arXiv:1909.00613v1 [math.PR] 2 Sep 2019
DJALIL CHAFAÏ, DAVID GARCÍA-ZELADA, AND PAUL JUNG
Abstract. We consider a planar Coulomb gas in which the external potential is gen-erated by a smeared uniform background of opposite-sign charge on a disc. This model can be seen as a two-dimensional Wigner jellium, not necessarily charge neutral, and with particles allowed to exist beyond the support of the smeared charge. The full space integrability condition requires low enough temperature or high enough total smeared charge. This condition does not allow at the same time, total charge neutrality and determinantal structure. The model shares similarities with both the complex Ginibre ensemble and the Forrester–Krishnapur spherical ensemble of random matrix theory. In particular, for a certain regime of temperature and total charge, the equilibrium measure is uniform on a disc as in the Ginibre ensemble, while the modulus of the farthest parti-cle has heavy-tailed fluctuations as in the Forrester–Krishnapur spherical ensemble. We also touch on a higher temperature regime producing a crossover equilibrium measure, as well as a transition to Gumbel edge fluctuations. More results in the same spirit on edge fluctuations are explored by the second author together with Raphael Butez.
Contents
1. Introduction 1
1.1. Basic two-dimensional potential theory 2
1.2. General planar Coulomb gases 2
1.3. Determinantal planar Coulomb gases 3
1.4. The Wigner jellium and Coulomb gases 5
1.5. The model and main results 6
2. Proofs 10 2.1. Proof of Lemma 1.1 10 2.2. Proof of Theorem 1.2 11 2.3. Proof of Theorem 1.3 13 2.4. Proof of Theorem 1.4 13 2.5. Proof of Theorem 1.5 15 References 16 1. Introduction
We use the identification C = R2. We denote by ℓC the Lebesgue measure on C. We denote by law→ and →, convergence in law and in probability, respectively, and weP annotate X ∼ µ to mean that the law of the random variable X is given by the probability distribution µ. We denote by P(C) the set of probability measures on C, equipped with the topology of weak convergence with respect to continuous and bounded test functions, and its associated Borel σ-field. This topology is metrized by the bounded-Lipschitz metric
dBL(µ, ν) = sup
nZ
f d(µ − ν) : kfk∞≤ 1, kfkLip≤ 1
o
Date: Summer 2019.
2000 Mathematics Subject Classification. Primary: 82B21; Secondary: 82D05; 81V45; 60F05.
Key words and phrases. Coulomb gas; jellium; Ginibre ensemble; Forrester–Krishnapur spherical en-semble; large deviation principle; Gumbel law; heavy tail; determinantal point process.
where f : C → R is measurable, kfk∞= supx|f(x)|, kfkLip= supx6=y |f(x)−f(y)||x−y| . Finally we define the Kullback–Leibler divergence or relative entropy of ν with respect to µ by
D(ν | µ) = Z
logdν
dµdν if ν ≪ µ and D(ν | µ) = +∞ otherwise.
1.1. Basic two-dimensional potential theory. We recall briefly some essential notions of two-dimensional potential theory. We refer, for instance, to [37, 49, 54, 11] for more details on the basic aspects of potential theory used in this note. The Coulomb kernel g in dimension two is given for all x ∈ R2, x 6= 0, by g(x) = − log |x|. It belongs to L1loc(ℓC) and constitutes the fundamental solution of the Laplace or Poisson equation, namely ∆g = −2πδ0 in the sense of Schwartz distributions on R2. In particular g is super-harmonic in the sense that ∆g ≤ 0. The Coulomb potential at point x ∈ C generated by a distribution of charges (say electrons) modeled by a probability measure µ on C such that g1Kc ∈ L1(µ) for some large enough compact set K is defined by
Uµ(x) = (g ∗ µ)(x) = Z
g(x − y)dµ(y) ∈ (−∞, +∞]. (1)
We have Uµ∈ L1loc(ℓC) and the identity ∆g = −2πδ0 gives the inversion formula
∆Uµ= −2πµ. (2)
In particular Uµ is super-harmonic in the sense that ∆Uµ ≤ 0. The Coulomb
(self-interaction) energy of the (distribution of charges) µ is defined when it makes sense by E(µ) = 12
Z Z
g(x − y)dµ(x)dµ(y) = 12 Z
Uµ(x)dµ(x). (3)
A set A ⊂ C has finite capacity when there exists a probability measure supported in A with finite Coulomb energy. When this is not the case, we say that the set has zero capacity.
Let V : C → R ∪ {+∞} be a lower semi-continuous function playing the role of an external potential, producing an external electric field −∇V . If V grows faster than g at infinity, the Coulomb energy EV with external field is defined by
µ ∈ P(C) 7→ EV(µ) = E(µ) + Z
V dµ.
It is lower semi-continuous with compact level sets, strictly convex, and it admits a unique minimizer called the equilibrium measure or Frostman measure denoted
µ∗= arg min P(C)EV.
This variational formula implies that there exists a constant c such that except on a set of zero capacity, we have Uµ∗ + V = c on the support of µ∗ while Uµ∗+ V ≥ c outside.
Combined with (2), we get, when V has Lipschitz weak first derivative, that dµ∗= ∆V
2π dℓC on the support of µ∗. (4) Furthermore µ∗ is compactly supported when V goes to +∞ at ∞ sufficiently fast. 1.2. General planar Coulomb gases. A planar Coulomb gas with n particles, potential V , and inverse temperature β ≥ 0 is the exchangeable Boltzmann–Gibbs probability measure Pn on Cn given by
dPn(x1, . . . , xn) =
e−βEn(x1,...,xn) Zn
where En(x1, . . . , xn) = X i<j g(xi− xj) + n n X i=1 V (xi) (6) and Zn= Z e−βEn(x1,...,xn)dℓC(x 1) · · · dℓC(xn). (7)
The Coulomb gas is well defined when Zn< ∞. It models a gas of unit charged particles,
or more precisely a random configuration of unit charged particles. We should keep in mind that we play here with electrostatics rather than with electrodynamics (no magnetic field). For all n, we define the random empirical measure
µn= 1 n n X k=1 δXn,k where Xn= (Xn,1, . . . , Xn,n) ∼ Pn.
We have P(µn ∈ A) = Pn(1nPnk=1δxk ∈ A) for any Borel subset A ⊂ P(C). In the low temperature regime β = βn with limn→∞nβn = ∞, (µn)n satisfies the following large
deviation principle: for all Borel subset A ⊂ P(C) with interior int(A) and closure clo(A), − inf int(A)EV + EV(µ∗) ≤ limn→∞ log P(µn∈ A) n2βn ≤ limn→∞log P(µn2βn∈ A) n ≤ − infclo(A)EV + EV (µ∗). (8) We refer to [38, 60, 6, 35, 18, 31] for these large deviation principles for Coulomb gases.
In the high temperature regime β = βnwith limn→∞nβn= κ ∈ (0, +∞), the same holds
true with EV formally replaced by EV + 1κD(· | ℓC) or, equivalently, formally replaced by
E +κ1D(· | νV) where νV has a density proportional to e−κV. We also have to replace µ∗by
the minimizer of EV+κ1D(· | ℓC). This is known as the crossover regime, which interpolates between νV and the minimizer of EV. The classical Sanov theorem corresponds formally
to this regime when we turn off the pair interaction by taking g = 0. This regime is considered in particular in [14, 9, 8, 18, 31, 2].
From (8), for all ε > 0, we get by taking A = {µ ∈ P(C) : dBL(µn, µ∗) > ε} that X
n
P(dBL(µn, µ∗) ≥ ε) < ∞,
which is a summable convergence in probability. By the Borel–Cantelli lemma, we obtain something known in the probabilistic literature as complete convergence, see for instance [66]. In particular, regardless of the way we define the random vectors Xn on the same
probability space, we have that almost surely, lim
n→∞dBL(µn, µ∗) = 0.
1.3. Determinantal planar Coulomb gases. It is useful to rewrite the density of the Coulomb gas Pn defined in (5), provided that Zn< ∞, as
e−βnPni=1V (xi) Zn
Y i<j
|xi− xj|β. (9)
This includes plenty of famous models from random matrix theory including the follow couple of models, and we refer to [32, 45, 39, 29, 28, 63, 64] for more information:
• Complex Ginibre ensemble. This corresponds to taking β = 2 and V = 1
2|·| 2.
The equilibrium measure is uniform on the unit disc, namely dµ∗= 1|·|≤1
π dℓC
in accordance with (4). This Coulomb gas describes the eigenvalues of a Gaussian random complex n × n matrix A with density proportional to e−Trace(AA∗)
where A∗ = ¯A⊤ is the conjugate-transpose of A. Equivalently, the entries of A are independent and identically distributed with independent real and imaginary parts having a Gaussian law of mean 0 and variance 1/(2n). This gas also appears in various other places in the mathematical physics literature, for instance as the modulus of the wave function in Laughlin’s model of the fractional quantum Hall effect [50], in the description of the vortices in the Ginzburg–Landau model of superconductivity [63], and in a model of rotating trapped fermions [48].
• Forrester–Krishnapur spherical ensemble. This corresponds to taking β = 2 and V = n + 1
2n log(1 + |x| 2). The equilibrium measure is heavy tailed and given by
dµ∗= 1
π(1 + |·|2)2dℓC.
The name of this gas comes from the fact that it is the image by the stereographical projection of the Coulomb gas on the sphere, with constant potential, onto to the complex plane. This Coulomb gas describes the eigenvalues of AB−1 where A and B are two independent copies of complex Ginibre random matrices. We can loosely interpret AB−1as a sort of matrix analogue of the Cauchy distribution since when A and B are 1 × 1 matrices, this is precisely a Cauchy distribution.
The case β = 2 has a remarkable integrable structure, called a determinantal structure, which provides exact solvability, see for instance [6, 57, 5, 20]. More precisely, if β = 2 then for all 1 ≤ k ≤ n, the k-th dimensional marginal distribution of the exchangeable probability measure Pn, denoted Pn,k, has density proportional to
(x1, . . . , xk) ∈ Ck7→ det[Kn(xi, xj)]1≤i,j≤k
where Kn is an explicit kernel which depends on n and V . Since Eµn has density Pn,1,
it follows in particular that the density of Eµn is proportional to x ∈ C 7→ Kn(x, x).
Following [44, 20], if β = 2 and if V is radially symmetric, say V = Q(|·|), then the point process of radii or moduli (this should be interpreted as a random multi-set)
{|Xn,1|, . . . , |Xn,n|}
has the same law as the point process {Yn,1, . . . , Yn,n} where Rn,1, . . . , Rn,nare independent
(and not identically distributed) random variables with Rk of density proportional to
t ∈ [0, +∞) 7→ t2k−1e−2nQ(t), 1 ≤ k ≤ n. (10) Following [61, 20, 43, 30], this allows for the asymptotic analysis of the modulus of the farthest particle of the Coulomb gas as n → ∞. In particular, one can analyze:
• Complex Ginibre ensemble. For this gas, the equilibrium measure has an edge and the modulus of the farthest particle tends to this edge, and the fluctuation is described by a Gumbel law. Namely, following [61, 20], if we define
an= 2√ncn and bn= 1 + 1 2 r cn n where cn= log(n) − 2 log log(n) − log(2π) then
max 1≤k≤n|Xn,k| P −→ n→∞1 and an( max1≤k≤n|Xn,k| − bn) law −→ n→∞G (11)
where G is the Gumbel law with cumulative probability distribution
t ∈ R 7→ G((−∞, t]) = e−e−t. (12)
• Forrester–Krishnapur spherical ensemble. For this gas, the modulus of the farthest particle tends to infinity and has a heavy tail. Namely, following [20, 43, 47].
max 1≤k≤n|Xn,k| P −→ n→∞+∞ and 1 √ n1≤k≤nmax |Xn,k| law −→ n→∞F (13)
where F is the probability distribution with cumulative distribution function t ∈ R 7→ F ((−∞, t]) = ∞ Y k=1 e−x−2k−1X j=0 x−2j j! 1x≥0, (14)
moreover this law is heavy tailed in the sense that 1 − F ((−∞, t]) = F ((t, +∞)) ∼
t→∞t
−2.
In random matrix theory and statistical physics, it is customary to speak about macro-scopic behavior for µn and about edge behavior for max1≤k≤n|Xn,k|.
1.4. The Wigner jellium and Coulomb gases. Let us consider n unit negatively charged particles (electrons) at positions x1, . . . , xnin C, lying in a positive background of
total charge α > 0 smeared according to a probability measure ρ on C with finite Coulomb energy c = E(ρ). We could alternatively suppose that the particles are positively charged (ions) and the background is negatively charged (electrons), this reversed choice would not affect the analysis of the model. The total energy of the system, counting each pair a single time, is given by
X i<j g(xi− xj) − α n X i=1 Uρ(xi) + α2c.
This matches (3) with V = −αnUρ. This observation leads us to define the jellium model
on S ⊂ C with background charge α > 0 and background distribution ρ with suppρ ⊂ S as being the Coulomb gas on the full space C, with potential V given by
V = (
−αnUρ on S
+∞ on Sc.
We say that the system is (charge) neutral. when α = n. We say that it is uniform when ρ is the uniform distribution on some compact subset of C. The great majority of jellium models studied in the literature are charge neutral and satisfy S = suppρ.
Conversely, a Coulomb gas with sub-harmonic potential V (meaning ∆V ≥ 0) can be seen as a jellium with background ρ = ∆V2παdℓC on S = C. When V is not sub-harmonic then ρ is no longer a positive measure but we can still interpret it as a background with opposite charge on {∆V < 0}. The famous example of the complex Ginibre ensemble is a Coulomb gas with potential V = |·|2, for which ∆V is constant, leading to an interpretation of this Coulomb gas as a degenerate jellium on the full space with Lebesgue background. The beautiful example of the Forrester–Krishnapur spherical ensemble is a Coulomb gas with potential V = (1 + 1/n) log(1 + |·|2), for which ∆V = 4(1 + 1/n)/(1 + |·|2)2, leading to an interpretation of this Coulomb gas as a jellium on the full space with a heavy tailed background. We can also consider such a background–potential inverse problem for the one-dimensional log-gases of random matrix theory, which can be seen as two-dimensional Coulomb gases confined to the real line, such as the Gaussian Unitary Ensemble. For
instance it follows from the discussion in [28, Section 1.4] that the logarithmic potential of the density x 7→ √ 2n π s 1 − x 2 2n1|x|<√2n is given on the interval S = [−√2n,√2n] by
x 7→ x 2 2 + n 2 logn 2 − 1 .
Let us give some useful historical comments. The jellium model was used around 1938 by Eugene P. Wigner in [65] for the modeling of electrons in metals, more than ten years before his renowned works on random matrices. This model was inspired from the Hartree– Fock model of quantum mechanics, see [34, 55, 56, 63], and [53, 52]. The term jellium was apparently coined by Conyers Herring since the smeared charge could be viewed as a positive “jelly”, see [40]. The model is also known as a one-component plasma with background. As already mentioned, usually charge-neutral jellium models are studied, and this is done typically after restricting the electrons to live on some compact support of positive background. The restriction ensures integrability of the energy and the interest is usually focused on the distribution/behavior of electrons in the “bulk” of the limiting system when the volume of the compact set goes to infinity (thermodynamic limit). There are some exceptions where the edge has been considered, for instance in [16]. Also, the edge of Laughlin states has been considered in [15, 33].
The case d = 3 is considered in [55], and quoting [55]: “It is also possible to consider the one- and two-dimensional versions of this problem, where the Coulomb potential |x|−1 is replaced by −|x| and − log |x|, respectively. In the one-dimensional, classical case, Baxter [7] calculated the partition function exactly. For that case, Kunz [46] showed that the one-particle distribution function exists and that it has crystalline ordering, i.e., the Wigner lattice exists for all temperatures. Brascamp and Lieb [12] showed the same to be true in the quantum mechanical case for one-component fermions when β is large enough. Although we do not deal with the one-dimensional problem here, our methods would apply in that case. In two dimensions there are difficulties connected with the long-range nature of the − log |x| potential, and we shall not discuss this here.” For more background literature on the jellium, see also [3, 41, 27, 1, 42, 22]. See in particular [51], for the fluctuations of non-neutral jelliums.
Historically, Coulomb gas models appeared naturally in statistics around 1920-1930 in the study of the spectrum of empirical covariance matrices of Gaussian samples. Nowadays we speak about the Laguerre ensemble and Wishart random matrices. This was almost ten years before the introduction of the jellium model by Wigner. In the 1950’s, Wigner rediscovered, by accident, these works by reading a statistics textbook, and this motivated him to use random matrices for the modeling of energy levels of heavy nuclei in atomic physics, see [21]. We refer to [10] for these historical aspects. The work of Wigner was amazingly successful, and he received in 1963 a Nobel prize in Physics “for his contributions to the theory of the atomic nucleus and the elementary particles, particularly through the discovery and application of fundamental symmetry principles.”. The term Coulomb gas is explicitly used by Dyson in his first seminal 1962 paper [26] and by Ginibre [32], while the term Fermi-gas was used earlier by Mehta & Gaudin [59] and also later by Dyson & Mehta [58].
1.5. The model and main results. In this note, we focus on a very simple planar jellium on the full space S = C, seen as a Coulomb gas Pn defined by (5) with
where ρ is the uniform probability distribution on the closed centered disc DR= {z ∈ C : |z| ≤ R}
of radius R > 0. We study the macroscopics and the edge of this planar Coulomb gas. We let α and β depend on n and we proceed in an asymptotic analysis as n → ∞. The potential V depends on n. Our analysis reveals that this special model shares similarities with the complex Ginibre and the Forrester–Krishnapur spherical ensembles.
We look at (a) when this system is well-defined, and when it is defined (b) global asymptotics at the level of the equilibrium measure, and (c) edge behavior in the sense of asymptotic analysis of the particle furthest from the origin.
We first need requirements under which the Boltzmann–Gibbs measures exist. The following lemma says that when the total charge of the background is high enough, the confinement effect on the gas is strong enough to define the Boltzmann–Gibbs measure. The condition is natural for Coulomb gases, see for instance [20]. Note that the condition does not allow, at the same time, both charge-neutrality and determinantal structure. Lemma 1.1 (Confinement or integrability condition). We have
Zn< ∞ if and only if α − n >
2 β − 1.
Moreover if this condition holds then Pn is a Coulomb gas with an external potential
V (x) = α 2n |x|2 R2 − 1 + 2 log R 1|x|≤R+α nlog |x|1|x|>R.
In particular, in the determinantal case β = 2, the condition on α reads α > n. On the other hand, in the neutral case α = n, the condition on β reads β > 2.
The proof of Lemma 1.1 is given in Section 2.1. A plot of V is provided in Figure 2.
Note that the condition does not depend on R.
The potential matches the one of the Ginibre ensemble when restricted to the disc of radius R, while it is similar to the one of the spherical ensemble when restricted to the region outside of the disc of radius R.
Theorem 1.2 (First order global asymptotics: low temperature regime). Suppose that both α = αn and β = βn may depend on n in such a way that
lim
n→∞nβn= ∞ and n→∞lim
αn
n = λ ≥ 1,
and, if λ = 1, that αn− n > β2n− 1 for n large enough. Then Zn< ∞ for n large enough
and Pn is well-defined by (5). Moreover, regardless of the way we define the sequence of
probability measures (Pn)n on the same probability space, we have that almost surely,
lim
n→∞dBL(µn, µ∗) = 0,
where µ∗ is the uniform distribution on DR/√
λ.
The proof of Theorem 1.2 is given is Section 2.2.
This low temperature regime contains the determinantal case β = 2.
The case λ < 1 is useless since Lemma 1.1 tells us in this case that Zn= ∞.
Theorem 1.3 (First order global asymptotics: high temperature regime). Suppose that both α = αn and β = βn may depend on n in such a way that
lim
n→∞nβn= κ > 0 and n→∞lim
αn
-1.5 -1 -0.5 0 0.5 1 1.5 -1.5 -1 -0.5 0 0.5 1 1.5 0 1 2 3 0.0 0.5 1.0 1.5 2.0
Figure 1. The top graphic is a plot of a simulation of Xn ∼ Pn, n = 8,
illustrating Theorem 1.2 and Theorem 1.5 in the case R = 2 and λ = 4. We used the algorithm from [17] with dt=.5 and T=10e6. About 10 inde-pendent copies were simulated and merged and we retained only the last 10% the trajectories. The bottom graphic shows a histogram of the radii of the same data together with the non asymptotic radial density for the complex Ginibre ensemble (dashed line, exact formula from determinantal structure) and radial density of equilibrium measure (solid line).
Then Zn < ∞ for n large enough and Pn is well-defined by (5). Moreover, regardless
of the way we define the sequence of probability measures (Pn)n on the same probability
space, we have that almost surely, lim
where µ∗ has a density ϕ that satisfies the following equation on its support ∆ log ϕ = 2πκϕ − λ1|·|≤R
πR2
. The proof of Theorem 1.3 is given in Section 2.3.
There should be a version in the critical case κ(λ − 1) = 2 but it is unclear for us that the functional used in our proof of Theorem 1.3 is well-defined. The proof of the large deviation principle may require a special proof. We should have −R
∆ log ϕdℓC = 4π. With the Gauss–Bonnet formula in mind, seeing −∆ log ϕ as a curvature suggests a space of Euler characteristic one, which could be thought as the unit disk.
Our last results concern the fluctuation of the edge, in other words the modulus of the farthest particle, in the determinantal case β = 2. We reveal a phase transition with respect to λ: the fluctuations are heavy tailed if λ = 1 and light tailed (Gumbel) if λ > 1. When λ = 1, we know from Theorem 1.2 that the equilibrium µ∗ is supported in DR. The farthest particle will then “feel” V outside DR, which is, according to Lemma
1.1, in this region, logarithmic, and resembles that of the Forrester–Krishnapur spherical ensemble. We can then expect that the fluctuations of the modulus of the farthest particle will be then heavy tailed, however the fluctuations law may differ from the one of the spherical ensemble. This intuition is entirely confirmed by the following theorem.
Theorem 1.4(Heavy-tailed edge). Suppose that β = 2 and α = αn= n + κn with κn> 0
and limn→∞κn = κ > 0, in such a way that in particular limn→∞αn/n = λ = 1. Then
Zn< ∞ by Lemma 1.1 and Pn is well-defined by (5). Moreover
max
1≤k≤n|Xn,k| law −→
n→∞L
where L is the law with cumulative distribution function given for all x ≥ R by L((−∞, x]) = ∞ Y k=0 1−Rx2(k+κ). In particular max 1≤k≤n|Xn,k| P −→ n→∞+∞.
The proof of Theorem 1.4 is given in Section 1.4. The law L in Theorem 1.4 has a heavy (right) tail.
Beyond the edge fluctuation, and following [13, proof of Theorem 2.4], it is actually possible to show that the whole (determinantal) point process converges as n → ∞ to a Bergman point process with explicit kernel (related to the erfc special function, see [36]). However the proof that we give of Theorem 1.4 follows a simpler scheme based on [44][20]. Note that in Theorem 1.4, L is supported in [R, +∞), and the asymptotic fluctuations at the edge are thus one-sided. In some sense the background produces here a hard edge. When λ > 1, we know from Theorem 1.2 that the equilibrium µ∗ is supported in DR/√
λ,
which is included in DR. This suggests that if the farthest particle sticks to the edge of the
limiting support, it will “feel” V inside DR, which is, according to Lemma 1.1, quadratic
and similar to the potential of a complex Ginibre ensemble. We can then expect that the fluctuations of the modulus of the farthest particle will be then light tailed and Gumbel distributed as for the complex Ginibre ensemble. This is confirmed by our last theorem. Theorem 1.5 (Gumbel edge). Suppose that β = 2 and that α = αn with limn→∞αn/n =
λ > 1. Then Zn< ∞ by Lemma 1.1 and Pn is well-defined by (5). Moreover, if we define
an= √nc n C and bn= C 1 +1 2 r cn n
where cn= log(n) − 2 log log(n) − log(2π) and C = √Rλ, then an( max 1≤k≤n|Xn,k| − bn) law −→ n→∞G
where G is the Gumbel law with cumulative distribution function G((−∞, x]) = e−e−x, x ∈ R. In particular we have max 1≤k≤n|Xn,k| P −→ n→∞ R √ λ. The proof of Theorem 1.5 is given in Section 2.5.
It is worth mentioning that following [30], one can pass from the heavy tailed law L of Theorem 1.4 to the Gumbel light tailed law G of Theorem 1.5. Namely, if for each κ > 0, ξκ is a random variable taking values in [1, +∞) with cumulative distribution function
t ∈ [1, +∞) 7→ P(ξκ≤ t) =
∞
Y k=0
(1 − t−2(k+κ)), and if εκ > 0 is the unique solution of εκexp(κεκ) = 1, then
2κξκ− 1 − εκ 2 law −→ κ→∞Gumbel.
A simulation study can be done using the algorithm in [17], see for instance Figure 1. Note that the edge of one-dimensional models is considered in [24, 25, 23].
2. Proofs 2.1. Proof of Lemma 1.1.
Proof of Lemma 1.1. Following for instance [62, Ch. 0, Example 5.7 ], we have Uρ(x) = 1 πR2 Z 2π 0 Z R 0 log 1 |x − reiθ|rdrdθ = 1 2 1 − |x| 2 R2 1|x|≤R− log|x| R1|x|>R− log R, (15) which is harmonic outside the support of ρ, in accordance with (2). Now we define
|x| V (x) = −αnUρ(x)
0
1 −2nα
Figure 2. Plot of the potential V = −αnUρas in Lemma 1.1 when R = 1.
Wρ= −Uρ and G(x, y) = g(x − y) + Wρ(x) + Wρ(y)
then X i<j G(xi, xj) = X i<j g(xi− xj) + (n − 1) n X i=1 Wρ(xi)
and e−βEn(x1,...,xn)= e−β P i<jg(xi−xj)+α Pn i=1Wρ(xi) = e−β P i<jG(xi,xj) n Y i=1 e−β(α−n+1)Wρ(xi).
The idea now is to show that the first exponential in the last display is bounded whereas the product of exponentials is integrable. Indeed, we shall use the following properties: (a) The function x 7→ |Wρ(x) − log |x|| is bounded for |x| ≥ 1;
(b) The function G is bounded from below (see Remark 2.1);
(c) For all closed set F and every compact set K such that F ∩ K = ∅, we have sup
(x,y)∈F ×K|G(x, y)| < ∞. By (a) and since Wρ is bounded from below, we have
x ∈ C 7→ e−β(α−n+1)Wρ(x) is integrable ⇐⇒ β(α − n + 1) > 2. Thus, using additionally (b), we obtain
β(α − n + 1) > 2 =⇒ Zn< ∞.
For the converse implication choose n − 1 pairwise disjoint compact sets K1, . . . , Kn−1in a neighborhood of the origin, say the open unit disk. Then, (c) implies that −βP
i<jG(xi, xj)
is bounded from below whenever (x1, . . . , xn−1) ∈ K1× · · · × Kn−1 and |xn| ≥ 1. As Wρ
is bounded from above in the unit disk there exists a constant C such that the integrand is bounded from below by
C |xn|β(α−n+1)
whenever (x1, . . . , xn−1) ∈ K1× · · · × Kn−1 and |xn| ≥ 1. We conclude that
β(α − n + 1) ≤ 2 =⇒ Zn= ∞.
Remark 2.1 (Confinement or integrability condition). The integrability condition in Lemma 1.1 can also be derived using the elementary inequality |a − b| ≤ (1 + |a|)(1 + |b|) valid for all a, b ∈ C, as in [20]. Namely, it givesQ
i<j|xi− xj| ≤ (Qni=1(1 + |xi|))n−1 since
each i appears exactly in n − 1 elements of {(i, j) : i < j}. Hence, for all x1, . . . , xn∈ C
such that |x1| > R, . . . , |xn| > R, we have, for some constants c′, c′′,
En(x1, . . . , xn) ≥ c′− X i<j log |xi− xj| + α n X i=1 log |xi| ≥ c′′− (α − (n − 1)) n X i=1 log |xi|.
Therefore Zn < ∞ if x 7→ 1/|x|β(α−(n−1)) is integrable at infinity with respect to the
Lebesgue measure on C, which holds when β(α − (n − 1)) > 2.
2.2. Proof of Theorem 1.2. We proceed as in [31] for the proof of 8. However we have to take into account the fact that the potential V = −αUρ depends on n.
Proof of Theorem 1.2. Suppose first that λ > 1. Define G : C × C → (−∞, ∞] by G(x, y) = g(x − y) + Wρ(x) + Wρ(y) and An= αn− n + 1 n − 4 nβn . Then, lim n→∞An= λ − 1 > 0
and e−βnEn(x1,...,xn) writes exp −βn X i<j g(xi− xj) + αn n X i=1 Wρ(xi) = exp −βn X i<j G(xi, xj) + Ann n X i=1 Wρ(xi) n Y i=1 exp [−4Wρ(xi)] ,
where Wρ= −Uρ. Let us define Hn: Cn→ (−∞, +∞] by
Hn(x1, . . . , xn) = 1 n2 X i<j G(xi, xj) + An 1 n n X i=1 Wρ(xi) and dσ(x) = e−4W ρ Zσ dℓC(x) where Zσ= Z C e−4Wρ(x)dℓC(x), so that the Coulomb gas law e−βnEndℓC(x) is proportional to
e−n2βnHndσ⊗n(x
1, . . . , xn).
Take any bounded continuous f : C → R. Then following for instance [31], we get that 1 n2βn log Z Cne −n2β n(f◦in+Hn)dσ⊗n(x 1, . . . , xn) = − inf µ∈P(Cn) n Eµ[f ◦ in+ Hn] + 1 n2βnD(µ | σ ⊗n)o,
where P(Cn) is the set of probability measures on Cn and where in(x1, . . . , xn) = 1 n n X i=1 δxi, so that it is natural to expect that
− inf µ∈P(Cn) n Eµ[f ◦ in+ Hn] + 1 n2βnD(µ | σ) o converges to − inf µ∈P(C) n f (µ) +1 2 Z C×CG(x, y)dµ(x)dµ(y) + (λ − 1) Z C Wρ(x)dσ(x) o
which is what exactly happens. We refer to [31] for the details. Suppose now that λ = 1. Define Hn: Cn→ (−∞, ∞] by
Hn(x1, . . . , xn) = 1 n2 X i<j G(xi, xj) where G(x, y) = g(x − y) + Wρ(x) + Wρ(y) and define γn= βn(αn− n + 1). and dσn(x) = e−γnWρ Zσn dℓC(x). The Coulomb gas law is proportional to
e−n2βnHndσ⊗n
As before, we notice that 1 n2βn log Z Cne −n2β n(f◦in+Hn)dσ⊗n n (x1, . . . , xn) = − inf µ∈P(Cn) n Eµ[f ◦ in+ Hn] + 1 n2βnD(µ | σ ⊗n n ) o , but now we remark that if dµ = ρ dℓC we have
1 nβnD(µ | σ ⊗n n ) = 1 nβn Z C ρ log ρ dℓC+ γn nβn Z C Wρdµ + γn nβn 1 γn log Z C e−γnWρdℓC if all these terms make sense (holds if ρ is bounded and compactly supported) and, thus,
lim n→∞ 1 nβnD(µ | σ ⊗n n ) = 0.
By the same arguments as before we may conclude.
2.3. Proof of Theorem 1.3. As for the proof of Theorem 1.2, we proceed as in [31] for the proof of 8, and we have to take into account the fact that the potential V = −αUρ
depends on n.
Proof of Theorem 1.3. Choose any δ > 0 such that 2 < δ < β(λ − 1) and define An= αn− n + 1 n − δ nβn . Then, lim n→∞An= λ − 1 − δ κ > 0 and we write exp −βn X i<j g(xi− xj) + αn n X i=1 Wρ(xi) = exp −βn X i<j G(xi, xj) + Ann n X i=1 Wρ(xi) n Y i=1 exp [−δWρ(xi)]
where Wρ= −Uρand G(x, y) = g(x − y) + Wρ(x) + Wρ(y) are as in the proof of Theorem
1.2. Furthermore, by following the same ideas as for the case λ > 1 in the proof of Theorem 1.2, we conclude that µn→ µ∗ as n → ∞, where µ∗ is the unique minimizer of
µ 7→ κ2 Z C×CG(x, y)dµ(x)dµ(y) + [κ(λ − 1) − δ] Z C Wρdµ + D µ | e−δW ρ Zσ dℓC.
We refer to [31] for the details. Then, still following [31], we can get from (4) that ∆ log ϕ = 2πκϕ − 2πκλ1πR|·|≤R2 .
2.4. Proof of Theorem 1.4.
Proof of Theorem 1.4. Since β = 2, Pn is determinantal and we can use (10) that we
rephrase as follows: the point process of the radii {|Xn,1|, . . . , |Xn,n|} has the same law as
the point process {|Yn,0|, . . . , |Yn,n−1|} where Yn,0, . . . , Yn,n−1 are independent (not
identi-cally distributed) complex random variables such that Yn,k ∼ an,k|z|2ke−2(n+κn)Wρ(z)dℓC(z) with an,k= Z C|z| 2ke−2(n+κn)V (z)dℓ C(z) −1
is a normalization constant, and Wρ= −Uρ. In particular, P( max 1≤k≤n|Xn,k| ≤ x) = n−1 Y k=0 1 − an,k Z Dc x |z|2ke−2(n+κn)V (z)dℓC(z) ! . (16)
By adding a constant, we shall suppose that Wρ(z) = log R if |z| = R. In that case
Wρ(z) = log |z| if |z| ≥ R while Wρ(z) > log |z| if |z| < R. (17)
Suppose that x > R. Then, by (16) and (17), P( max 1≤k≤n|Xn,k| ≤ x) = n−1 Y k=0 1 − an,k Z Dc x |z|2(k−n−κn)dℓ C(z) . Using the change of indexes k → n − k − 1 we obtain
P( max 1≤k≤n|Xn,k| ≤ x) = n−1 Y k=0 1 − an,n−k−1 Z Dc x |z|−2(k+κn+1)dℓC(z). The limit can be calculated by using Lebesgue’s dominated convergence theorem.
Domination. First we observe the domination 1 − bn,n−k−1 Z Dc x |z|−2(k+κn+1)dℓC(z) ≤ 1 − a n,n−k−1 Z Dc x |z|−2(k+κn+1)dℓC(z) (18) where bn,n−k−1= Z Dc R |z|−2(k+κn+1)dℓC(z)−1. But the left-hand side of (18) can be calculated explicitly as
1 − bn,n−k−1 Z Dc x |z|−2(k+κn+1)dℓC(z) = 1−R x 2(k+κn)
and, if we choose ε ∈ (0, κ), then, for n large enough κ − ε < κn so that
1−Rx2(k+κ−ε)≤ 1−Rx2(k+κn). Since ∞ Y k=0 1−Rx2(k+κ−ε)> 0 we have a domination from below of our product.
Pointwise convergence. Now, let us see the convergence of the terms. The coefficient a−1n,n−k−1= Z C|z| 2(n−k−1)e−2(n+κn)Wρ(z)dℓC(z) = Z C|z| −2(k+1)e−2κnWρ(z)e−2n(Wρ(z)−log |z|)dℓ C(z) has an integrand which converges to
|z|−2(k+1)e−2κWρ(z)1
|z|≥R = |z|−2(k+κ+1)1|z|≥R
and that is dominated by, for instance, |z|−2(k+1)e−(κ+ε)Wρ(0)e−2(κ−ε)Wρ(z) which is inte-grable. So, lim n→∞a −1 n,n−k−1 = Z Dc R |z|−2(k+κ+1)dℓC(z) and, by evaluating the integrals,
lim n→∞ 1 − a−1n,n−k−1 Z Dc x |z|−2(k+κn+1)dℓC(z)= 1−R x 2(k+κ) .
Lebesgue’s dominated convergence theorem. Having dominated each term from below and having proved the convergence of each term we apply Lebesgue’s dominated
conver-gence theorem and obtain the desired result.
2.5. Proof of Theorem 1.5.
Proof of Theorem 1.5. For simplicity, we give the proof when α = αn= nλ, λ > 1. From
the formula for V = −αnUρ given by Lemma 1.1, we get that V = VGin on DR, where
VGin= λ 2R2 |·|
2.
Let PnGinbe the Boltzmann–Gibbs probability measure on Cndefined by (5) with potential VGin, which is a (scaled) complex Ginibre ensemble. It follows that for any event
A ⊂ DnR= DR× · · · × DR= {x ∈ Cn: max(|x1|, . . . , |xn|) ≤ R} we have Pn(A) = ZnGin Zn PnGin(A). (19)
From Theorem 1.3, the limiting distribution µ∗ under Pn is supported in DR/√λ. Now
λ > 1 implies DR/√
λ (DR. For an arbitrary ε ∈ (0, R(1 − 1/
√
λ)], let us define the event An= DnR/√λ+ε = n x ∈ Cn: max(|x1|, . . . , |xn|) ≤ R √ λ+ ε o ⊂ DRn.
Let an, bn and G be as in Theorem 1.5 and let ξ ∼ G. Let us define
Mn= an(max(|x1|, . . . , |xn|) − bn).
Then it is known, see [61, 20], that lim
n→∞P
Gin
n (Acn) = 0 and n→∞lim EPGin
n (e iθMn) = E PGin n (e iθξ), θ ∈ R. (20)
Lemma 2.2 (Partition functions).
lim
n→∞
ZnGin Zn
= 1. (21)
Proof of Lemma (2.2). Since ZnPn(An) = ZnGinPnGin(An) and since limn→∞PnGin(An) = 1
from (20), the desired statement is actually equivalent to lim
n→∞Pn(An) = 1. (22)
But from (10), we obtain, denoting V = Q(|·|) and r = R/√λ + ε, Pn(An) = n Y k=1 1 − cn,k Z ∞ r t2k−1e−2nQ(t)dt where c−1n,k= Z ∞ 0 t 2k−1e−2nQ(t)dt. It is possible to follow this elementary route and to evaluate the limit of this product by evaluating the integrals. Actually (22) is a weak consequence of [4], which is itself a refine-ment of [19, Theorem 1.12]. Note that [4, Theorem 1] deals with potentials not necessarily rotational invariant, and provides a quantitative exponential upper bound on the proba-bility. Note also that condition (vi) of [4] translates in our context toR
CUρe2λUρdℓC< ∞ which holds precisely when λ > 1. When λ = 1, Theorem 1.4 shows that (22) does not hold, so that λ > 1 is the optimal condition on λ under which (22) can hold.
Now, using (19), (21), and (20), we obtain |EPn(e iθMn) − E Pn(e iθMn1 An)| = |EPn(e iθMn1 Ac n)| ≤ Pn(A c n) = ZnGin Zn PnGin(Acn) −→ n→∞0.
Next, and similarly, using (19), EPn(e iθMn1 An) = ZnGin Zn EPGin n (e iθMn1 An) = ZnGin Zn EPGin n (e iθMn) −Z Gin n Zn EPGin n (e iθMn1 Ac n), and using (21) and (20) we obtain
lim n→∞|EPn(e iθMn1 An) − E(e iθξ )| = 0. Finally we have obtained as expected that
lim
n→∞EPn(e
iθMn) = lim
n→∞EPnGin(e
iθMn) = E(eiθξ), θ ∈ R.
Remark 2.3. An alternative and purely equivalent way to formulate the proof of Theorem 1.5 is to note first that (19) is indeed equivalent to state that
Pn(· | DnR) = PnGin(· | DnR). (23)
Next, we may deduce from (19), (21), and (20) that lim
n→∞Pn(D n
R) = limn→∞PnGin(DRn) = 1.
which implies using (23) and Lebesgue’s dominated convergence theorem that lim n→∞ Pn(· | DnR) − PnGin(· | DnR) = 0 weakly in P(C), which implies in turn using again (20) that
lim
n→∞EPn(e
iθMn) = lim
n→∞EPnGin(e
iθMn) = E(eiθξ), θ ∈ R. References
[1] M. Aizenman, S. Jansen & P. Jung – “Symmetry breaking in quasi-1D Coulomb systems”, Annales
Henri Poincaré 11 (2010), no. 8, p. 1453–1485. 6
[2] G. Akemann & S.-S. Byun – “The high temperature crossover for general 2D Coulomb gases”, preprint arXiv:1808.00319v1, 2018. 3
[3] Alastuey, A. & Jancovici, B. – “On the classical two-dimensional one-component Coulomb plasma”, J. Phys. France 42 (1981), no. 1, p. 1–12. 6
[4] Y. Ameur – “A localization theorem for the planar Coulomb gas in an external field”, arXiv preprint arXiv:1907.00923v1, 2019. 15
[5] Y. Ameur, H. Hedenmalm & N. Makarov – “Random normal matrices and Ward identities”, Ann.
Probab. 43 (2015), no. 3, p. 1157–1201. 4
[6] G. W. Anderson, A. Guionnet & O. Zeitouni – An introduction to random matrices, Cambridge Studies in Advanced Mathematics, vol. 118, Cambridge University Press, Cambridge, 2010. 3, 4 [7] R. J. Baxter – “Statistical mechanics of a one-dimensional Coulomb system with a uniform charge
background”, Mathematical Proceedings of the Cambridge Philosophical Society 59 (1963), no. 4, p. 779–787. 6
[8] R. J. Berman – “Kähler–Einstein metrics emerging from free fermions and statistical mechanics”, J.
High Energy Phys.(2011), no. 10, p. 106, 31. 3
[9] T. Bodineau & A. Guionnet – “About the stationary states of vortex systems”, Ann. Inst. H.
Poincaré Probab. Statist. 35 (1999), no. 2, p. 205–237. 3
[10] O. Bohigas & H. A. Weidenmüller – “History—an overview”, in The Oxford handbook of random
matrix theory, Oxford Univ. Press, Oxford, 2011, p. 15–39. 6
[11] C. Bordenave & D. Chafaï – “Around the circular law”, Probab. Surv. 9 (2012), p. 1–89. 2 [12] H. J. Brascamp & E. H. Lieb – “Some inequalities for Gaussian measures and the long-range order
of the one-dimensional plasma”, in Inequalities: Selecta of Elliott H. Lieb (M. Loss & M. B. Ruskai, éds.), Springer Berlin Heidelberg, Berlin, Heidelberg, 2002, p. 403–416. 6
[13] R. Butez & D. Garcia-Zelada – “Extremal particles of two-dimensional Coulomb gases and random polynomials on a positive background”, preprint arXiv:1811.12225v1, 2018. 9
[14] E. Caglioti, P.-L. Lions, C. Marchioro & M. Pulvirenti – “A special class of stationary flows for two-dimensional Euler equations: a statistical mechanics description”, Comm. Math. Phys. 143 (1992), no. 3, p. 501–525. 3
[15] T. Can, P. Forrester, G. Téllez & P. Wiegmann – “Singular behavior at the edge of laughlin states”, Physical Review B 89 (2014), no. 23, p. 235137. 6
[16] , “Exact and asymptotic features of the edge density profile for the one component plasma in two dimensions”, Journal of Statistical Physics 158 (2015), no. 5, p. 1147–1180. 6
[17] D. Chafaï & G. Ferré – “Simulating Coulomb and log-gases with hybrid Monte Carlo algorithms”,
J. Stat. Phys. 174 (2019), no. 3, p. 692–714. 8, 10
[18] D. Chafaï, N. Gozlan & P.-A. Zitt – “First-order global asymptotics for confined particles with singular pair repulsion”, Ann. Appl. Probab. 24 (2014), no. 6, p. 2371–2413. 3
[19] D. Chafai, A. Hardy & M. Maïda – “Concentration for coulomb gases and coulomb transport inequalities”, Journal of Functional Analysis 275 (2018), no. 6, p. 1447–1483. 15
[20] D. Chafaï & S. Péché – “A note on the second order universality at the edge of Coulomb gases on the plane”, J. Stat. Phys. 156 (2014), no. 2, p. 368–383. 4, 5, 7, 9, 11, 15
[21] D. Chafaï – “Wigner about level spacing and Wishart”, blogpost http://djalil.chafai.net/blog/2014/09/26/wigner-about-level-spacing-and-wishart/, 2014. 6
[22] F. D. Cunden, P. Facchi, M. Ligabò & P. Vivo – “Universality of the third-order phase transition in the constrained Coulomb gas”, J. Stat. Mech. Theory Exp. (2017), no. 5, p. 053303, 18. 6
[23] A. Dhar, A. Kundu, S. N. Majumdar, S. Sabhapandit & G. Schehr – “Exact Extremal Statistics in the Classical 1D Coulomb Gas”, Phys. Rev. Lett. 119 (2017), p. 060601. 10
[24] A. Dhar, A. Kundu, S. N. Majumdar, S. Sabhapandit & G. Schehr – “Extreme statistics and index distribution in the classical 1d coulomb gas”, Journal of Physics A: Mathematical and
Theoretical 51 (2018), no. 29, p. 295001. 10
[25] A. Dhar, A. Kundu, S. N. Majumdar, S. Sabhapandit & G. Schehr – “Extreme statistics and index distribution in the classical 1d Coulomb gas”, J. Phys. A 51 (2018), no. 29, p. 295001, 32. 10 [26] F. J. Dyson – “Statistical theory of the energy levels of complex systems. i”, Journal of Mathematical
Physics 3 (1962), no. 1, p. 140–156. 6
[27] P. J. Forrester – “Exact results for two-dimensional Coulomb systems”, Physics Reports 301 (1998), no. 1, p. 235 – 270. 6
[28] , Log-gases and random matrices, London Mathematical Society Monographs Series, vol. 34, Princeton University Press, Princeton, NJ, 2010. 3, 6
[29] P. J. Forrester & M. Krishnapur – “Derivation of an eigenvalue probability density function relating to the Poincaré disk”, J. Phys. A 42 (2009), no. 38, p. 385204, 10. 3
[30] D. García-Zelada – “Edge fluctuations for a class of two-dimensional determinantal Coulomb gases”, preprint arXiv:1812.11170v1, 2018. 4, 10
[31] , “A large deviation principle for empirical measures on Polish spaces: Application to singular Gibbs measures on manifolds”, preprint arXiv:1703.02680v3 to appear in Annales de l’Institut Henri Poincaré, Probabilités et statistique, 2019. 3, 11, 12, 13
[32] J. Ginibre – “Statistical ensembles of complex, quaternion, and real matrices”, J. Mathematical Phys.
6 (1965), p. 440–449. 3, 6
[33] A. Gromov, K. Jensen & A. G. Abanov – “Boundary effective action for quantum hall states”,
Physical review letters 116 (2016), no. 12, p. 126802. 6
[34] G. F. Guiuliani & G. Vignale – Quantum Theory of the Electron Liquid, Cambridge University Press, 2008. 6
[35] A. Hardy – “A note on large deviations for 2D Coulomb gas with weakly confining potential”,
Electron. Commun. Probab. 17 (2012), p. no. 19, 12. 3
[36] H. Hedenmalm & W. A. – “Off-spectral analysis of Bergman kernels”, preprint, 2018. 9 [37] L. L. Helms – Potential theory, second éd., Universitext, Springer, London, 2014. 2
[38] F. Hiai & D. Petz – The semicircle law, free random variables and entropy, Mathematical Surveys and Monographs, vol. 77, American Mathematical Society, Providence, RI, 2000. 3
[39] J. B. Hough, M. Krishnapur, Y. Peres & B. Virág – Zeros of Gaussian analytic functions and
determinantal point processes, University Lecture Series, vol. 51, American Mathematical Society, Providence, RI, 2009. 3
[40] R. Hughes – “Theoretical practice: the Bohm-Pines quartet”, Perspectives on science 14 (2006), no. 4, p. 457–524. 6
[41] B. Jancovici, J. L. Lebowitz & G. Manificat – “Large charge fluctuations in classical Coulomb systems”, Journal of Statistical Physics 72 (1993), no. 3, p. 773–787. 6
[42] S. Jansen & P. Jung – “Wigner crystallization in the quantum 1d jellium at all densities”,
Commu-nications in Mathematical Physics 331 (2014), no. 3, p. 1133–1154. 6
[43] T. Jiang & Y. Qi – “Spectral radii of large non-Hermitian random matrices”, J. Theoret. Probab. 30 (2017), no. 1, p. 326–364. 4, 5
[44] E. Kostlan – “On the spectra of Gaussian matrices”, Linear Algebra Appl. 162/164 (1992), p. 385– 388, Directions in matrix theory (Auburn, AL, 1990). 4, 9
[45] M. Krishnapur – “From random matrices to random analytic functions”, Ann. Probab. 37 (2009), no. 1, p. 314–346. 3
[46] H. Kunz – “The one-dimensional classical electron gas”, Annals of Physics 85 (1974), no. 2, p. 303 – 335. 6
[47] B. Lacroix-A-Chez-Toine, A. Grabsch, S. N. Majumdar & G. Schehr – “Extremes of 2d Coulomb gas: universal intermediate deviation regime”, J. Stat. Mech. Theory Exp. (2018), no. 1, p. 013203, 39. 5
[48] B. Lacroix-A-Chez-Toine, S. N. Majumdar & G. Schehr – “Rotating trapped fermions in two dimensions and the complex ginibre ensemble: Exact results for the entanglement entropy and number variance”, Phys. Rev. A 99 (2019), p. 021602. 4
[49] N. S. Landkof – Foundations of modern potential theory, Springer-Verlag, New York-Heidelberg, 1972, Translated from the Russian by A. P. Doohovskoy, Die Grundlehren der mathematischen Wis-senschaften, Band 180. 2
[50] R. B. Laughlin – “Elementary theory: the incompressible quantum fluid”, in The Quantum Hall
Effect (R. E. Prange & S. M. Girvin, éds.), Springer, 1987, p. 233–301. 4
[51] D. Levesque, J.-J. Weis & J. Lebowitz – “Charge fluctuations in the two-dimensional one-component plasma”, Journal of Statistical Physics 100 (2000), no. 1-2, p. 209–222. 6
[52] M. Lewin, E. H. Lieb & R. Seiringer – “Statistical mechanics of the uniform electron gas”, J. Éc.
polytech. Math. 5 (2018), p. 79–116. 6
[53] , “A floating Wigner crystal with no boundary charge fluctuations”, preprint arXiv:1905.09138v1, 2019. 6
[54] E. H. Lieb & M. Loss – Analysis, second éd., Graduate Studies in Mathematics, vol. 14, American Mathematical Society, Providence, RI, 2001. 2
[55] E. H. Lieb & H. Narnhofer – “The thermodynamic limit for jellium”, Journal of Statistical Physics
12 (1975), no. 4, p. 291–310. 6
[56] E. H. Lieb & R. Seiringer – The stability of matter in quantum mechanics, Cambridge University Press, Cambridge, 2010. 6
[57] M. L. Mehta – Random matrices, second éd., Academic Press, Inc., Boston, MA, 1991. 4
[58] M. L. Mehta & F. J. Dyson – “Statistical theory of the energy levels of complex systems. v”, Journal
of Mathematical Physics 4 (1963), no. 5, p. 713–719. 6
[59] M. L. Mehta & M. Gaudin – “On the density of eigenvalues of a random matrix”, Nuclear Physics
18 (1960), p. 420–427. 6
[60] D. Petz & F. Hiai – “Logarithmic energy as an entropy functional”, in Advances in differential
equations and mathematical physics (Atlanta, GA, 1997), Contemp. Math., vol. 217, Amer. Math. Soc., Providence, RI, 1998, p. 205–221. 3
[61] B. Rider – “A limit theorem at the edge of a non-Hermitian random matrix ensemble”, J. Phys. A
36 (2003), no. 12, p. 3401–3409, Random matrix theory. 4, 15
[62] E. B. Saff & V. Totik – Logarithmic potentials with external fields, Grundlehren der Mathema-tischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 316, Springer-Verlag, Berlin, 1997, Appendix B by Thomas Bloom. 10
[63] S. Serfaty – Coulomb gases and Ginzburg-Landau vortices, Zurich Lectures in Advanced Mathemat-ics, European Mathematical Society (EMS), Zürich, 2015. 3, 4, 6
[64] , “Systems of points with Coulomb interactions”, Eur. Math. Soc. Newsl. (2018), no. 110, p. 16–21. 3
[65] E. Wigner – “Effects of the electron interaction on the energy levels of electrons in metals”, Trans.
Faraday Soc. 34 (1938), p. 678–685. 6
[66] J. E. Yukich – Probability theory of classical Euclidean optimization problems, Lecture Notes in Mathematics, vol. 1675, Springer-Verlag, Berlin, 1998. 3
(DC) CEREMADE, Université Paris-Dauphine, PSL University, France.
E-mail address: mailto:djalil(at)chafai.net
URL: http://djalil.chafai.net/
(DGZ) CEREMADE, Université Paris-Dauphine, PSL University, France.
URL: https://davidgarciaz.wixsite.com/math (PJ) KAIST, Daejon, Korea.
E-mail address: mailto:pauljung(at)kaist.ac.kr
URL: http://mathsci.kaist.ac.kr/~pauljung/
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