of [SS5, SS6, RouSe, PetSer]. More precise

32

SYLVIA SERFATY

results including rigidity down to the microscopic scales and separation of points canbe found in [RNSe, PRN].
As is usual and as alluded to before, as a by-product of the large deviation principle weobtain the order N term in the expansion of the partition function.
Corollary 6.3 (Next-order expansion and thermodynamic limit). Under the same assumptions, we have, as N → ∞:• In the logarithmic cases Log1, Log2ββ N log N
log ZN,β = − N 2 IV (µV ) +− N min F β + N (|Σ| − 1) + o((β + 1)N ).
22d• In the Coulomb cases Coul
2
β
(6.9)
log ZN,β = − N 1+ d IV (µV ) − N min F β + N (|Σ| − 1) + o((β + 1)N ).
2
The scaling properties allow to rewrite the previous expansion in the cases Log1, Log2 as(6.8)(6.10)ββ N log N
log ZN,β = − N 2 IV (µV ) +− N C(β, d)
2
2
d
ˆ
β
−N 1−
µV (x) log µV (x) dx + o((β + 1)N ),2dΣ

where C(β, d) is a constant depending only on β and the dimension d, but independent of thepotential.
In the particular case of Log1 with a quadratic potential V (x) = x2 , the equilibrium
measure is known to be Wigner’s semi-circular law and the limiting process at the microscopic
scale when centered around a point x ∈ (−2, 2) (let us emphasize that here there is no
averaging) has been identified for any β > 0 in [VV,KN1]. It is called√ the sine-β point process1
and we denote it by Sineβ (x) (so that Sineβ (x) has intensity 2π 4 − x2 ). For β > 0 fixed,
the law of these processes do not depend on x up to rescaling and we denote by Sineβ thecorresponding process with intensity 1.
A corollary of our main result is then a new variational property of Sineβ : it minimizes
the sum of β/2 times the renormalized energy (with background 1) and the specific relative
entropy.
6.3. Proof structure. As mentioned above, combining Propositions 5.3 and 6.2, we naturally obtain the LDP upper bound. Obtaining the LDP lower bound is significantly harder,
and is based on the following quasi-continuity of the interaction, in the following sense.
For any integer N , and δ > 0 and any P̄ ∈ Ps (Σ × Config), let us definen~ N ∈ (Rd )N ,(6.11) TN (P̄ , δ) := XN

(1− 2d )+~N) +FNµV (XoN
log N 1Log1,Log2 ≤ W(P̄ , µV ) + δ .
dProposition 6.4. Let P̄ ∈ Ps,1 (Σ × Config). For all δ1 , δ2 > 0 we have
(6.12)


1log QN,β {EmpN ∈ B(P̄ , δ1 )} ∩ TN (P̄ , δ2 ) ≥ −ent[P̄ |Π1 ] − cω,Σ .
N →∞ Nlim infMICROSCOPIC DESCRIPTION OF LOG AND COULOMB GASES

33

Let us compare with Proposition 6.2. The first inequality of (6.5) implies (taking A =B(P̄ , δ1 ) and using the definition of Q̄N,β as the push-forward of QN,β by EmpN ) that
1
log QN,β {EmpN ∈ B(P̄ , δ1 )} ≥ −ent[P̄ |Π1 ] − cω,Σ .
N →∞ Nlim inf

To obtain Proposition 6.4, which is the hard part of the proof, we thus need to show thatthe event TN (P̄ , δ2 ) has enough volume in phase-space near P̄ . This relies on the screening
construction which we now describe.
6.4. Screening and consequences. The screening procedure allows, starting from a given
configuration or set of configurations, to build configurations whose energy is well controlled
and can be computed additively in blocks of large microscopic size.
Let us zoom everything by N 1/d and work in the rescaled coincidence set Σ0 = N 1/d Σ. It
will be split into cubes K of size R with R independent of N but large. This can be done
except in a thin boundary layer which needs to be treated separately (for simplicity we will
not discuss this).
In a cube K (or in K × R in the Log1 case), the screening procedure takes as input a given
configuration whose electric energy in the cube is not too large,´ and replaces it by a better
configuration which is neutral (the number of points is equal to K µV 0 ), and for which there
is a compatible electric field in K whose energy is controlled by the initial one plus a negligible
error as R → ∞. The point configuration is itself only modified in a thin layer near ∂K, andthe electric field is constructed in such a way as to obtain E · ~ν = 0 on ∂K. The configurationis then said to be screened, because in some sense it then generates only a negligible electricfield outside of K.
Starting from a given vector field, we not only construct a modified, screened, electric field,
but a whole family of electric fields and configurations. This is important for the LDP lower
bound where we need to exhibit a large enough volume of configurations having well-controlledenergy.
Let us now get into more detail.
6.4.1. Compatible and screened electric fields. Let K be a compact subset of Rd with piecewise
C 1 boundary (typically, K will be an hyperrectangle or the support Σ), C be a finite point
configuration in K, µ be a nonnegative density in L∞ (K), and E be a vector field.
• We say that E is compatible with (C, µ) in K provided(6.13)− div E = cd (C − µ)in K.

• We say that E is compatible with (C, µ) and screened in K when (6.13) holds andmoreover(6.14)

E · ~ν = 0 on ∂K,
where ~ν is the outer unit normal vector.

´´
We note that by integrating (6.13) over K and using (6.14), we find that K C = K µ. In
other words, screened
with neutral ´systems, in which´ electric fields are necessarily associated´
the positive charge K C is equal to the negative charge K µ. In particular K µ has to be an
integer.

34

SYLVIA SERFATY

6.4.2. Computing additively the energy. Vector fields that are screened can be naturally“pasted" together: we may construct point configurations by elementary blocks (hyperrectangles) and compute their energy additively in these blocks. More´precisely, assume that space ispartitioned into a family of hyperrectangles K ∈ K such that K dµ0V are integers. We would
like to construct a vector field E (K) in each K such that(

(6.15)−div E (K) = cd (CK − µ0V ) in K
E (K) · ~ν = 0
on ∂K(where ~ν is the outer unit normal to K) for some discrete set of points CK ⊂ K, and with
ˆ|Eη(K) |2
K

well-controlled.
When solving (6.15), we could take E (K) to be a gradient, but we do not require it, in fact
relaxing this constraint is what allows to paste together the vector fields. Once the relations(6.15) are satisfied on each K, we may paste together the vector fields E (K) into a uniquevector field E tot , and thediscrete sets of points CK into a configuration C tot . The cardinality´
of C tot will be equal to Rd dµV 0 , which is exactly N .
We may thus obtain a configuration of N points, whose energy we may try to evaluate.
The important fact is that the enforcement of the boundary condition E (K) · ~ν = 0 on eachboundary ensures that− div E tot = cd C tot − µV 0

(6.16)

in Rd

holds globally. In general, a vector field which is discontinuous across an interface has a distributional divergence concentrated on the interface equal to the jump of the normal derivative,
but here by construction the normal components coincide hence there is no divergence createdacross these interfaces.
Even if the E (K) ’s were gradients, the global E tot is in general no longer a gradient. This
0 generated by thedoes not matter however, since the energy of the local electric field ∇HN
tot
0
finite configuration C (and the background µV ) is always smaller, as stated in
Lemma 6.5. Let Ω be a compact subset of Rd with piecewise C 1 boundary, let N ≥ 1, let~ N be in (Rd )N . We assume that all the points of X
~ N belong to K and that Ω ⊂ Σ. Let E beX
a vector field such that(

(6.17)−div E = cd
E · ~ν = 0

Then, for any η ∈ (0, 1) we haveN0i=1 δx0i − µVˆ

(6.18)
Rd
0

P



in Ω
on ∂Ω.

ˆ
0|∇HN,η
|2 ≤|Eη |2 .
KµV
Proof. The point is that ∇HN,η
is the L2 projection of any compatible E onto gradients, andthat the projection decreases the L2 norm. The truncation procedure can be checked not toaffect that property.
MICROSCOPIC DESCRIPTION OF LOG AND COULOMB GASES

We thus have

ˆ
Rd

35

ˆ
0
|∇HN,η
|2 ≤X|Eη(K) |2 ,K∈K K

and the energy FNµV (C tot ) can indeed be computed additively over the cells (at least for an
upper bound, which is precisely what we care about at this stage). This is the core of themethod: by passing to the electric representation, we transform a sum of pairwise interactions
into an additive (extensive) energy.
The screening theorem states that this program can be achieved while adding only a negligible error to the energy, modifying only a small fraction of the points, and preserving thegood separation. It also allows, when starting from a family of configurations, to build families of configurations which occupy enough volume in configuration space (i.e. losing only anegligible logarithmic volume).
6.5. Generating microstates and conclusion. To prove Proposition 6.4, we generate configurations “at random" whose volume is evaluated by Proposition 6.2, and then modify them
via the screening procedure.
Since we want the global configurations to approximate (after averaging over translations)a given tagged point process P̄ , we draw the point configuration in each hypercube K ∈ K
jointly at random according to a Poisson point process, and standard large deviations results
imply that enough of the averages ressemble P̄ . Using Proposition 6.2 they will still occupy
a volume bounded below by −ent[P̄m |Π1 ] − cω,Σ .
These configurations then need to be modified by the screening procedure, while checking
that these modifications do not alter much their phase-space volume, their energy, and keepthem close to the given tagged process P̄ . They also need to go through another modification,
which we call regularization and which separates pairs of points that are too close (while
decreasing the energy and not altering significantly the volume). This is meant to ensure thatthe error between Wη and W can be made small as η → 0 uniformly among the configurations.
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Courant Institute of Mathematical Sciences, New York University, 251 Mercer st, New
York, NY 10012
E-mail address: serfaty@cims.nyu.edu