THE TWO-DIMENSIONAL COULOMB GAS FLUCTUATIONS THROUGH A SPECTRAL GAP YACIN AMEUR CHRISTOPHE CHARLIER AND JOAKIM CRONVALL

2025-05-06 0 0 1.2MB 47 页 10玖币
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THE TWO-DIMENSIONAL COULOMB GAS: FLUCTUATIONS THROUGH A
SPECTRAL GAP
YACIN AMEUR, CHRISTOPHE CHARLIER, AND JOAKIM CRONVALL
Abstract. We study a class of radially symmetric Coulomb gas ensembles at inverse temperature
β= 2, for which the droplet consists of a number of concentric annuli, having at least one bounded
“gap” G, i.e., a connected component of the complement of the droplet, which disconnects the
droplet. Let nbe the total number of particles. Among other things, we deduce fine asymptotics
as n→ ∞ for the edge density and the correlation kernel near the gap, as well as for the cumulant
generating function of fluctuations of smooth linear statistics. We typically find an oscillatory
behaviour in the distribution of particles which fall near the edge of the gap. These oscillations are
given explicitly in terms of a discrete Gaussian distribution, weighted Szeg˝o kernels, and the Jacobi
theta function, which depend on the parameter n.
1. Introduction
1.1. Coulomb droplets with spectral gaps. In recent years, much work has been done relating to
statistical properties of two-dimensional Coulomb gas ensembles near the edge of a connected droplet.
Typically these works have focused on properties near the “outer boundary”, i.e., the boundary of the
unbounded component Uof the complement of the droplet. See for example [1,6,7,8,13,22,23,24,
33,35,36,38,43,49,50,55].
In the present work, we study a class of radially symmetric ensembles (at inverse temperature
β= 2), for which the droplet Sconsists of a finite number of concentric annuli, having at least
one bounded “spectral gap” G, i.e., a component of the complement C\Swhich disconnects S. A
schematic picture, of a typical droplet under study, is given in Figure 1.
Similar to what goes on near the outer boundary, we shall find that the particles that fall near
the edge ∂G of a spectral gap tend to form a strongly correlated “field”, but with an additional
uncertainty built into it, since there are two disjoint boundary components near which each individual
particle could fall. Among other things, we shall quantify the additional uncertainty in terms of a
discrete Gaussian distribution, which varies (or “oscillates”) with the total number of particles. In a
sense, we thus obtain new two-dimensional counterparts to results in the multi-cut regime found in
e.g. [32,20,28].
It is worth remarking that we here exclusively study droplets with ordinary “soft edges”. This means
that the particle density varies continuously in a neighbourhood of the boundary of S, with a quick
but smooth (error-function type) drop-off in the direction of the complement. A very different but
yet somewhat parallel setting, with “hard edges” where the density vanishes in a highly discontinuous
manner, is studied in [5,27].
2010 Mathematics Subject Classification. 60B20; 60G55; 41A60; 33E05; 30C40; 31A15.
Key words and phrases. Coulomb gas; spectral gap; soft edge; fluctuations; Gaussian free field; discrete normal
distribution; Jacobi theta function; weighted Szeg˝o kernel.
1
arXiv:2210.13959v1 [math-ph] 25 Oct 2022
2 YACIN AMEUR, CHRISTOPHE CHARLIER, AND JOAKIM CRONVALL
S
G
S
U
bN
r2
r1
Figure 1. The gap G={r1<|z|< r2}disconnects the droplet S. The domain
U={|z|> bN} ∪ {∞} is the component of ˆ
C\Scontaining ∞.
1.1.1. Some potential theoretic preliminaries. We begin by recalling some general principles of weighted
potential theory, with respect to an arbitrary admissible (not necessarily rotationally symmetric) ex-
ternal potential, i.e., a function
Q:C→R∪ {+∞}
whose properties are specified below.
Given a compactly supported unit (positive) Borel measure on C(i.e. µ(C) = 1) we define its
weighted logarithmic energy by
IQ[µ] = ZC2
log 1
|z−w|dµ(z)dµ(w) + µ(Q),
where we write µ(Q) = RCQ dµ. If we think of µas a blob of charge, the first term represents
the self-interaction energy, and the second one gives the energy from interaction with the external
potential.
Here and in what follows, the potential Qis assumed to be lower semicontinuous, finite on some
set of positive capacity, and “large” near infinity in the sense that Q(z)−2 log |z|→∞as |z|→∞.
By standard results (cf. e.g. [57]) there then exists a unique equilibrium measure σon Cminimizing
IQover all compactly supported unit Borel measures on C.
The support of σis termed the droplet and denoted S=S[Q]. Assuming (as we will) that Qis
C2-smooth in a neighbourhood of S, we have by Frostman’s theorem (see [57, Theorem II.1.3]) that
dσ = ∆Q·1SdA,(1.1)
where we use the conventions
∆ = ∂¯
∂=1
4(∂xx +∂yy), dA =1
πdxdy.
(Here and in what follows, ∂=1
2(∂
∂x −i∂
∂y ) and ¯
∂=1
2(∂
∂x +i∂
∂y ) are the usual complex derivatives
with respect to z=x+iy.)
Note that Qis subharmonic on S(since σis a measure).
Given a measure µ, we write Uµfor the usual logarithmic potential
Uµ(z) = ZC
log 1
|z−w|dµ(w).(1.2)
FLUCTUATIONS THROUGH A SPECTRAL GAP 3
It is known (see [57, Theorem I.1.3]) that there exists a constant γ=γQ(“Robin’s constant”) such
that the equilibrium measure satisfies
(1.3) Q+ 2Uσ=γon S
and
(1.4) Q+ 2Uσ≥γon C\S.
We denote by
ˇ
Q(z) = γ−2Uσ(z)
the so-called obstacle function, which is a subharmonic function [57, Theorem 0.5.6] satisfying ˇ
Q=Q
on S,ˇ
Q≤Qon C(see also Figure 2) and
ˇ
Q(z) = 2 log |z|+O(1),as z→ ∞.
Moreover, ˇ
Qis harmonic on C\Sand globally C1,1-smooth, i.e., its gradient is Lipschitz continuous
(this directly follows from (1.2)). In the sense of distributions, ∆ ˇ
Q= ∆Q·1S.
Many of the above facts are easy to understand by a Perron-family argument: let SH1be the family
of all subharmonic functions s:C→Rsuch that s≤Qeverywhere on Cand s(z)≤2 log |z|+O(1)
as z→ ∞. By a simple application of the maximum principle (e.g. [6,57]) ˇ
Q(z) is the envelope
(1.5) ˇ
Q(z) = sup{s(z) ; s∈SH1}.
We shall make a few further mild assumptions on our potential Q. First of all, we shall assume
that the Laplacian ∆Qis strictly positive on the boundary ∂S. It is also convenient to assume that
we have strict inequality Q > ˇ
Qin the complement C\S, i.e., S=S∗where S∗={Q=ˇ
Q}is
the contact set. We remark that for smooth potentials, it might happen that the complement C\S
has infinitely many components, but if Qis real-analytic near ∂S there are only finitely many such
components due to Sakai’s theory in [56].
With these preliminaries out of the way, we specialize to a class of radial potentials such that the
droplet has one or several “gaps”.
potential
obstacle function
0.5 1.0 1.5 2.0 2.5
0.5
1.0
1.5
2.0
2.5
Figure 2. The blue and orange curves are radial cross-sections of the graphs of Q
and ˇ
Q, respectively, with Q=a|z|6−b|z|4+c|z|2,a= 0.1, b= 0.8 and c= 1.8. Here
N= 1, and the black dots are a0= 0, b0=r1,a1=r2, and b1(see (1.7)).
4 YACIN AMEUR, CHRISTOPHE CHARLIER, AND JOAKIM CRONVALL
1.1.2. Class of potentials. Let Qbe a potential obeying the assumptions above and which is radially
symmetric,
Q(z) = q(|z|), q : [0,+∞)→R.(1.6)
We remark that already the case when q(r) is a polynomial in r2is quite rich, and a concretely
minded reader may think of this class in what follows.
It is easily seen that the connected components of the droplet are then closed concentric annuli
(some of which might be circles). To avoid “degenerate” cases, we will assume that the droplet Sis
a finite union of annuli:
(1.7) S=∪N
j=0{aj≤ |z| ≤ bj}
where 0 ≤a0< b0< a1< b1<···.
To focus on the main case of interest, we shall assume that N≥1 and thus that there is a “gap”,
i.e., a component of C\Sof the form
G={r1<|z|< r2}
where we write r1=bjand r2=aj+1 for some jbetween 0 and N−1.
In addition to our above assumptions, we will generally (unless the opposite is made explicit)
assume that Qis C6-smooth in some neighbourhood of ∂G.
We shall study one- and two-point correlations in a neighbourhood of the closure of a gap, especially
near the boundary circles |z|=r1and |z|=r2. We remark that we can also regard the unbounded
component of the complement, U={|z|> bN} ∪ {∞} as a spectral gap.
Asymptotics near the outer boundary |z|=bNhave been well studied, for example in [1,6,8,43,
49,38] and references therein, but we shall nevertheless find some new contributions also for this case.
1.1.3. Determinantal point processes. Given an n-point configuration {zj}n
1we define the Hamiltonian
(1.8) Hn=X
j6=k
log 1
|zj−zk|+n
n
X
j=1
Q(zj).
With dAn(z1, . . . , zn) = dA(z1)···dA(zn) the normalized Lebesgue measure in Cn, we then consider
the Gibbs probability measure
(1.9) dPn=1
Zn
e−HndAn,
where Zn=RCne−HndAnis the partition function.
The Coulomb gas in external potential Q(at inverse temperature β= 2) is a sample {zj}n
1, picked
randomly with respect to Pn.
For k≤nwe define the k-point correlation function as the unique continuous function on Ck
satisfying (with {zj}n
1a random sample and Enexpectation with respect to Pn)
En[f(z1, . . . , zk)] = (n−k)!
n!ZCk
f Rn,k dAk
for all compactly supported continuous functions fon Ck.
For each fixed k≥1, we have Johansson’s convergence theorem [30,44,41]
1
nkRn,k dAk→dσ(z1). . . dσ(zk),(n→ ∞),
in the weak sense of measures on Ck.
FLUCTUATIONS THROUGH A SPECTRAL GAP 5
As is well-known (see e.g. [53,57]) the process {zj}n
1is determinantal, i.e., there exists a correlation
kernel Kn(z, w) such that Rn,k(z1, . . . , zk) = det(Kn(zi, zj))k
i,j=1.
The kernel Kn(z, w) is only determined up to multiplication by cocycles cn(z, w) = hn(z)/hn(w)
where hnis a non-vanishing measurable function. We fix a canonical choice in the following way.
Let Wn⊂L2=L2(C, dA) be the subspace of all weighted (holomorphic) polynomials on Cof the
form
Wn={p=P·e−n
2Q:Pis a holomorphic polynomial of degree ≤n−1},
where the norm in L2is defined by kfk2=RC|f|2dA.
The canonical correlation kernel Knis just the reproducing kernel for the space Wn, i.e.,
Kn(z, w) =
n−1
X
j=0
pj(z)pj(w)
kpjk2,
where {pj}n−1
0is the orthogonal basis of Wnconsisting of the weighted monomials
pj(z) = zje−n
2q(r),(r=|z|).
Following Mehta [53], we denote the 1-point function by
(1.10) Rn(z) := Kn(z, z).
1.1.4. Error-function asymptotics. For ease of reference, we note the following fact.
Proposition 1.1. Suppose that Qis radially symmetric, is C2-smooth in a neighbourhood of S, and
strictly subharmonic on ∂G. Let pbe a boundary point of Sand let n1(p)be the unit normal to ∂S
pointing out of S. Then, as n→ ∞, we have, uniformly for tin compact subsets of C,
(1.11) Rn(p+t
p2n∆Q(p)
n1(p)) = n∆Q(p)1
2erfc t+o(n),
where erfc is the usual complementary error function
(1.12) erfc t=2
√πZ+∞
t
e−s2ds.
Remark on the proof. As far as we are aware, error-function asymptotics of the above type was first
noted for the case of the Ginibre ensemble in [37]. Universality of the asymptotic in (1.11) for a large
class of potentials was settled in [43], which however only discusses asymptotic at an outer boundary
∂U. On the other hand, Proposition 1.1 is immediate from [8, Theorem 1.8], which applies also for
other boundary components.
In what follows, we shall find and exploit a subleading term in (1.11), which typically turns out to
be of order √n. (A result in this direction appears in [49]; more on that below.)
1.1.5. Twin peaks. Since the obstacle function ˇ
Q(z) is harmonic and radially symmetric in the gap
G={r1<|z|< r2}there are constants Aand Bsuch that
ˇ
Q(z) = A+Blog r, r1≤r=|z| ≤ r2.
(This follows by a well-known theorem on harmonic functions on annuli, see e.g. [10,57].)
Since ˇ
Q=Qon ∂G we have
(1.13) A=q(r1)−Blog r1=q(r2)−Blog r2.
摘要：

THETWO-DIMENSIONALCOULOMBGAS:FLUCTUATIONSTHROUGHASPECTRALGAPYACINAMEUR,CHRISTOPHECHARLIER,ANDJOAKIMCRONVALLAbstract.WestudyaclassofradiallysymmetricCoulombgasensemblesatinversetemperature =2,forwhichthedropletconsistsofanumberofconcentricannuli,havingatleastonebounded\gap"G,i.e.,aconnectedcomponento...

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