Portugal. Math. (N.S.)
Vol. xx, Fasc. , 200x, xxx–xxx

Portugaliae Mathematica
c European Mathematical Society

Large Systems with Coulomb Interactions :Variational Study and Statistical Mechanics
Sylvia Serfaty

Abstract. Systems with Coulomb and logarithmic interactions arise in various settings:
an instance is the classical Coulomb gas which in some cases happens to be a randommatrix ensemble, another is vortices in the Ginzburg-Landau model of superconductivity,
where one observes in certain regimes the emergence of densely packed point vortices
forming perfect triangular lattice patterns named Abrikosov lattices, a third is the studyof Fekete points which arise in approximation theory. In this review, we describe tools
to study such systems and derive a next order (beyond mean field limit) “renormalizedenergy” that governs microscopic patterns of points. We present the derivation of thelimiting problem and the question of its minimization and its link with the Abrikosov
lattice and crystallization questions. We also discuss generalizations to Riesz interactionenergies and the statistical mechanics of such systems. This is based on joint works with
Etienne Sandier, Nicolas Rougerie, Simona Rota Nodari, Mircea Petrache, and Thomas
Leblé.
Mathematics Subject Classification (2010).82B05, 82B21, 82B26, 15B52.
Keywords. Coulomb systems, Coulomb gases, log gases, Large Deviations Principle,
Abrikosov lattices, Fekete points, random matrices.

1. Introduction and motivations
We are interested in the following class of energies
Hn (x1 , . . . , xn ) =Xg(xi − xj ) + nnXV (xi )

(1.1)

i=1

i6=j

where x1 , . . . , xn are n points in Rd and the interaction kernel g is given by either
g(x) =

1|x|d−2d ≥ 3,(1.2)

2

Sylvia Serfatyorg(x) = − log |x| in dimension d = 2.

(1.3)

Later we will also discuss generalizations tog(x) =

1|x|smax(0, d − 2) ≤ s < d, d ≥ 1,

(1.4)

or
g(x) = − log |x| in dimension d = 1,

(1.5)

that can be treated with slight modifications. We are interested in the asymptotics
n → ∞ of the minimum of Hn . One notes that in the cases (1.2)–(1.3), g is amultiple of the Coulomb kernel in dimension d, and there is a constant cd depending
only on d such that−∆g = cd δ0 ,(1.6)
where δ0 is the Dirac mass at the origin.
We now review various motivations for studying such systems.
1.1. Fekete points. Fekete points arise in interpolation theory as the points
minimizing interpolation errors for numerical integration [SaTo]. They are often
studied on manifolds, such as the d-dimensional sphere, and then correspond tosets of n points which maximizeY|xi − xj |.
i6=jEquivalently they minimize−X

log |xi − xj |.

i6=j

In Euclidean space, one also considers “weighted Fekete points” which maximizeYP|xi − xj |e−n i V (xi )i6=jor equivalently minimize
−

X
i6=j

log |xi − xj | + nnXV (xi )

i=1

which in dimension 2 corresponds exactly to the minimization of our Hamiltonian
Hn in the particular case (1.3). They also happen to be zeroes of orthogonal
polynomials, see [Si].

Coulomb systems

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
X
i6=j

1
|xi − xj |s(1.7)

for all possible s, hence a motivation for (1.4). On all these matters we refer to[SaTo], the review paper [SK, BHS] and the forthcoming monograph [BHS].
1.2. Statistical mechanics. The study of Hn is also interesting for understanding the associated Gibbs measuredPn,β (x1 , . . . , xn ) =

1
Zn,β

1

e− 2 βHn (x1 ,...,xn ) dx1 . . . , dxn(1.8)

where β > 0 represents an inverse temperature and Zn,β is the partition function of
the system, i.e. a number that normalizes Pn,β to a probability measure on (Rd )n .
This corresponds to the Gibbs measure of a classical “Coulomb gas system” (or alog gas in cases (1.5)–(1.3)) (cf. [Forr]), by extension we can also call it a “Rieszgas” in the case (1.4). Such systems have been studied in the physics literature[SM, JLM, LiLe, LN, PeSm]. They can be considered as a toy model for matter,with classical particles. As always with such statistical mechanics ensembles, one
would like to understand the behavior in terms of the temperature: are there
critical temperatures corresponding to phase transitions for which the nature of
the states changes?
1.3. Random matrix theory. The study of (1.8) has attracted a lot of attention
due to its connection with random matrix theory. As first noticed by [Wi, Dy], inthe particular cases (1.5)–(1.3) the Gibbs measure (1.8) also corresponds to the lawof the eigenvalues (which can be computed algebraically) of some famous randommatrix ensembles:•

when (1.3), with 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.

•

when (1.5), with 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 the Gaussian Unitary Ensemble.

•

when (1.5), β = 1 and V (x) = x2 /2, (1.8) is the law of the (real) eigenvaluesof an n × n real symmetric matrix with normal Gaussian iid entries. This iscalled the Gaussian Orthogonal Ensemble.

•

when (1.5), β = 4 and V (x) = x2 /2, (1.8) is the law of the eigenvalues of ann × n quaternionic symmetric matrix with normal Gaussian iid entries.

4

Sylvia Serfaty

One thus observes in these ensembles the phenomenon of “repulsion of eigenvalues”:
they repel each other logarithmically, i.e. like two-dimensional Coulomb particles.
The particular choice of β = 2 makes these determinantal point processes because then the law can be rewritten

2
Pn
1 Y
|xi − xj | e−n i=1 V (xi ) dx1 . . . dxn
Zn,β i<jwhere a square Vandermonde determinant appears. This allows to compute algebraically a lot of quantities in this particular case, such as the partition functions(when V is x2 ), the limiting processes at the microscopic scale, etc, and there isa large literature on this. In [BEY1, BEY2], Bourgade-Erdös and Yau manage tounderstand the case (1.5) for all β and general V , and they show the universality(after suitable rescaling) of the microscopic behavior and local statistics of thepoints, i.e. the fact that they are essentially independent of V .
1.4. Vortices in condensed matter physics. Interaction energies of the form
(1.1) in the case (1.3) also arise as effective interaction energies for vortices inmodels from condensed matter physics: the Ginzburg-Landau model of superconductivity and the Gross-Pitaevskii functionals for superfluids and Bose-Einsteincondensates. In this spirit, the mathematical study of such vortices started with
Bethuel-Brezis-Hélein [BBH] who studied the simplified functionalZ(1 − |u|2 )21|∇u|2 +Eε (u) =2 Ω
2ε2
where u is a function from a two-dimensional (bounded simply connected) domainΩ to the complex plane C, which is prescribed to take boundary values u = g withg a map from ∂Ω to S1 of nonzero topological degree n. Bethuel, Brezis and Hélein
analyzed minimizers of Eε under this boundary condition, and showed that they
have n zeroes (or vortices) of topological degree 1, at locations xε1 , . . . , xεn . These
points tend as ε → 0, to minimize a “renormalized energy”
XX
W (x1 , . . . , xn ) = −
log |xi − xj | +R(xi , xj )
i6=j

i,j

where R is a regular function depending on the boundary data g. They also provedthat
min Eε ∼ πn|log ε| + min W as ε → 0,
where the leading order term πn|log ε| corresponds to the “self-interaction” of all
the vortices, and the second order term min W governs the vortex locations.
The original Ginzburg-Landau model of superconductivity contains a gauge
and an applied magnetic field:Z1(1 − |u|2 )2Gε (u, A) =|∇A u|2 + |∇ × A − hex |2 +
.
(1.9)
2 Ω
2ε2

5

Coulomb systems

Here A : Ω → R2 is the gauge of the magnetic field, ∇A = ∇ − iA is the covariant
derivative, h := ∇ × A = ∂2 A1 − ∂1 A2 is the induced magnetic field in the sample.
The constant parameters are hex , the intensity of the external magnetic field, andε a material constant, which is often small. Associated to this functional are theGinzburg-Landau equations:

1
2
2


 −(∇A ) u = ε2 u(1 − |u| ) in Ω−∇⊥ h = hiu, ∇A ui in Ω(GL)h = hex on ∂Ω∇A u · ν = 0 on ∂Ω,where ∇⊥ denotes the operator (−∂2 , ∂1 ), ν is the outer unit normal to ∂Ω andha, bi is the scalar product in C as identified with R2 .
The analysis of [BBH] was first generalized to the model with gauge, still with
fixed boundary conditions, in [BR]. In the true physics model, vortices arise dueto the hex parameter, with no prescribed boundary data. In the experiments andphysics predictions, it is observed that when hex is above a first critical field Hc1 oforder |log ε|, then vortices start to appear. Their number increases as hex is further
increased, and they tend to form perfect triangular Abrikosov lattices, named after
the physicist Abrikosov who first predicted them.
Several of these features have been proven rigorously in a series of works on
the vortices in this Ginzburg-Landau model, which are summarized in [SS1]. (Inthat reference one can also find a detailed introduction to the functional, as well
as references to the mathematics and physics literature.) To analyze the vorticesin (1.9) one defines the vorticity of a configuration (u, A) asµ(u, A) = ∇ × hiu, ∇A ui + ∇ × A.
This is the gauge-invariant analogue of the standard vorticity, such as the one
defined in fluids. One can show that in the asymptotics ε → 0, for configurations
whose energy is reasonably controlled one hasµ(u, A) ' ∇ × hiu, ∇ui ' 2πXdi δxi(1.10)iwhere xi are the vortex centers and di their integer degrees (all possibly dependingon ε). This is not exact, however it can be given some rigorous meaning in somefunctional space in the asymptotics ε → 0 (cf. [SS1, Chap. 6]). A more true
statement is that the right hand side is a sum of approximate Diracs, smeared out(ε)
at the scale ε, which we will denote by δxi . Taking the curl (or the vector productwith ∇) of the second equation in (GL) leads to−∆h = ∇ × hiu, ∇A ui = µ(u, A) + ∇ × A

6Sylvia Serfaty

or in other terms to what is called the London equation:P(ε)−∆h + h ' 2π i di δxi in Ωh = hexon ∂Ω.

(1.11)In an electrostatic analogy, h is thus like a Coulomb (or more accurately Yukawa)
potential generated by the point vortices, which behave like (smeared out) point
charges. Assuming for simplicity that all degrees are +1 (which is true for energy
minimizers), we may then write with (1.11) thatZX
h − hex =GΩ (x, y)(2πδx(ε)− hex )iΩiwhere GΩ is the kernel of −∆ + I with Dirichlet boundary condition i.e.
(−∆x GΩ + GΩ = δy in ΩGΩ = 0on ∂Ω.

(1.12)

Of course GΩ (x, y) ∼ − log |x − y| + R(x, y) where R is some regular remainder, so
GΩ behaves essentially like the two-dimensional Coulomb kernel. One has |u| ' 1
and |∇A u|2 ' |∇h|2 as ε → 0 by using the second equation in (GL), and then one
may formally rewrite (1.9) asZ1|∇h|2 + |h − hex |2Gε (u, A) '
2 Ω
ZZ
X
X
1
=
GΩ (x, y)(2πδx(ε)− hex )(x)(2πδx(ε)− hex )(y) dx dy
i
i
2
i
i
X
' πn|log ε| − π
log |xi − xj | + remainder terms. (1.13)
i6=jHere the term πn|log ε| comes from the diagonal terms i = j, i.e. the self interactionof the smeared out Dirac masses, the logarithmic terms come from the leading orderof GΩ and the remainder terms from the next order terms of GΩ , which are regular.
We thus see that everything happens formally as if the vortices were a system ofpoints with logarithmic interactions as in (1.3). The works [SS1, SS3] make that
analogy rigorous.

2. The leading order behavior of Hn
The leading order behavior of Hn is well understood since [Cho], and the limit (ormean-field limit) is
ZZZ
E(µ) =
g(x − y) dµ(x) dµ(y) +V (x) dµ(x)(2.1)
Rd ×RdRd

Coulomb systems

7

defined over P(Rd ), the space of probability measures on Rd . Finding the minimumof E is also known as the “capacitor problem” in potential theory and was firstconsidered by Gauss and solved by Frostman in the 30’s [Fro].
Theorem 2.1 (Frostman). If V is continuous and lim|x|→∞ V /2 + g = +∞, thenE has a unique minimizer µV among probability measures. Moreover• µV has compact support of positive measure• it is uniquely characterized by the fact that there exists a constant c such that
( µ
in Rd
h V + V2 ≥ c(2.2)
hµV + V2 = cq.e. on Supp(µV )wherehµV = g ∗ µV .

(2.3)

This measure µV is called the equilibrium measure. The uniqueness easily
comes from observing that E is stricly convex on P(Rd ). The characterization ofµV comes from making variations of the form (1 − t)µV + tν with ν ∈ P(Rd ) andletting t → 0. “q.e.” means quasi-everywhere RRor except on a set of capacity 0 (acompact set E is of capacity zero if inf µ∈P(E)g(x − y) dµ(x) dµ(y) = +∞).
Important examples are the case where V (x) = |x|2 with (1.2) or (1.3), thenµV = |B11 | 1B1 . This can be guessed by taking formally the Laplacian of (2.2) on
the support of µV which yields −∆hµV = µV = ∆(|x|2 /2) = 1 there. In randommatrix theory, in the case (1.3), this corresponds to the so-called circular law.
We will always assume that Σ := Supp(µV ) is compact with a C 1 boundary,
and also that µV has a density (still denoted µV (x)) which is bounded above andC 1 on Σ and behaves like a power of the distance to Σ (cf. [PeSe] for precise
assumptions). We will also denoteζ = hµV +

V
−c
2

(2.4)

with c the constant in (2.2). Then ζ ≥ 0 in Rd and ζ = 0 in Σ quasi-everywhere(and everywhere as soon as V is regular enough).
Proposition 2.2 (Γ-convergence of Hn ). Assume (x1 , . . . , xn ) 1 arePsuch thatnHn (x1 , . . . , xn ) ≤ Cn2 , then up to extraction of a subsequence we have n1 i=1 δxi *µ ∈ P(Rd ) (for the weak-* topology on probabilities), andlim infn→∞

Hn (x1 , . . . , xn )≥ E(µ).
n2

Conversely, P
given µ ∈ P(Rd ) with E(µ) < ∞, there exists a sequence of (x1 , . . . , xn )n1such that n i=1 δxi * µ andlim supn→∞
1 everywhere we really mean x

Hn (x1 , . . . , xn )≤ E(µ).
n2

1,n , . . . , xn,n i.e. the whole configuration depends on n

8Sylvia Serfaty

immediately deduce that if for all n, (x1 , . . . , xn ) minimizes Hn , thenPWeni=1 δxi * µV , where µV is the unique minimizer of E as above, and we musthave
min Hnlim= E(µV ).
(2.5)
n→∞
n2
1
n

This settles the leading order behavior of the minimizers of Hn : their macroscopic
behavior is to resemble µV .
In the case with temperature, i.e. (1.8), it is striking that this behavior persists.
In fact it was proven in [PH, BZ, BG, CGZ] that Pn,β admits a Large Deviation
Principle (LDP) at speed n2 and rate function β2 (E − min E).
Definition 2.3. One says that a sequence of Borel probability measures (Pn )nadmits an LDP at speed an with rate function I if for every Borel set E,− inf I ≤ lim inf
◦

E

n→∞

log Pn (E)log Pn (E)≤ lim sup≤ − inf I.
an
an
n→∞
E

In our case, this means roughly that if n1
a neighborhood of that event behaves likePn

i=1 δxi * µ, then the probability of2β

e−n 2 (E(µ)−E(µV )) .
Since µV is the only minimizer of E, all configurations which converge to µ 6= µV
have exponentially small probability. This means that even with temperature (withthe scaling of temperature chosen here), configurations macroscopically resembleµV .
For the proof of Proposition 2.2 and of the LDP, we refer to [Ser, Chap. 2].

3. Expanding Hn to next order
The goal is then to understand what governs the next order term in the asymptoticsof Hn . This term will at the same time give us information on the microscopic(vs. macroscopic previously) arrangements of the points. We expect that typicalconfigurations of low energy have n points distributed on (or near) the set Σ.
Since Σ is a bounded set of dimension d, we can thus expect the typical distance
between points to be n−1/d : this is the microscopic lengthscale. We will thus blow
up configurations at that lengthscale. For simplicity we present the computationsin the Coulomb cases.
PnHere we expand the Hamiltonian by viewing the point distribution νn :=i=1 δxi as a perturbation of nµV :νn = nµV + (νn − nµV ).

(3.1)

9

Coulomb systems

Inserting the splitting (3.1) into the definition of Hn , one finds that if the points
x1 , . . . , xn are distinct, and denoting 4 for the diagonal of Rd ,Hn (x1 , . . . , xn )=Xg(xi − xj ) + nnXV (xi )

i=1

i6=j

ZZ

Z
g(x − y)dνn (x)dνn (y) + n

=

V dνn
Z
2
2
= n
V dµV
g(x − y)dµV (x)dµV (y) + n
4c
ZZ
Z
+ 2n
g(x − y)dµV (x)d(νn − nµV )(y) + n V d(νn − nµV )4cZZ+g(x − y)d(νn − nµV )(x)d(νn − nµV )(y).
(3.2)
4c

ZZ

4c

We now recall that ζ was defined in (2.4) so that we may rewrite the middle linein the right-hand side of (3.2) as
ZZ2nZg(x − y)dµV (x)d(νn − nµV )(y) + n V d(νn − nµV )4c
ZZV= 2n (hµV + )d(νn − nµV ) = 2n (ζ + c)d(νn − nµV )
2
Z
Z
Z
Z
2
= 2n ζdνn − 2nζdµV + 2nc d(νn − nµV ) = 2n ζdνn .

The last equality is due to the facts that ζ = 0 q.e. on the support of µV and that
νn and nµV have the same mass n. We also have to notice that since µV has a
L∞ density with respect to the Lebesgue measure, it does not charge the diagonal
4 (whose Lebesgue measure is zero) and we can include it back in the domain
of integration. By that same argument, one may recognize in the first line of theright-hand side of (3.2) the quantity n2 E(µV ).
We may thus rewrite (3.2) as
Hn (x1 , . . . , xn ) = n2 E(µV ) + 2nnXζ(xi )i=1

ZZ
g(x − y)d(νn − nµV )(x)d(νn − nµV )(y). (3.3)+
4cNote that this is an exact relation, valid for any configuration of distinct points.
The first term in the right-hand side gives the leading order, i.e. the energy of theequilibrium measure. In the second term, ζ plays the role of an effective confining
potential, which is active only outside of Σ (recall ζ ≥ 0, and ζ = 0 in Σ). The lastterm in the right-hand side is the most interesting, it measures the discrepancy

10Sylvia Serfaty

between the diffuse equilibrium measure µV and the discrete empirical measure1n νn . It is an electrostatic (Coulomb) interaction between a “negatively charged
background” −nµV and the n positive discrete charges at the points x1 , . . . , xn .
In the sequel, we will express this energy term in another fashion, and show that
it is indeed a lower-order term.
To go further, we need to introduce hn , the potential generated by the distribution of charges νn − nµV , defined byZ
hn := g ∗ (νn − nµV ) = g(· − y)d(νn − nµV )(y).
(3.4)
Note that hn decays at infinity, because the charge distribution νn − nµV is
compactly supported and has zero total charge, hence, when seen from infinity
1)
behaves like a dipole. More precisely, hn decays like ∇g at infinity, that is O( rd−1
12
and its gradient ∇hn decays like the second derivative D g, that is O( rd ) (indimension 1, like 1/r and 1/r2 ). Formally, using Green’s formula (or Stokes’
theorem) and the definitions, one would like to say that, at least in dimensiond ≥ 2,ZZ
Z
g(x − y)d(νn − nµV )(x)d(νn − nµV )(y) = hn d(νn − nµV )
4c
Z
Z
1
1
= hn (− ∆hn ) ≈|∇hn |2 . (3.5)
cd
cdThis 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 pairwise interactions of all the charges and “background” by an integral(extensive) quantity, which is easier to handle in some sense. However, (3.5) does
not make sense because ∇hn fails to be in L2 due to the presence of Dirac masses.
Indeed, nearR each atom xi of νn , the vector-field ∇hn behaves like ∇g and theintegrals B(0,η) |∇g|2 are divergent in all dimensions. Another way to see this isthat the Dirac masses charge the diagonal 4 and so 4c cannot be reduced to thefull space.
To remedy this, we introduce truncated potentials, and a “renormalized” wayof computing the integral. Given η > 0, setfη (x) = (g(x) − g(η))+(3.6)

and observe that fη solves(η)−∆fη = cd (δ0 − δ0 )(η)

where δ0 denotes the uniform measure of mass 1 on ∂B(0, η). For hn as in (3.4),
we then define the truncated potentialhn,η (x) = hn (x) −

n
X
i=1

fη (x − xi )

(3.7)

11

Coulomb systems

and note that it solves−∆hn,η = cd (nX

− nµV ).
δx(η)
i

(3.8)

i=1

We then have the following
Lemma 3.1.
 Z

ZZ
1
2
g(x − y)d(νn − nµV )(x)d(νn − nµV )(y) = lim|∇hn,η | − ng(η) .
η→0 cd Rd
4c
Proof. Let us compute the right-hand side of this relation. Let us choose R so that
all the points are in B(0, R−1) in Rd , and η small enough that 2η < mini6=j |xi −xj |.
Since hn,η = hn (defined in (3.4)) at distance ≥ η from the points, by Green’s
formula and (3.7), we have
Z
Z
Z
∂hn
2
−
|∇hn,η | =hn,η ∆hn,η
hn
∂ν
BR
BR
∂BR
!
Z
Z
X
∂hn
(η)
+ cd
=
hn,η
hn
δxi − nµV . (3.9)
∂ν
BR
∂BR
i
In view of the decay of hn at infinity mentioned above, the boundary integral tends
to 0 as R → ∞. We thus find
!
Z
Z
n
X
2
(η)
|∇hn,η | = cdhn,ηδxi − nµV
Rd

Rd

i=1

Z
hn −

= cd
Rd

n
X

!
fη (· − xi )

i=1

n
X

!
δx(η)
− nµV
i

. (3.10)i=1(η)

Since fη (· − xi ) = 0 on ∂B(xi , η) = Supp(δxi ) and outside of B(xi , η), and since
the balls B(xi , η) are disjoint, we may write
!
Z
Z
Z X
n
n
X
2
(η)
|∇hn,η | = cdhn
δxi − nµV − ncdfη (· − xi )µV .
Rd

Rd

Rd i=1i=1

Let us now use (temporarily) the notation hin (x) = hn (x) − g(x − xi ) (for the
potential generated by the distribution bereft of the point xi ). The function hin isR(η)
regular near xi , hence hin δxi → hin (xi ) as η → 0. It follows that
!
Z
Z X
n
n
X
(η)
cd
hn
δxi − nµV − ncdfη (x − xi )µV
Rd

Rd i=1i=1= ncd g(η) + cd

n
X
i=1

hin (xi ) − ncdZRdhn µV + O(n2 kµV kL∞ )Z|fη | + o(1).
B(0,η)(3.11)

12

Sylvia Serfaty

We can check that

R

1
lim
η→0 cdB(0,η)

Z|fη | → 0 as η → 0, sonX2|∇hn,η | − ng(η) =Rdhin (xi ) − nZ
hn µV .

(3.12)

Rd

i=1

Now, from the definitions it is easily seen thatZihn (xi ) =g(xi − y)d(νn − nµV )(y),(3.13)

Rd \{xi }from which it follows thatZZ
g(x − y)d(νn − nµV )(x)d(νn − nµV )(y)
4c

=

n Z
X
i=1

Z
g(xi −y)d(νn −nµV )(y)−nRd \{xi }hn µV =RdnX

hin (xi )−ni=1Z
hn µV .
Rd

In view of (3.12), we conclude that the formula holds.
Combining (3.3) and Lemma 3.1, we obtain2

Hn (x1 , . . . , xn ) = n E(µV ) + 2nnXζ(xi ) + lim

η→0

i=1

1
cd

Z

2



|∇hn,η | − ng(η) .
Rd(3.14)
The final step consists in rescaling this quantity, as announced, by changing x intox0 = n1/d x. We let µ0V (x0 ) = µV (x) be the blown-up density of the equilibrium
measure, Σ0 = n1/d Σ and setnXδx0i − µ0Vh0n = g ∗(3.15)
i=1

and as above
h0n,η = g ∗

n
X


(η)
δx0 − µ0V ,i

i=1which of course satisfynX−∆h0n = cd
δx0i − µ0Vand− ∆h0n,η = cd

i=1

n
X


(η)
δx0 − µ0V .

(3.16)ii=1

Changing variables in (3.14) yields
Proposition 3.2. For any n, any (x1 , . . . , xn ), we havenXn
log n)1d=2
2
i=1
 Z

n2−2/d1+lim|∇h0n,η |2 − cd g(η) . (3.17)
cd η→0 n Rd

Hn (x1 , . . . , xn ) = n2 E(µV ) + 2nζ(xi ) + (−

13

Coulomb systems

We have thus obtained a completely algebraic splitting of the energy, valid
for all configurations for fixed n, which separates the leading order term n2 E(µV )from terms which are expected to be of next order. This result was obtained in[SS4, SS5, RouSe], and its analogue for (1.4) in [PeSe]. We will now focus on
studying the asymptotics of

 Z
1
0
2
|∇hn,η | − cd g(η) .
(3.18)Fn (x1 , . . . , xn ) = lim
η→0 n Rd
A nice feature of the quantity defining Fn is its almost monotonicity:
Lemma 3.3. If α < η, we have
Z
Z
1
1
0
2
|∇hn,η | − cd g(η) ≤|∇h0n,α |2 − cd g(α) + oη (1),
n Rd
n Rd
where the oη (1) depends only on d and kµV kL∞ .
The proof is based on integration by parts similarly as in Lemma 3.1. It canbe found in [Ser, Chap. 3].

4. The renormalized energy
When taking limits in (3.16), if the blow-up was centered at a point x0 , we are led
to solutions of relations of the formX−∆h = cd
Np δp − m
in Rd
(4.1)
p∈Λ

where Np ∈ N∗ and Λ is a discrete (infinite) set of points. Here m is a constant,equal to µV (x0 ) (indeed, when centered around x0 , the density µ0V converges tothe constant µV (x0 )) since µV was assumed to be a continuous density. We callAm the class of vector fields E = ∇h with h satisfying a relation of the form (4.1).
To each such h naturally corresponds as in (3.7) a truncated potentialXhη := h −Np fη (x − p),p∈Λwhich satisfies−∆hη = cd X

Np δp(η) − m .

(4.2)

p∈Λ

In view of (3.18), it is then quite natural to define
Definition 4.1 (Renormalized energy). For ∇h ∈ Am and 0 < η < 1, we defineZ1|∇hη |2 − mcd g(η)(4.3)
Wη (∇h) = lim sup|KR | KRR→∞

14

Sylvia Serfaty

with KR = [−R, R]d , andW(∇h) = lim Wη (∇h).

(4.4)

η→0

We note that Wη is in fact monotone (nonincreasing) in η just as in Lemma
3.3, so that the limit exists, thus Wη ≥ W1 for any η ≤ 1, while W1 is easily seen
to be bounded below by −mcd g(1). Therefore W is bounded below on Am by aconstant depending only on m and d.
The constant m is acting like a uniform negative background charge which
neutralizes the points, and also corresponds to the average density of points. In
fact we can prove that if W(∇h) < ∞ thenPp∈Λ∩KR Np
= m.
(4.5)
lim
R→∞
|KR |
This follows from the fact that a relation of the form (4.1) allows to estimate thediscrepancy between the number of points and the volume via the energy itself:
one integrates (4.2) (applied for some η < 1 small but fixed) against a cut-off
function χR equal to 1 in KR and vanishing outside KR+1 . Green’s theorem thenallows to find


Z
Z
X
1
(η)


∇χR · ∇hη
(4.6)
χRNp δ p − m =cdp∈Λwhich is o(Rd ) as R → ∞. Using the Cauchy-Schwarz inequality, the right-handside can be bounded above byq
C Rd−1 |KR |(Wη (∇h) + mcd g(η))
which is o(Rd ) as R → ∞. On the other hand, since η < 1,Z
X
X
X
Np ≤ χ R
Np δp(η) ≤p∈Λ∩KR−2p∈ΛNpp∈Λ∩KR+2

hence the left-hand side of (4.6) is easily seen to be equivalent tom|KR |, and we conclude that (4.5) holds.

Pp∈Λ∩KR Np −We also have the following scaling property of W : if E ∈ Am then Ê :=·) ∈ A1 andm1/d−1 E( m1/d
W(E) = m2−2/d W(Ê) − (2πm log m) 1d=2 .

(4.7)

Thus it suffices to study W on A1 . On this class we can show (as seen just above)that it is bounded below, and also that it has a minimizer. The big open question
is to identify the minimum and the minimizers.

15

Coulomb systems

If the configuration Λ is periodic, or equivalently if it lives on a torus T ofvolume N and ifnX−∆h = cd (δai − 1) in T
(4.8)
i=1

with possible repetitions in the ai , then we can compute W in a more explicit form:
Lemma 4.2. Assume (4.8) holds. If some ai is repeated then W(E) = +∞,otherwise


g
c2d X
2
G(ai − aj ) + cd lim G −(4.9)W(∇h) =
x→0
N
cd
i6=j

where G is the solution on the torus of−∆G = δ0 −1.
N

The function G is the Green function of the torus, and behaves like cgd near
the origin. Up to a constant, the value of W just consists of a sum of pairwiseinteractions, but now computed with a periodic Green’s function, which naturally
includes a neutralizing background.
Pn
Proof. We may write h(x) = cd i=1 G(x − ai ). ThenZ
ZW(∇h) = lim lim sup − |∇hη |2 − cd g(η) = lim − |∇hη |2 − cd g(η)η→0 R→∞

η→0 T

KRby periodicity. We then writeZ|∇hη |2 = −TZhη ∆hη .
TPn
Pn
We may then insert that hη (x) = cd i=1 G(x−ai )− i=1 fη (x−xi ) and −∆hη =Pn(η)
cd ( i=1 δxi − 1) and expand exactly as in the proof of Lemma 3.1, to obtain the
result.
The particular case where N = 1, i.e. there is only one point per period, corresponds to a configuration which is exactly a (Bravais) lattice Λ (with fundamental
cell normalized to 1). Then the formula above reduces tog2
W = cd lim G −x→0
cd
and this can be computed by expanding G in Fourier series. One finds that
G(x) =

Xk∈Λ∗ \{0}e2iπk·x.
4π 2 |k|2

16

Sylvia Serfaty

The right-hand side is an Eisenstein series. Using this formula one can prove (cf.
[SS3]) that in dimension 2, if Λ1 and Λ2 are two lattices of unit volume, thenW(Λ1 ) − W(Λ2 ) = lim

s→0

X
p∈Λ∗
1 \{0}

1−|p|2+sXp∈Λ∗
2 \{0}

1= lim ζΛ∗1 (s) − ζΛ∗2 (s),s→0|p|2+s(4.10)
where ζΛ (s) is called the Epstein zeta function of the lattice Λ. The minimizationof W among lattices is then solved via the following result due to Cassels, Rankin,
Ennola, Diananda, Montgomery (for a nice proof see [Mon]):
Theorem 4.3. Assume d = 2 and s > 0. Then Λ 7→ ζΛ (s) is uniquely minimized
among lattices of volume 1 by the triangular lattice (i.e. the one based on eiπ/3 ).
It follows from (4.10) that in dimension 2, W is uniquely minimized amongvolume 1 lattices, by the triangular lattice. This reconnects to the Abrikosov
lattice that was observed in superconductivity, and leads us to conjecture that thetriangular lattice achieves a global minimum of W. Note that [BS] showed that
this conjecture is equivalent to a conjecture of [BHS].
In dimensions larger than 2, the minimization of the ζ function is not understood, and so even the minimization of W among lattices is not sorted out. It isfor example reasonable to believe that in dimension 3, the minimum is achievedby the BCC (body-centered cubic) lattice, see [SaSt]. For this, and more generally
all questions on crystallization, we also refer to the recent review [BL].

5. The screening result and analysis of minimizers of Hn
5.1. Screening. The screening procedure is a way to localize the energy, whichis by nature nonlocal in the point configuration: the electric potential h at any
point depends a priori on the configuration everywhere. The idea is to cut thedomain into cubes, and modify the configurations near the boundary of each such
cubes in such a way that the energy becomes equal to the sum of the energies on
the subcubes. For that we need to relax the problem and instead of working withelectric potentials h satisfying (4.1), work with electric fields E = ∇h, which thensatisfyX−div E = cd
Np δp − min Rd ,
(5.1)
p∈Λ

(this idea originates in [ACO]). Relaxing the constraint that E has to be a gradient,
it is then possible to glue together two electric fields on adjacent cubes keeping a
relation of the form (5.1), provided that their normal components coincide on thecommon boundary (then no divergence is created across the interface). The goalis thus to modify electric fields in such a way that their normal components always
coincide, by making them vanish on the boundaries. The energy of a vector field
constructed by such a pasting becomes additive in the pasted pieces, i.e. essentially17

Coulomb systems

local. At the end one may recover a gradient vector field by L2 projection onto
gradients, which naturally only decreases the energy.
The modification of the configuration in each cube is achieved through thefollowing screening proposition:
Proposition 5.1. Given E ∈ A1 with W(E) < ∞, satisfying
X−div E = cd (Np δp − 1).
p∈ΛGiven R such that |KR | ∈ N, and given ε > 0, η > 0 there exists Λ̂ a configurationof points and Ê a vector field (both possibly also depending on η) defined in KR
and satisfying Ê = E in KR(1−ε) (hence Λ̂ = Λ there too)X
 −div Ê = cd
δp − 1
in KR
(5.2)
p∈Λ̂

 Ê · ν = 0
on ∂KR
andZ
KR|Êη |2 ≤Z|Eη |2 + εg(η)Rd .

(5.3)

KR

The way to understand this is that given E ∈ A1 and KR , we keep E preservedin a large subcube, and use the thin layer near the boundary to completely modify
the configuration and place points “by hand” in such a way that they cancel theeffect of what is happening inside (hence the name “screening”), and a negligible
energy is added. The points in the layer compensate the oscillation of E on theboundary of the subcube and also make the whole configuration globally neutral.
Indeed, the boundary condition Ê · ν = 0 implies by integrating (5.2) over KR andusing Green’s theorem, that the number of points in KR must equal |KR |.
This screening allows to efficiently obtain upper bounds on the minimal energy
by constructing vector fields by truncating vector fields on cubes KR , applying
Proposition 5.1 and pasting together the results.
The screening result has several consequences, that were explored in [RNSe] inthe case (1.3). Since it allows to modify boundary traces of vector fields withoutchanging the energy too much, it proves that minA1 W is also equal to the limit
as R → ∞ of the minimum of W over KR -periodic configurations, and also to(Z!
X2
lim lim min − |∇hη | − cd g(η), −∆h = cd
Np δp − 1 in KR
η→0 R→∞KRpand ∂ν h = ϕ on ∂KR }(5.4)

for reasonable given boundary data ϕ.
In other words, boundary effects are negligible in the overall energy, and tocompute min W, it would suffice to compute the minimum over periodic configurations, for which the formula (4.9) is available, and then take the limit of large
period.

18

Sylvia Serfaty

5.2. Minimizers of Hn in the case (1.3). The screening also allows to get thefollowing result of equidistributions of points and energy (it was written in the case(1.3) but should work in all Coulomb cases):
Theorem 5.2 ([RNSe]). Assume (1.3). Let (x1 , . . . , xn ) ⊂ (R2 )n minimize Hn ,then•for all i, xi ∈ Σ•

we have rigidity of the number of points: letting x0i = n1/d xi and K` (a) =[a − `, a + `]d , if ` ≥ c > 0 and dist(K` (a), ∂Σ0 ) ≥ nβ/2 (β < 1), we have
lim sup #{x0i ∈ K` (a)} −
n→∞

•

Z

µ0V (x) dx ≤ C`.

(5.5)

K` (a)

we have equidistribution of energyZ
lim sup lim
n→∞

η→0

K` (a)|∇h0n,η |2 − cd #{x0i ∈ K` (a)}g(η) −ZK` (a)

min W dxAµ0 (x)V2≤ o` (` ). (5.6)
This result is based on a comparison argument. Let (x1 , . . . , xn ) be a minimizer,
let us blow up (at scale n1/d ) and consider En = ∇h0n the electric field that it
generates. If one examines a microscopic box K` (a) = [a − `, a + `]d ⊂ Σ0 , one
can delete En in that box, and replace it by a vector field of choice, obtained byapplying Proposition 5.1 to a minimizer of W (with the right density i.e µV (x)),
thus making a new point configuration. By comparison, the energy of the new
total vector field should be larger than the original one (since it was a minimizer),and this should say that the energy of the original En in the box is (5.6). In orderto make this reasoning rigorous one has to use Proposition 5.1 to glue together
the old and new vector fields. One also has to find, by a mean value argument, agood boundary of the cube on which En is well behaved. This cannot be done at
small scales a priori but the reasoning has to be applied iteratively at smaller andsmaller scales and bootstrapped until one gets to scale ` = O(1). Gluing together
the old vector field En outside K` (a) and the new one inside K` (a) will not produce
a gradient vector field, but as above, we may project it later onto gradients (inL2 ) while decreasing the energy. Once (5.6) is proven, (5.5) follows essentially byintegrating (3.16) over the given cube, integrating by parts and using the controlof (5.6) to control the boundary terms.
A result analogous to (5.5) is proven in [AOC] by very different methods, butthere is no result of the type (5.6).
Theorem 5.2 naturally implies an asymptotic expansion to next order of theminimum of Hn . However we will present that result below in the more general
setting of all dimensions.

Coulomb systems

19

6. Gamma-convergence approach
The approach outlined for Theorem 5.2 works for true minimizers of Hn , but
it is also of interest (in particular for studying the case with temperature) toobtain information for generic configurations. This is done via a Γ-convergenceapproach: in this section, we will describe how to obtain lower bounds for genericconfigurations. In view of (3.17), it suffices to study Fn given by (3.18). We
note that the integral defining Fn is given in a large (even infinite) domain. To
bound it from below we introduced [SS3, SS4] a general abstract method whichallows to get “lower bounds for 2-scale energies”, and was inspired by Varadhan.
In the present context, given a configuration (x1 , . . . , xn ) (or really a sequence ofconfigurations depending on n), we let Pn be the push forward of the normalized
Lebesgue measure on Σ by
x 7→ (x, ∇h0n (n1/d x + ·))where h0n is given by (3.15). This defines a probability measure on the set of (points
in Σ, vector fields) which can be thought of as a “tagged electric field process”,
where for each vector field, we keep as a tag the memory of the point where it
was blown-up. We let in be the map (x1 , . . . , xn ) 7→ Pn , which embeds (Rd )ninto this space of probability measures. To obtain a lower bound for Fn , we may
naturally assume that Fn ≤ C along the sequence, where C is independent of n.
It is then not too difficult to show that, Fn being coercive enough, this implies
that the sequence (Pn )n is tight, and thus up to extraction it converges to someprobability measure P . We may also check that P satisfies by construction of Pn
the following properties:• the first marginal of P is the normalized Lebesgue measure• the second marginal of P is translation-invariant• for P -a.e. (x, E) we have E ∈ Aµ0 (x) .
VWe say such probability measures are admissible. Defining then for any E in someAmZfη (x, E) = −|Eη |2 − cd µV (x)g(η)
B(0,1)

where to each E ∈ Am we may naturally associate an Eη via
XEη = E −Np ∇fη (· − p).
p

We may compute that by definition of the push-forward, the fact Rthat the firstmarginal of Pn is the normalized Lebesgue measure on Σ, and that µV = 1,ZZ
cd1
1y∈B1 |∇h0n,η |2 (n1/d x + y) dy dx −g(η)
fη (x, E) dPn (x, E) = −|Σ|Σ |B(0, 1)|
Z
1
cd
≤ 0
|∇h0n,η |2 (z) dz −g(η)|Σ | {dist(z,Σ0 )≤1}
|Σ|

20

Sylvia Serfaty

where we used the change of variables z = n1/d x + y and Fubini’s theorem. Since|Σ0 | = n|Σ| we deduce thatZ
1
Fn (x1 , . . . , xn ).
fη (x, E) dPn (x, E) ≤|Σ|
The weak convergence of Pn to P and the continuity of fη allows to take the limitn → ∞ in this expression and obtainZ
lim inf Fn (x1 , . . . , xn ) ≥ |Σ| fη (x, E) dP (x, E).
n→∞Next, we exploit the fact that P is translation-invariant in its second variable. The
multi-parameter ergodic theorem (cf. [Bec]) states that it implies thatZZfη (x, E) dP (x, E) = fη∗ (x, E) dP (x, E)
whereZfη∗ (x, E) := −fη (x, E(λ + ·)) dλ.

KR(It is part of the theorem that the limit exists). Computing and using Fubini’s
theorem again easily gives thatZfη∗ (x, E) = lim − |Eη |2 − cd µV (x)g(η) = Wη (E)R→∞ K
R

for E ∈ AµV (x) . We have thus obtained thatZ
lim inf Fn (x1 , . . . , xn ) ≥ |Σ|n→∞Wη (E) dP (x, E).

We may then use Fatou’s theorem to take the η → 0 limit and obtainZf ).
lim inf Fn (x1 , . . . , xn ) ≥ |Σ| W(E) dP (x, E) := W(P
n→∞

(6.1)

Combining with (3.17), we have obtained a general lower bound for Hn . This lower
bound is expressed as an average of W over all blown-up centers, and an averageover all blown-up profiles of the configuration (like a Young measure). Using thethird property of admissible measures, we may easily compute thatZfmin
W(P ) =
min W dx.
P admissible

Σ AµV (x)Also by scaling (4.7) we deduce that
Z

Z
2−2/d
f
min
W(P ) = min W µV (x)dx +µV (x) log µV (x) dx 1d=2 .
P admissible

A1

(6.2)

Coulomb systems

21

The final step consists in showing that this minimum can be asymptoticallyachieved by some sequence of n-point configurations.
To prove that, we split Σ0R(the blow-up of Σ) into cubes of size R on which µ0V is integer. We paste in eachcube a minimizer of W which has first been truncated and screened via Proposition
5.1 and then rescaled so as to make it have the right density µ0V . As mentioned
above, once such screened vector fields have been pasted together, one may estimate
the energy of the underlying point configuration by projecting the global vector
field onto gradients. This can only decrease the energy, and we conclude that thedesired minimum can be achieved. The final result is
Theorem 6.1 ([SS4, RouSe]). Assume we are in the cases (1.2) or (1.3). As
n → ∞ we havenf + o(n2−2/d ), (6.3)
log n 1d=2 + n2−2/dminWmin Hn = n2 E(µV ) −P admissible
2
f given by (6.2). In addition, if (x1 , . . . , xn ) ∈ (Rd )n minimize Hn ,with min W
f i.e.
letting Pn = in (x1 , . . . , xn ), up to extraction Pn * P with P a minimizer W,
P -a.e. (x, E), E minimizes W over AµV (x) .

7. Generalization to the Riesz case
As mentioned at the beginning, the approach we described can be extended beyond
the Coulomb case to the case of Riesz interaction potentials as in (1.4) and to thecase of one-dimensional logarithmic interactions as in (1.5). This was done in [SS5]for the case (1.5) and in [PeSe] for the case (1.4). It was a crucial ingredient in the
Coulomb case that the sum of pairwise interaction could be transformed via (3.17)
into a quantity which is extensive in space and local in hn . This relied on the
Coulomb nature of the potential, more precisely the fact that g was the kernel of a
local operator. This is no longer the case for (1.4) and (1.5), however these kernels
can be seen as the kernels of local operators via the Caffarelli-Silvestre extension
formula for fractional Laplacians. In that procedure one embeds the space Rd into
Rd+1 by writing
Rd+1 = {X = (x, y), x ∈ Rd , y ∈ R}.
One then considers the local operator −div (|y|γ ∇·) (which is elliptic, thus with agood regularity theory) when the space Rd is extended by one dimension to
Rd+1 = {X = (x, y), x ∈ Rd , y ∈ R},
and div and ∇ act on Rd+1 . Let g be as in (1.4). Then one has that givena measure µ on Rd and denoting by δRd the uniform measure on Rd seen as asubspace of Rd+1 , the potentialZh := g ∗ (µδRd ) =g(X − X 0 ) (µδRd )(X 0 )
Rd+k

22

Sylvia Serfaty

is the solution in Rd+1 of−div (|y|γ ∇h) = cd,s µδRdforγ =s−d+1(7.1)

and cd,s a constant depending only on d and s. The same is true in the case (1.5) bytaking s = 0 in the formula (7.1). In that case γ = 0 and h is really the harmonic
extension to the plane of the potential defined on the line. One may then write in
all cases (1.4) or (1.5)
ZZ(g ∗ µ)µ = cd,s|y|γ |∇h|2 .
Rd

Rd+1

One still defines fη = (g − g(η))+ which makes sense in Rd+1 and one sets(η)δ0:= div (|y|γ ∇fη ) + δ0 .

With the help of this formula, the whole approach described in the previous sections
then works identically, replacing the Laplacians by the operators −div (|y|γ ∇·) andthe integrals over Rd by integrals over Rd+1 with weight |y|γ . For example the class
Am is defined as the set of gradient vector fields E over Rd+1 such thatX
−div (|y|γ E) = cd,s Np δp − mδRd  in Rd+1p∈Λ

where Λ is a discrete subset of Rd+1 . The renormalized energy is then defined asZ1|y|γ |Eη |2 − cd,s mg(η).
W(E) = lim lim sup
η→0 R→∞ |KR | K ×R
R
The analogue of Theorem 6.1 is then the following (in which one should understands as being 0 in the case (1.5)):
Theorem 7.1 ([SS5],[PeSe]). Assume we are in the cases (1.5) or (1.4). As
n → ∞ we have
f + o(n1+s/d ),min Hn = n2 E(µV ) − n log n 1d=1,g=− log + n1+s/dmin
W
P admissible

(7.2)
with

Z
W(P ) =

min W dx

Σ AµV (x)

and
Z
min

P admissiblef ) = min W
W(PA1µV (x)1+s/d dx+ZµV (x) log µV (x) dx 1d=1,g=− log .

In addition, if (x1 , . . . , xn ) ∈ (Rd )n minimize Hn , letting Pn = in (x1 , . . . , xn ), up
f i.e. P -a.e. (x, E), E minimizes W
to extraction Pn * P with P a minimizer W,
over AµV (x) .

23

Coulomb systems

8. Application to the statistical mechanics
The analysis described in the last sections allows to get without much more worksome information on the case with temperature, this is what was done in [SS4,SS5, RouSe, PeSe]. Indeed, combining (3.17) with (6.1) we obtain a general lower
bound on Hn which we may then insert into (1.8) to get




β n
β 1+s/dflog n 1d=1,2,g=− log + n
Zn,β ≤ exp n E(µV ) +
min W
2
2 d
2
Z
Pn
e−βn i=1 ζ(xi ) dx1 . . . dxn
2β(Rd )nand since ζ → 1Σ this can be written
log Zn,ββ nββf+o(βn1+s/d )+O(n).
log n 1d=1,2,g=− log + n1+s/d min W≤ n2 E(µV )+
2
2 d
2
(8.1)
This is already a nontrivial bound (new in many cases), which can be complemented
without too much effort with a bound from below. However, it does not give an
optimal estimate up to o(n). Such an estimate can be provided by a stronger
result, obtained with Thomas Leblé: in [LS], we obtained a full Large Deviations
Principle which characterizes the behavior of the system at the microscopic scale
for all β. To obtain a nontrivial result, it is better to rescale temperature in (1.8)
and consider instead
dPn,β (x1 , · · · , xn ) =

1
Zn,β

β

−s

e− 2 n d Hn (x1 ,...,xn ) dx1 . . . dxn

xi ∈ R d .

(8.2)

Our result is expressed in terms of tagged point processes instead of tagged
electric field processes as in Section 6. First, for a given infinite configuration ofpoints C and a given m > 0 we may define a renormalized energy on points vianoX
Wm (C) = inf W(E), E ∈ Am , −div E = cd
δp − m .
p∈C(This can be done in cases (1.5)–(1.4) as well). For each (x1 , . . . , xn ), we thenconsider P¯n the push-forward of the normalized Lebesgue measure on Σ by
x 7→ (x, θn1/d x (x01 , . . . , x0n ))where θλ represents the action of translating by λ a configuration. Such measures
are again tight under good energy bounds, and converge up to extraction. As in
Section 6, the first marginal of P̄ is the normalized Lebesgue measure on Σ, and24

Sylvia Serfaty

the second marginal of P̄ is translation invariant. The measure P̄ can also be1dx|Σ ⊗ P̄ x . We then define
disintegrated (i.e. sliced) into |Σ|Fβ (P̄ ) =

β

Z Z

cd,s

Σ

WµV (x) dP̄ x dx +Z

ent[P̄ x |Poisson] dx

Σwhere ent[P |Poisson] is the specific relative entropy of the point process P with respect to the Poisson point process of intensity 1 (it is a large volume limit analogueof the usual relative entropy).
The main result is
Theorem 8.1 ([LS]). The push forward of Pn,β by jn : (x1 , . . . , xn ) 7→ P̄n satisfies
an LDP with speed n and rate function Fβ − inf Fβ .
Roughly speaking this means that
Pn,β (P̄n ' P̄ ) ' e−n(Fβ (P̄ )−inf Fβ )hence the Gibbs measure Pn,β concentrates on minimizers of Fβ . This minimization problem corresponds to some balancing (depending on β) between a term
based on W, which prefers order of the configurations (and expectedly crystallization), and an entropy term which measures the distance to the Poisson process,thus prefers microscopic disorder and decorrelation between the points. As β → 0,or temperature gets very large, the entropy term dominates and one can prove[Le2] that the minimizer of Fβ converges to the Poisson process. On the contrary,when β → ∞, the W term dominates, and prefers regular configurations (conjecturally, lattices). In dimension 1 where the minimum of W is known to be achievedby the lattice, this can be made into a complete proof of crystallization as β → ∞(cf. [Le1, Le2]). When β is intermediate then both terms are important and one
does not expect crystallization nor complete decorrelation.
This result has several consequences. The first one is that the limiting point
processes obtained in random matrix models: the sine-beta and Ginibre point
processes, can be characterized as minimizing cβd W1 + ent(·|Poisson) (defined forthe logarithmic interaction) among stationary point processes of intensity 1.
The second is the existence of a thermodynamic limit, i.e. an order n expansionof log Zn,β .
Corollary 8.2 (Thermodynamic limit, [LS]).
slog Zn,β = −βn2− dE(µV ) − nβ min Fβ + o((β + 1)n)2

in the cases (1.4); and in the cases (1.5)–(1.3)
log Zn,β = −βn2
βnE(µV ) +log n − nβ min Fβ + o((β + 1)n)
2
2d

(8.3)

25

Coulomb systems

or more explicitly

log Zn,β = −



βn2
βn
1
1
E(µV ) +log n − nβ min
W1 + ent[·|Poisson]
2
2d
2
β

Z
1
1
− nβ
−
µV (x) log µV (x) dx + o((β + 1)n). (8.4)β
2dΣHere the o(1) tend to zero as n → ∞ independently of β.
This provides an asymptotic expansion of the free energy (i.e. − β1 log Zn,β ) up
to order n, where the order n term itself has the structure of a free energy. This
formulae are to be compared with the recent results of [Shc13, BG13b, BG13a,
BFG13] in the dimension 1 logarithmic case. In both logarithmic cases, we also
recover in (8.4) the cancellation of the order n term when β = 4 in dimension 2and β = 2 in dimension 1 that was first observed in [Dy, Part II, section II] and[ZW06]. Such an expansion is completely new in the case (1.4).
The proof of Theorem 8.1 requires a thorough reworking of the problem, but
still relies on the two crucial ingredients described above: the asymptotic expansionof Hn and the screening result. To prove an LDP, one needs to obtain an upperbound and a lower bound for Pn,β (jn (x1 , . . . , xn ) ∈ B(P̄ , ε)). By classical large
deviations theorems (à la Sanov), one has1
log |{(x1 , . . . , xn ) ∈ Σn , jn (x1 , . . . , xN ) ∈ B(P̄ , ε)}|ε→0 n→∞ nZ
lim lim=−ent(P̄ |Poisson) dx.

(8.5)

Σ

In other words, the specific relative entropy corresponds to the (logarithm of the)volume in phase-space occupied by configurations whose P̄n = jn (x1 , . . . , xn ) is
close to P̄ . One then wishes to insert the splitting (3.17)–(3.18) into the explicit
form for Pn,β (jn (x1 , . . . , xn ) ∈ B(P̄ , ε)). The lower bound (6.1) combined with
(8.5) then allows to obtain an upper bound for Pn,β (jn (x1 , . . . , xn ) ∈ B(P̄ , ε)). To
obtain a lower bound is much more delicate, due to the need to take the n → ∞limit in Fn and the lack of continuity of W. In order to achieve it, we examine
configurations of n points that are drawn at random according to a Bernoulli
process in Σ (and by (8.5) we know how to evaluate the volume in phase-space thatthey occupy), and we show that we may modify each of them, using the screening
result, and a procedure for separating pairs of points that are too close to eachother, so that the resulting set of configurations still occupies enough logarithmic
volume in phase space (we lose volume, but not too much) and so that their Fn isclose to W(P̄ ). For details, as well as open questions and perspectives, we refer to
[LS].

26

Sylvia Serfaty

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Received
Sylvia Serfaty, Paris Sorbonne Universités, UPMC Univ. Paris 6, UMR 7598 Laboratoire Jacques-Louis Lions, Paris, F-75005 France& Courant Institute, New York University, 251 Mercer st, NY NY 10012, USA.
E-mail: serfaty@ann.jussieu.fr