Repulsion in low temperature $\beta$-ensembles

0.13 MB·

by  Yacin Ameur

REPULSION IN LOW TEMPERATURE β-ENSEMBLES
YACINAMEUR
7
1
0 Abstract. We prove a result on non-clustering of particles in a two-
2
dimensional Coulomb plasma, which holds provided that the inverse

b temperature β satisfies β > 1. As a consequence we obtain a resulton
e crystallizationasβ :theparticleswill,onamicroscopicscale,appear
F →∞
at acertaindistance fromeachother. Theestimationofthis distanceis

8 connected to Abrikosov’sconjecture that the particlesshouldfreezeup
accordingtoahoneycomblatticewhenβ .
→∞
]
R
P
Consider a large but finite system of identical point-charges ζ n in the
h. plane C, in the presence of an external field nQ, such that Q(ζ){ iis}1"large”
t
a near ζ = . The system is picked randomly from the Boltzmann-Gibbs
m ∞
distributionatinversetemperatureβ > 1,
[

1
2 dP(β)(ζ) = e βHn(ζ)dA n(ζ), ζ = (ζ ,...,ζ ) Cn.
v n (β) − ⊗ 1 n ∈
Z
6 n
9
7 HereHn isthetotalenergy
4
0 n
1
1. Hn(ζ1,...,ζn) = log +n Q(ζj),
ζ ζ
0 Xj,k | j− k| Xj=1
7
1
: dA = dxdy/π is Lebesguemeasure on C divided by π. The constantZ(β) =
v n
Xi e−βHndA⊗n istheso-calledpartitionfunctionoftheensemble.
R Arandomsample ζ nmightbetermed"Coulombgas”,"one-component
r { j}1
a plasma”,or"β-ensemble”. Forbrevity,weuse"system”asasynonym.
Itiswell-knownthatthesystemtends,onaverage,tofollowFrostman’s
equilibriummeasureinexternalpotentialQ. Thesupportoftheequilibrium
measureisacompactsetwhichwecallthedroplet.
The rough approximation afforded by the equilibrium measure is too
crude to reveal details on a microscopic scale. However, it is believed
on physical grounds that the particles should be evenly spread out in the
interiorofthedroplet,withanon-trivialbehaviourneartheboundary-the
Halleffect. Everythingofimportancegoesoninthevicinityofthedroplet.
In this note, we prove that the distance betweenneighbouring particles
at a given location in the plane is large with high probability. Further, the
distance tends to increase with β, and as β , we recover formally the
→ ∞
separationtheoremforFeketesetsfromthepapers[1,6].
1
2 YACINAMEUR
Formulationofresults
Let Q : C R + be a suitable function of sufficient increase near
→ ∪{ ∞}
;preciseconditionsaregivenbelow. WecallQtheexternalpotential.
∞Let µ be a compactly supported Borel probability measure on C. The
weightedlogarithmicenergyofµisdefinedby
1
IQ[µ] = x log dµ(ζ)dµ(η)+ Qdµ.
ζ η ZC
C2 | − |
Assuming that Q obeys some natural conditions recalled below there is a
unique compactly supported probability measure σ which minimizes I .
Q
This is Frostman’s equilibrium measure in external field Q. The support
S = suppσisknownasthedroplet,andtheequilibriummeasuretakesthe
form(see[25])
dσ(ζ) = χ (ζ)∆Q(ζ)dA(ζ),
S
where we write ∆ = ∂∂¯ for 1/4 times the standard Laplacian; χ is the
S
characteristicfunctionofthesetS.
Remark. Let ζ n bearandomsamplewithrespecttoP(β). WriteE(β)forthe
{ j}1 n n
expectation with respect to P(β). It is well-known that E(β)[1 n f(ζ )]
n n n j=1 j →
σ(f)asn foreachcontinuousboundedfunction f onC.PSee[18,21].
→∞
The preceding remark shows that, in a sense, the equilibrium measure
givesafirstapproximationtothemacroscopicbehaviourofthesystem. We
here want to study microscopic properties. For this, we could fix a point
p C, which might depend on n, and zoom on it at an appropriate rate.
∈
However,for technical reasonsit is easierto choosethe coordinatesystem
so that p = 0. In other words, 0 will in the following denote the origin of
ann-dependentcoordinatesystemwhichcanbeobtainedfromsomestatic
referencesystembyrigidmotion.
LetD = D (0) denotethediskcenter0radiusr. Bythemicroscopic scale
r r
r at0wemeantheradiussuchthat
n
n ∆QdA = 1.
ZD
rn
We allow for any situation such that r is well-defined. This is a mild
n
restriction. Indeed, we always have ∆Q 0 on S, since σ is a probability
≥
measure. By our assumptions below, this implies that r is always well-
n
definedif0isintheinteriorofS. Also,ifwehave∆Q> 0onsomeportionof
∂S,thenr iswell-definedwhen0isinsomeneighbourhoodofthatportion.
n
Sincethebehaviourofthegasisofinterestonlyinaneighbourhoodofthe
droplet,wecanthusessentiallytreatallcasesofinterest.
Given a sample ζ n, we rescale about 0 and consider the process z n
{ j}1 { j}1
where
(1) z = r 1ζ .
j −n j
LOWTEMPERATUREβ-ENSEMBLES 3
WedenotebyP(β)theimageofP(β) underthemap(1)andwriteE(β)forthe
n n n
correspondingexpectation. AlsoletD = D betheunitdisk.
1
Let F be the event that at least one of the z falls in D. Denote η =
n j n
P(β)(F ). Passing to a subsequence, we can assume that the sequence η
n n n
is convergent. It is perfectly possible that η 0, meaning that nothing
n
→
of interest goes on in the vicinity of 0. (Cf. concluding remarks.) A more
interestingpossibilityisthatη isboundedbelowbysomepositiveconstant
n
η.
Givenarandomsample z n F wedefinewedefineanumbers by
{ j}1 ∈ n 0
s = minmin z z , ( z n F ).
0 zj∈D k,j | j− k| { j}1 ∈ n
Thus s is the largest rescaled distance from a particle in D to its nearest
0
neighbour. Werefertos asthespacingofthesample,inthevicinity ofthe
0
point0.
Wearenowpreparedtoformulateourmainresults. Thefollowingresult
showsthatthestrengthofrepulsiontendstoincreasewithβ.
Theorem. Suppose that β > 1 and fix n 1. Then there is a constant c =
0
≥
c(n ,β)> 0suchthatifn n ,η η> 0,and0 < ǫ < 1,then
0 0 n
≥ ≥
(2) P(β)( s c (ǫη)2(β11) F ) 1 m ǫ,
n { 0 ≥ · − }| n ≥ − 0
1
wherem0 = 16c−2(ǫη)−β−1. Moreover, givenanyβ0 > 1,ccanbechosenindepen-
dentofβwhenβ β .
0
≥
Thelefthandsidein(2)shouldbeunderstoodasaconditionalprobability
giventhatF hasoccurred.
n
Thenextresultgivesakindofcrystallizationasβ .
→∞
Corollary. Supposethatinfη > 0and∆Q(0) const.> 0. Letcbeanynumber
n
≥
withc< 1/(8√e). Then
lim liminfP(β)( s > c F )= 1.
n 0 n
β n { } |
→∞ →∞
WeobtainCorollarybyestimatingthebestconstantcin(2)asn ,β .
0
→∞
Corollary should be compared with the asymptotic lower bound 1/√e
for thedistance betweenFeketepointsobtained in [1, Theorem1]. In fact,
our method of proof is somewhat related to the approach in [1, 6], see
concludingremarksbelow.
Inthepresentcontext,Abrikosov’sconjecturestatesthat underthecon-
ditions in Corollary, the system z n should more and more resemble a
{ j}1
honeycomblattice as n,β . Thedistancebetweenneighbouringparti-
→ ∞
clesinthislatticecanbecomputed,leadingnaturallytotheconjecturethat
the"right”boundforcinCorollaryshouldbec< 21/23 1/4. Cf. [1].
−
Herearepreciseassumptionstobeusedintheproofsbelow: (i)Q : C
R + isl.s.c.;(ii)theinteriorofthesetΣ := Q < isnon-empty;(i→ii)
∪{ ∞} { ∞}
Qisreal-analyticonIntΣ;(iv)liminf Q(ζ)/log ζ2 > 1;(v)S IntΣ.
ζ
→∞ | | ⊂
4 YACINAMEUR
In addition, we freely use the following notation: The dA-measure of a
subsetω Cisdenoted ω. ByW wemeanthesetofweightedpolynomials
n
⊂ | |
f = pe nQ/2 where p is a holomorphic polynomial of degree at most n 1.
−
−
We denote averages by g = 1 gdA. D (ζ) denotes the disc center ζ
ω ω ω r
radiusrandwewriteD>= D (0|).|R
r r
Proofsofthemainresults
SupposethattheTaylorexpansionof∆Qabout0takestheform
∆Q= P+"higherorder terms”
where P 0, P . 0, and P is homogeneous of some degree 2k 2. The
≥ −
existenceofsuchaPisofcourseaconsequenceofthereal-analyticityofQ.
Following[8]wewriteτ forthepositiveconstantsuchthat
0
2π
1
τ 2k = P(eiθ)dθ.
−0 2πk Z
0
Notethatτ canbecastintheform
0
∆kQ(0)
τ 2k = .
−0 k[(k 1)!]2
−
ThisfollowseasilybyexpressingPasapolynomialinζandζ¯.
Using τ , we conveniently express the microscopic scale to a negligible
0
error,asfollows
r = τ n 1/2k(1+O(n 1/2k)), (n ).
n 0 − −
→∞
Notethatifk = 1thenτ = 1/ ∆Q(0).
0
Asin[8]wedefineaholomporphicpolynomialH by
2
(3) H(ζ):= Q(0)+2∂Q(0)ζ+ + ∂2kQ(0)ζ2k.
··· (2k)!
WewillalsousethedominanthomogeneouspartofQat0,i.e.,thefunction
∂i∂¯jQ(0)
Q (ζ) = ζiζ¯j.
0
i!j!
i+j=X2k,i,j 1
≥
Thepointisthatwehavethecanonicaldecomposition(cf. [8])
Q(ζ)= ReH(ζ)+Q (ζ)+O(ζ2k+1), (ζ 0).
0
| | →
Below we fix a large integer n . The following Bernstein-typelemma is
0
anelaborationof[1,Lemma2.1].
Lemma1. Suppose that n n . If f W and f(0) , 0thenthere isaconstant
0 n
≥ ∈
K = K(n )suchthat
0
f (0) Kr 1 f .
|∇| | | ≤ −n ?D | |
rn
If∆Q(0) const.> 0thenwecantakeK(n )= 4√e(1+o(1)),(n ).
0 0
≥ → ∞
LOWTEMPERATUREβ-ENSEMBLES 5
Proof. Denoteh = ReHwhereHisthepolynomialin(3). Alsowrite
∂i∂¯jQ(0)
q = | |.
0
i!j!
i+j=X2k,i,j 1
≥
Sincer = τ n 1/2k(1+O(n 1/2k))wehave
n 0 − −
(4) ζ r nQ(ζ) h(ζ) q nζ2k+O(n 1/2k) C ,
n 0 − n
| |≤ ⇒ | − | ≤ | | ≤
whereC = τ2kq +Cn 1/2k.
n 0 0 −
Nownotethat
f (ζ) = p (ζ) n∂Q(ζ)p(ζ)e nQ(ζ)/2
′ −
|∇| | | | − |
and
d
pe nh/2 (ζ) = pe nH/2 (ζ) .
− −
Insertingζ = 0itisn(cid:12)(cid:12)(cid:12)(cid:12)∇ow(cid:16)| s|eenth(cid:17)at (cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)dζ (cid:16) (cid:17) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)
d
f (0) = pe nH/2 (0) .
−
|∇| | | (cid:12)dζ (cid:12)
(cid:12) (cid:16) (cid:17) (cid:12)
(cid:12) (cid:12)
WenowapplyaCauchyestimate(cid:12)todeducethati(cid:12)fr /2 r r then
(cid:12) (cid:12) n n
≤ ≤
d pe H/2 (0) = 1 p(ζ)e−nH(ζ)/2 dζ 2 pe nh/2 dζ.
− −
By(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)d(ζ4)(cid:16)thelas(cid:17)tin(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)tegr2aπli(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)sZd|ζ|o=mr inateζd2by (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ πr2n Z|ζ|=r| | | |
eCn/2 pe nQ/2 dζ = eCn/2 f dζ.
−
Zζ=r| | | | Zζ=r| || |
| | | |
Itfollowsthat
4eCn/2 rn
f (0) dr f dζ Kr 1 f ,
|∇| | | ≤ πr3n Zrn/2 Z|ζ|=r| || | ≤ −n ?Drn | |
where K = 4sup eCn/2 . If ∆Q(0) const. > 0 then k = 1 and τ2q = 1,
whichgivesK n4≥en10/2{+C/√}n0. ≥ 0 0 (cid:3)
≤
TheweightedLagrangeinterpolationpolynomialsassociatedwithacon-
figuration ζ n ofdistinctpointsaredefinedby
{ j}1
ℓj(ζ) =  (ζ ζi)/ (ζj ζi) e−n(Q(ζ)−Q(ζj))/2, (j = 1,...,n).
− − ·
Notethatℓ Yi,Wj andℓ (Yζi,)j = δ . 
j n j k jk
∈
Now let ζ n be a random sample from P(β). Then ℓ (ζ) is a random
{ j}1 n j
variable which dependson the sample and on ζ. In the next few lemmas,
wefixanindex j,1 j n.
≤ ≤
6 YACINAMEUR
Lemma2. SupposethatUisameasurablesubsetofCoffinitemeasure U. Then
| |
(5) E(β) χ (ζ ) ℓ (ζ)2βdA(ζ) = U.
n U j j
" ·ZC| | # | |
Proof. We shall use the following identity, whose verification is left to the
reader
ℓj(ζ)2βe−βHn(ζ1,...,ζj,...,ζn) = e−βHn(ζ1,...,ζ,...,ζn).
| |
BythisandFubini’stheorem,integratingfirstinζ ,weimmediatelyget
j
dA(ζ)E(β) ℓ (ζ)2β χ (ζ )
n j U j
ZC | | ·
h i
= dA(ζ ) dP(β)(ζ ,...,ζ,...,ζ )= U,
j n 1 n
ZU ZCn | |
(cid:3)
proving(5).
In the sequel, we assume that n n and recall the constant K = K(n )
0 0
≥
provided by Lemma 1. We will write K(n ,ζ) the same constant with 0
0
replaced by ζ and let r (ζ) be the microscopic scale at ζ. Finally, we fix a
n
suitable,largeenough,constantM;wemaytakeM = 3forexample.
ItiseasytoseethatthereisaconstantT = T(M) 1suchthatifζ D
≥ ∈ Mrn
then T 1r (ζ) r (0) Tr (ζ). If ∆Q(0) const. > 0 we might take
− n n n
≤ ≤ ≥
T = 1+o(1)asn .
→∞
Lemma3. Supposethatβ 1/2. Then
≥
1
(6) r2E(nβ)"χDrn(ζj)·ZC|ℓj(ζ)|2βdA(ζ)# = 1.
n
Moreover, ifn n ,K = sup K(ζ),andr = r (0),then
≥ 0 ζ∈DMrn n n
1
(7) r2E(nβ)"χDrn(ζj)·ZD |∇|ℓj|(ζ)|2βdA(ζ)# ≤ T2β+4K2βr−n2β.
n Mrn
Proof. Theidentity(6)followsfromLemma2withU = D .
rn
To prove (7) we fix a non-zero f W and assume that f(ζ) , 0 where
n
ζ D . ByLemma1andJensen’s∈inequality,wehaveforallβ 1/2that
∈ Mrn ≥
f (ζ)2β K2βr (ζ) 2β f 2β.
|∇| | | ≤ n − ?Drn(ζ)(ζ)| |
LOWTEMPERATUREβ-ENSEMBLES 7
Applyingthiswith f = ℓ andtakingexpectations,weget
j
E(nβ)"ZD |∇|ℓj|(ζ)|2βdA(ζ)·χDrn(ζj)#
Mrn
≤ K2βZDMrn dA(ζ)rn(ζ)−2β−2ZDrn(ζ)(ζ) dA(η)E(nβ)hχDrn(ζj)|ℓj(η)|2βi
≤ T2β+2K2βr−n2β−2ZCdA(η)E(nβ) hχDrn(ζj)|ℓj(η)|2βiZDTrn(η) dA(ζ)
= T2β+4K2βr−n2β+2,
(cid:3)
whereweused(6)inthelaststep.
Inthefollowing,weletzandζdenotecomplexvariablesrelatedvia
z = r 1ζ.
−n
Weshallusetherandomfunctions̺ definedby
j
̺ (z) = ℓ (ζ) = ℓ (r z).
j j j n
| | | |
Thus̺ (z )= δ where z n istherescaledprocess.
j k jk { k}1
Lemma4. Letβ 1/2. Thenwithnotationasabove
≥
(8) E(nβ)(cid:20)χD(zj)k∇̺jkL22ββ(DM)(cid:21) ≤ T4(TK)2β.
Proof. Theinequality(7)saysthat
1
r2E(nβ)"χDrn(ζj)ZD (rn|∇|ℓj|(ζ)|)2βdA(ζ)# ≤ T4(TK)2β.
n Mrn
(cid:3)
Rescalingweimmediatelyobtain(8).
Supposethat β > 1. We will use Morrey’s inequality, which assertsthat
for all real-valued f in the Sobolev space W1,2β(D ), all z,w D , we
M ∈ M/√2
have
(9) |f(z)− f(w)| ≤ Ck∇fkL2β(DM)|z−w|1−1/β.
See[11,Corollary9.12]anditsproof. Infact,theproofin[11,p. 283]shows
that(9)holdswithC = C (1 1/β) 1 whereC 2π1/2β.
0 − 0
− ≤
WearenowreadytofinishtheproofofTheorem.
Recall that F denotesthe eventthat at least oneparticle hits D and fix
n
anarbitrary j, 1 j n. AssumingthatP(β)(F ) η > 0, wededucefrom
n n
≤ ≤ ≥
Lemma4thefollowinginequalityfortheconditionalexpectation
T4(TK)2β
(10) E(nβ)(cid:18)χD(zj)k∇̺jkL22ββ(DM) (cid:12) Fn(cid:19)≤ η .
Fixǫ > 0andrecallthats denotesthe(cid:12)(cid:12)distancefromapointin z n D
to its closestneighbour, wh0erewe assume that z n F . We mu{stj}p1r∩ove
{ j}1 ∈ n
thats c(ǫη)1/2(β 1) with(conditional)probabilityatleast1 ǫ.
0 −
≥ −
8 YACINAMEUR
Foreachλ > 0wehavebyChebyshev’sinequalityand(10)
1
P(nβ)(cid:18)(cid:26)χD(zj)k∇̺jkL22ββ(DM) > λ(cid:27) (cid:12) Fn(cid:19) ≤ λE(nβ)(cid:18)χD(zj)k∇̺jkL22ββ(DM) (cid:12) Fn(cid:19)
(cid:12)(cid:12) T4(TK)2β (cid:12)(cid:12)
.
≤ ηλ
Wenowsetλ= T4(TK)2βη 1ǫ 1. Pickasample z n F andsupposethat
− − { j}1 ∈ n
zj ∈ D. Then k∇̺jkL2β(DM) ≤ λ1/2β = T2/β(TK)(ǫη)−1/2β with probability at
least1 ǫ.
−
Now let z be a closest neighbour to z . In view of (9) there is another
k j
constantC = C (1 1/β) 1 suchthat,withprobabilityatleast1 ǫ,wehave
0 −
− −
either z z M/√2or
j k
| − |≥
(11) 1 = ̺ (z ) ̺ (z ) CT1+2/βK(ǫη) 1/2βz z 1 1/β,
j j j k − j k −
| − |≤ | − |
i.e., z z c(ǫη)1/2(β 1) wherewemaychose
j k −
| − |≥
(12) c= (CT1+2/βK) β/(β 1) = (1 1/β)β/(β 1)(C T1+2/βK) β/(β 1).
− − − 0 − −
−
Wehaveshownthat
(13) min z z c(ǫη)1/2(β 1)
j k −
k,j | − | ≥
withprobabilityatleast1 ǫ. Theformula(12)showsthatccanbechosen
−
independentofβwhenβ β > 1. TheproofofTheoremiscomplete.
0
≥
Lemma 5. Let ND be the number of particles which fall in D. Also define
r0 = c(ǫη)1/2(β−1)/2 and m0 = 4/r20. Then ND ≤ m0 and s0 ≥ c(ǫη)1/2(β−1) with
probability atleast1 m ǫ.
0
−
Proof. Suppose that at least m particles, denoted z ,...,z , fall in D. For
1 m
1 j mlet E betheeventthat thediskD (z )contains nopointz with
≤ ≤ j r0 j k
1 k n,k , j. ThenP(β)(Ec) ǫ,whereEc isthecomplementaryevent.
≤ ≤ n j ≤ j
Wehaveshownthat
P(β)( m E )= 1 P(β)( m Ec) 1 mǫ.
n ∩j=1 j − n ∪j=1 j ≥ −
ItfollowsthatifatleastmparticlesfallinDthenwithprobabilityatleast
1 mǫ there are m disjoint disks of radius r inside D . Comparing areas
0 2
w−eseethatmr2 4,i.e.,m m . (cid:3)
0 ≤ ≤ 0
1
Thelemmasaysthatifm0 = m0(β,ǫ,η) = 16c−2(ǫη)−β−1 then
(14) P(β)( s c(ǫη)1/2(β 1) F ) 1 ǫm .
n 0 − n 0
{ ≥ }| ≥ −
ThisprovesTheorem.
LOWTEMPERATUREβ-ENSEMBLES 9
Nowassumethat∆Q(0) const. > 0. ThenbyLemma1,theconstantK
≥
theremightbetakenasK = 4√e(1+o(1))whilewemaytakeT = 1+o(1)as
n . HenceC T1+2/βK 8√e(πT4)1/2β(1+o(1)),andso,ifc(β)denotesthe
0
→ ∞ ≤
largestconstantsuchthattheestimate(13)holdsasymptotically,asn ,
0
→∞
then liminf c(β) 1/(8√e) and limsup m (β,ǫ,η) 210e. It is now
β→∞ ≥ β→∞ 0 ≤
clearfrom(14)thatCorollaryisaconsequenceofTheorem.
q.e.d.
Concludingremarks
Consider the quantities η = P(β)(F ). It is of interestto know when η
n n n n
is boundedbelow. For this, weformthe quantitiesα = r 2D S. That
is, α isthefraction ofthearea ofthemicro-disc D nwhic−nh f|allrsn ∩insi|dethe
n rn
dropletS. Weconjecturethatη isboundedbelowifα isboundedbelow.
n n
More generally, if we fix a positive R and put αR = (Rr ) 2D S. It
n n − | Rrn ∩ |
seemsplausible thatη is boundedbelowif and onlyif thesequenceαR is
n n
boundedbelowforsomeR 1. Thishasbeenverifiedinthedeterminantal
≥
situation β = 1 in many cases. See the papers [4, 5, 8]. See also [1] for the
caseβ = . Wewillreturntothesematterslater[2].
∞
A configuration ζ n is said to be a Fekete configuration in external
{ j}1
potentialQiftheenergyH (ζ ,...,ζ )isminimal. Wethinkofsuchconfig-
n 1 n
urationsasfrozenparticlesystems,attemperature1/β = 0;theyarewholly
contained in the droplet, see [25]. To our knowledge, uniform separation
betweenFeketepointswasfirstshownin[6].
TheanalysisofFeketeconfigurationsin[1,6]dependsontheinequality
ℓ 1 for the associated weighted Lagrange polynomials ℓ . This bound
j j
| | ≤
playsasimilarrolewhenβ = astheL2β-estimateinLemma2doesinthe
∞
presentcase.
TheideaofusinganL2β-boundonLagrangesectionsoccursin[15]. The
contextthereisdifferent,butinaway,wehaveelaboratedonthisideahere.
It is possible to extend Corollary to the "bulk singular case” where
∆Q(0) = 0. However, the constant c must be modified, depending on the
dominant termsin the Taylorexpansionof∆Q about 0. This is so because
theconstantsT,Kdependonthoseterms.
In the determinantal case β = 1, limiting point fields z have been
{ j}∞1
identified in many cases. The study of universality in this setting was
initiated in the paper [4]. When β > 1, the determinantal structure is lost,
andtheproblemofcalculatinglimitingpointfieldsremainsachallenge. The
question is perhaps especially intriguing when we rescale about a regular
boundarypointofS,oraboutsomeotherkindofspecialpoint,cf. [5,7,24].
At a regular boundary point, it seems plausible that the distribution
should be translation invariant in the direction tangent to the boundary,
i.e., in a suitable coordinate system, the distribution depends only on x =
Rez. The Hall effect is believed to give rise to certain irregularities in the
10 YACINAMEUR
distribution,which aretobelocatedslightlytotheinsideoftheboundary.
See [13] and the references there for a discussion of this fascinating topic.
While our results provide more and more information when β gets very
large, the results in [13], by contrast, seem to be more accurate when β is
close to 1. A corresponding analysis was performed earlier in the bulk in
theprogenitorpaper[19];see[14,16,20,23]formorerecentdevelopments.
In the case of"moderatelysized” β, 1 β , neitherof themethods
≪ ≪ ∞
seem to give very clear pictures of the situation. However, the paper [12]
suggeststhataphase-transition("freezing”)shouldtakeplaceafteracertain
finite value β = β . The study of existence and possible size of melting
0
temperature1/β iscurrentlyanactiveareaofresearch.
0
It is sometimes useful to consider n-dependentpotentials V = Q+h/n
n
where Q is as before and h is a fixed smooth function. See e.g. [3, 9, 21].
Our theorygeneralizeswithoutdifficultytothissituation. Onecan obtain
furthergeneralizationsbyrelaxingtheassumptionofreal-analyticityofthe
underlyingpotentialQ. Weleavedetailstotheinterestedreader
By the "hard edge β-ensemble” in external potential Q, we mean the
ensemble obtained by redefining Q to be + outside of the droplet. Cf.
∞
[4, 5, 26]for thecaseβ = 1. The questionofspacingsin thissettingwill be
takenupelsewhere.
WefinallywanttogiveafewremarkspertainingtoWard’sidentity. This
is a relation connecting the one- and two-point functions of a β-ensemble
which has been part of the physical mainstream for quite some time. In
thepresentcontext,itwasusedsystematicallybyWiegmannandZabrodin
andtheirschool,anditisanimportanttoolinconformalfieldtheory[22].
Inthepaper[3],Ward’sidentitywasusedtogivearelativelysimpleproof
of Gaussian field convergence of linear statistics of a β = 1 ensemble. Re-
centlyin[9]themethodwas,asitseems,extendedtogeneralβ-ensembles.
The one-dimensional counterpart to Ward’s identity is called the loop
equation. Itisconvenienttousedifferentnamesdependingondimension;
whilethetwoidentitiesarederivedinsimilarways,theyhavequitedifferent
analytical properties. The loop equation (or "fundamental relation”) was
usedin[19]tostudyfluctuationsinone-dimensionalβ-ensembles. See[10]
andthereferencesformorerecentdevelopmentsinthearea.
ThemicroscopicversionofWard’sidentitywasintroducedfairlyrecently
in[4]. ItiscalledWard’sequation. See[4,Section7.7]forthegeneralcaseof
β-ensembles. It is natural to ask how Ward’s equation fits into thepresent
context. Wehopetocomebacktothislateron.
References
[1] Ameur,Y.,AdensitytheoremforweightedFeketesets,Int.Math.Res.Notices2016,doi:
10.1093/imrn/rnw161.
[2] Ameur,Y.,Inpreparation.
[3] Ameur,Y.,Hedenmalm,H.,Makarov,N.,Wardidentitiesandrandomnormalmatrices,
Ann.Probab.43(2015),1157–1201.Cf.arXiv:1109.5941v3foradifferentversion.