OVERCROWDING AND SEPARATION ESTIMATES FOR THE COULOMB GAS ERIC THOMA

2025-04-30 0 0 696.86KB 43 页 10玖币

侵权投诉

OVERCROWDING AND SEPARATION ESTIMATES FOR THE

COULOMB GAS

ERIC THOMA

Abstract. We prove several results for the Coulomb gas in any dimension d≥2 that follow

from isotropic averaging, a transport method based on Newton’s theorem. First, we prove

a high-density Jancovici-Lebowitz-Maniﬁcat law, extending the microscopic density bounds

of Armstrong and Serfaty and establishing strictly sub-Gaussian tails for charge excess in

dimension 2. The existence of microscopic limiting point processes is proved at the edge of

the droplet. Next, we prove optimal upper bounds on the k-point correlation function for

merging points, including a Wegner estimate for the Coulomb gas for k= 1. We prove the

tightness of the properly rescaled kth minimal particle gap, identifying the correct order in

d= 2 and a three term expansion in d≥3, as well as upper and lower tail estimates. In

particular, we extend the two-dimensional “perfect-freezing regime” identiﬁed by Ameur and

Romero to higher dimensions. Finally, we give positive charge discrepancy bounds which are

state of the art near the droplet boundary and prove incompressibility of Laughlin states in

the fractional quantum Hall eﬀect, starting at large microscopic scales. Using rigidity for

ﬂuctuations of smooth linear statistics, we show how to upgrade positive discrepancy bounds

to estimates on the absolute discrepancy in certain regions.

1. Introduction

1.1. The setting. For d≥2, the d-dimensional Coulomb gas (or one-component plasma)

at inverse temperature β∈(0,∞) is a probability measure on point conﬁgurations XN=

(x1, . . . , xN)∈(Rd)Ngiven by

N,β(dXN) = 1

Zexp −βHW(XN)dXN(1.1)

where dXNis Lebesgue measure on (Rd)N,Zis a normalizing constant, and

HW(XN) = 1

1≤i6=j≤N

g(xi−xj) +

i=1

W(xi) (1.2)

is the Coulomb energy of XNwith conﬁning potential W. The kernel gis the Coulomb

interaction given by

g(x) = (−log |x|if d= 2

|x|d−2if d≥3.(1.3)

While we will rarely require it, we have in mind the scaling W=VN:= N2/dV(N−1/d·) for

a potential Vsatisfying certain conditions. The normalization in VNis chosen so that the

typical interstitial distance is of size O(1), i.e. the Coulomb gas PVN

N,β is on the “blown-up”

scale. However, unless otherwise stated, we will work only under the assumption that ∆W

exists and is bounded from above on Rdand (1.1) is well-deﬁned, though see Remark 1.9 for

Date: February 20, 2023.

2020 Mathematics Subject Classiﬁcation. 82B05, 60G55, 60G70, 49S05.

arXiv:2210.05902v2 [math-ph] 21 Feb 2023

2 ERIC THOMA

comments on how this can be loosened signiﬁcantly. For some results, we will need additional

assumptions on W.

Up to normalization, the kernel ggives the repulsive interaction between two positive point

charges, and so the Coulomb gas exhibits a competition between particle repulsion, given by

the ﬁrst sum in (1.2), and particle conﬁnement, given by the second sum in (1.2). The behavior

of XNat the macroscopic scale (i.e. in a box of side length O(N1/d)) is largely dictated by

the equilibrium measure µeq, a compactly supported probability measure on Rdsolving a

variational problem involving V, in the sense that the empirical measure N−1PN

i=1 δN−1/dxi

is well-approximated weakly by µeq with high probability as N→ ∞. In particular, the

rescaled points condense on the droplet, i.e. the support of µeq. Even on mesoscales O(Nα),

0< α < 1/d, the equilibrium measure gives a good approximation for particle density. Letting

µN

eq be deﬁned by µN

eq(A) = Nµeq(N−1

dA) for measurable A⊂Rd, one can form the random

ﬂuctuation measure

ﬂuct(dx) =

i=1

δxi(dx)−µN

eq(dx),(1.4)

which, despite being of size O(N) in total variation, is typically of size O(1) when acting on

smooth functions (e.g. [Ser20,LS18,AS21,BBNY19]).

Most of the time, we will work with the more general probability measure PW,U

N,β deﬁned by

PW,U

N,β (dXN)∝exp −βHW,U (XN)dXN,HW,U (XN) = HW(XN) + U(x1, . . . , xN),(1.5)

where U=UN: (Rd)N→Ris symmetric, superharmonic and locally integrable in each

variable xi, and such that the measure PW,U

N,β is well-deﬁned. Measures of this form capture

behavior of the gas under conditioning. For example, the Coulomb gas (1.1) disintegrates

along xnto PW,U

N−1,β with U(XN−1) = PN−1

i=1 g(xi−xN). They also play an important role in

the study of the fractional quantum Hall eﬀect; see Section 1.6 for further discussion as well

the surveys [Rou22b,Rou22a].

We will apply a transport-type argument, called isotropic averaging, to give upper bounds

for PW,U

N,β on a variety of events, all concerning the overcrowding of particles. This terminology

and a similar method was ﬁrst used in [Leb21], but a technical issue limited its applicability.

Our main contribution is to demonstrate that the method has wide-ranging applicability by

giving relatively short and intuitive solutions to several open problems. We believe that it

will be an important tool in future studies of the Coulomb gas.

1.2. A model computation. The basic idea behind isotropic averaging will be motivated

through the following model computation. We will refer to this computation, in more general

forms, throughout the paper.

We start by deﬁning certain isotropic averaging operators. Given an index set I ⊂

{1,2, . . . , N}and a rotationally symmetric probability measure νon Rd, we deﬁne

IsoI,ν F((xi)i∈I ) = ˆ(Rd)I

F((xi+yi)i∈I )Y

i∈I

ν(dyi)

for any nice enough function F: (Rd)I→R. We also consider the operator IsoI,ν acting on

functions of XN, or more generally any set of labeled coordinates, by convolution with νon

OVERCROWDING IN COULOMB GASES 3

the coordinates with labels in I. For example, we have

IsoI,ν F(XN) = ˆ(Rd)I

FXN+ (yi1I(i))N

i=1Y

i∈I

ν(dyi)

by convention, and IsoI,ν F(x1) = F(x1) if 1 6∈ I, otherwise IsoI,ν F(x1) = F∗ν(x1).

An important observation is that the kernel gis superharmonic everywhere and harmonic

away from the origin, and thus we have the mean value inequality

IsoI,ν g(xi−xj)≤g(xi−xj)∀i, j. (1.6)

In our physical context, the isotropic averaging operator replaces each point charge xi,i∈ I,

by a charge distribution shaped like νcentered at xi. Newton’s theorem implies that the

electric interaction between two disjoint, radial, unit charge distributions is the same as the

interaction between two point charges located at the respective centers. More generally, if

the charge distributions are not disjoint, then the interaction is more mild than that of the

point charge system (this is because g(r) is decreasing in rand the electric ﬁeld generated by

a uniform spherical charge is 0 in the interior).

Consider an event Ewhich we wish to show to be unlikely. For deﬁniteness, we let Ebe

the event “Br(z) contains at least 2 particles” for some ﬁxed r1 and z∈Rd. By a union

bound, we have

PW,U

N,β (E)≤X

i<j

PW,U

N,β (E{i,j}) = N

2!PW,U

N,β (E{1,2}), E{i,j}:= {xi∈Br(z)}∩{xj∈Br(z)}.

(1.7)

We can bound the likelihood of E{1,2}by comparing each XN∈E{1,2}to the weighted family

of conﬁgurations generated by replacing x1and x2by unit charged annuli of inner radius 1/2

and outer radius 1. Letting νbe the uniform probability measure supported on the centered

annulus Ann[1/2,1](0) ⊂Rd, we have by Jensen’s inequality

PW,U

N,β (E{1,2}) = 1

ZˆE{1,2}

e−βHW,U (XN)dXN≤e−β∆

ZˆE{1,2}

e−βIso{1,2},ν HW,U (XN)dXN(1.8)

≤e−β∆

ZˆE{1,2}

Iso{1,2},ν e−βHW,U (XN)dXN

for

∆ = inf

XN∈E{1,2}HW,U (XN)−Iso{1,2},ν HW,U (XN).

We can then consider the L2((Rd){1,2})-adjoint of Iso{1,2},ν , which we call Iso∗

{1,2},ν , to bound

PW,U

N,β (E{1,2})≤e−β∆

Zˆ(Rd)N

Iso∗

{1,2},ν 1E{1,2}(XN)e−βHW,U (XN)dXN(1.9)

=e−β∆EW,U

N,β [Iso∗

{1,2},ν 1E{1,2}].

We call the above calculation, namely (1.8) and (1.9), the model computation. There are now

two tasks: (1) give a lower bound for ∆ and (2) give an upper bound for the expectation of

Iso∗

{1,2},ν 1E{1,2}.

Regarding task (1), we expect ∆ will be large: two particles initially clustered in Br(z) are

replaced by annular charges of microscopic length scale, which interact mildly. It is a simple

calculation to see the pairwise interaction between the charged annuli is bounded by g(1/2)

4 ERIC THOMA

(with the abuse of notation g(x) = g(|x|)). Regarding the potential term PN

i=1 W(xi) within

HW,U (XN), it will increase by at most a constant after isotropic averaging since ∆W≤C.

The superharmonic term Udoes not increase. Therefore, we have ∆ ≥g(2r)−C.

Regarding task (2), since Iso{1,2},ν is a convolution by ν⊗2, we have

Iso∗

{1,2},ν 1E{1,2}(XN) = Iso{1,2},ν 1E{1,2}(XN)≤ kνk2

L∞k1E{1,2}(·,·, x3, . . . , xN)kL1(R2)≤Cr2d.

Moreover, we have Iso∗

{1,2},ν 1E{1,2}(XN) = 0 if x1or x2is not in B1+r(z)⊂B2(z). Thus

EW,U

N,β [Iso∗

{1,2},ν 1E{1,2}]≤Cr2dPW,U

N,β ({x1, x2∈B2(z)}).

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

PW,U

N,β (E)≤Ce−βg(2r)r2dN2PW,U

N,β ({x1, x2∈B2(z)}).(1.10)

The probability appearing in the RHS will be bounded by CN−2by our microscopic local

law Theorem 1, which is proved using a separate isotropic averaging argument, and we see

that the probability of Eis bounded by Cr2de−βg(2r). This is optimal in d= 2, but can

be improved to Cr3d−2e−βg(2r)in d≥3 (see Theorem 3). The CN−2r2dbound for the

probability of E{1,2}comes from the decrease in phase space volume available to x1and x2

from the full macroscopic scale of O(N) volume per particle to a speciﬁc sub-microscopic ball

of O(rd) volume upon restricting to E{1,2}. In d≥3, the polynomial singularity of ggenerates

additional eﬀective constraints on x1and x2within Br(z).

We remark that our technique exhibits perfect localization and gives quantitative estimates

with computable constants. In particular, it is robust to certain types of conditioning and

randomization of the ball Br(z), as well as allowing to prove disparate phenomena on vastly

diﬀerent scales. It can also be generalized to use operators other than IsoI,ν , as in the proof

of Theorem 4where we give both upper and lower bounds on the minimal inter-particle

diﬀerence. For the lower bound, we must apply our model computation with a “mimicry”

operator deﬁned in Proposition 4.3. The method, in particular techniques for estimating

∆, can be made very precise, as in Theorem 5. Our model computation bears resemblance

to the Mermin-Wagner argument from statistical physics [MW66]. It is also similar to an

argument of Lieb, which applies only to ground states (β=∞) and was generously shared

and eventually generalized and published in [NS15,RS16,PS17].

Notation. We identify PW,U

N,β with the law of a point process X, with the translation between

XNand Xgiven by X=PN

i=1 δxi. All point processes will be assumed to be simple. We

also deﬁne the “index” process Xgiven by X(A) = {i:xi∈A}for measurable sets A. For

example, we have E={X(Br(z)) ≥2}and E{1,2}={{1,2} ⊂ X(Br(z))}for the events E

and E{1,2}considered in this subsection.

1.3. JLM laws. Introduced in [JLM93], Jancovici-Lebowitz-Maniﬁcat (JLM) laws give the

probability of large charge discrepencies in the Coulomb gas. The authors considered an

inﬁnite volume jellium and approximated the probability of an absolute net charge of size

much larger than R(d−1)/2in a ball of radius Ras R→ ∞. The jellium is a Coulomb gas with

a uniform negative background charge, making the whole system net neutral in an appropriate

sense. Since the typical net charge in BR(0) is expected to be of order R(d−1)/2(see [MY80]),

the JLM laws are moderate to large deviation results and exhibit tail probabilities with very

strong decay in the charge excess. The arguments of [JLM93] are based on electrostatic

principles and consider several diﬀerent regimes of the charge discrepancy size.

OVERCROWDING IN COULOMB GASES 5

We are interested in a rigorous proof of the high density versions of the JLM laws. These

versions apply when X(BR(z)) exceeds the expected charge µeq(BR(z)) by a large multiplica-

tive factor C. They predict that

Pjell({X(BR(z)) ≥Q})∼





exp −β

4Q2log Q

Q0if d= 2,

exp −β

4RQ2if d= 3,(1.11)

for Q0=|BR(z)|. The prediction applies for QRdas R→ ∞

Our main results prove the high density JLM law upper bounds in all dimensions in the

ultra-high positive charge excess regime. We do so for PW,U

N,β , a Coulomb gas with potential

conﬁnement and superharmonic perturbation, though the result holds also for the jellium

mutatis mutandis. We note that our result does not require R→ ∞. Indeed, we have found

it very useful at R= 1 as a local law upper bound valid on all of Rd, an extension of the

microscale local law in [AS21] which is only proved for zsuﬃciently far into the interior of

the droplet and under other more restrictive assumptions. Note that although we do not

obtain a sharp coeﬃcient on Q2in the exponent of the d≥3 case, it could be improved with

additional eﬀort in Proposition 2.1.

Theorem 1 (High Density JLM Law).For any R≥1, integer λ≥100, and integer Q

satisfying

Q≥





Cλ2R2+Cβ−1

log( 1

4λ)if d= 2,

CRd+Cβ−1Rd−2if d≥3,(1.12)

we have

PW,U

N,β ({X(BR(z)) ≥Q})≤(e−1

2βlog( 1

4λ)Q2+C(1+βλ2R2)Qif d= 2,

e−2−dβR−d+2Q(Q−1) if d≥3,(1.13)

and the result remains true if zis replaced by x1. The constant Cdepends only on the

dimension and the upper bound for ∆W. In particular if d= 2 and Q≥Cβ,W R2, we may

choose λ=qQ

R2to see

N,β({X(BR(z)) ≥Q})≤e−β

4logQ

R2Q2+CβQ2+CQ.

Remark 1.1. The physical principles leading to the law (1.11) focus on the change of free

energy between an unconstrained Coulomb gas and one constrained to have charge Qin BR(z).

For the constrained gas, the most likely particle conﬁgurations involve a build up of positive

charge on an inner boundary layer of BR(z)and a near vacuum outside of BR(z)which

“screens” the excess charge. Since the negative charge density is bounded (in a jellium by

deﬁnition and in PW,U

N,β by ∆W≤C), the negative screening region must be extremely thick

when QRd. The self-energy of the negative screening region is the dominant contributor

to the (1.11) bounds in [JLM93]. In our proof, we apply an isotropic averaging operator that

moves the particles within BR(z)to the bulk of the vacuum region, extracting a large average

energy change per particle, thus providing a diﬀerent perspective on the JLM law.

Remark 1.2. Theorem 1applies to small β > 0. In particular, one sees that charge excesses

of order T Rd,T1, become unlikely as soon as R≥C−1β−1/2. For this particular estimate

type, Theorem 1therefore improves the minimal eﬀective distance given in [AS21, Theorem

1] in dimensions d= 2 and d≥5(R≥Cβ−1/2(log β−1)1/2and R≥Cβ 1

d−2−1, respectively).

文档加载中……请稍候！
如果长时间未打开，您也可以点击刷新试试。

下载文档到电脑，查找使用更方便

10 玖币 0人已下载

立即下载

摘要：

OVERCROWDINGANDSEPARATIONESTIMATESFORTHECOULOMBGASERICTHOMAAbstract.WeproveseveralresultsfortheCoulombgasinanydimensiond2thatfollowfromisotropicaveraging,atransportmethodbasedonNewton'stheorem.First,weproveahigh-densityJancovici-Lebowitz-Manicatlaw,extendingthemicroscopicdensityboundsofArmstrongan...

展开>> 收起<<

OVERCROWDING AND SEPARATION ESTIMATES FOR THE COULOMB GAS ERIC THOMA.pdf

共43页,预览5页

还剩页未读，继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

OVERCROWDING AND SEPARATION ESTIMATES FOR THE COULOMB GAS ERIC THOMA

相关推荐

开通VIP享超值会员特权

作者详情

相关内容

热门标签

举报选择: