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THE AFFINE ENSEMBLE: DETERMINANTAL POINT PROCESSES

ASSOCIATED WITH THE ax +bGROUP

LU´

IS DANIEL ABREU, PETER BALAZS, AND SMILJANA JAKˇ

SI´

C

Abstract. We introduce the aﬃne ensemble, a class of determinantal point processes

(DPP) in the half-plane C+associated with the ax +b(aﬃne) group, depending on an

admissible Hardy function ψ. We obtain the asymptotic behavior of the variance, the

exact value of the asymptotic constant, and non-asymptotic upper and lower bounds for the

variance on a compact set Ω ⊂C+. As a special case one recovers the DPP related to the

weighted Bergman kernel. When ψis chosen within a ﬁnite family whose Fourier transform

are Laguerre functions, we obtain the DPP associated to hyperbolic Landau levels, the

eigenspaces of the ﬁnite spectrum of the Maass Laplacian with a magnetic ﬁeld.

1. Introduction

Determinantal point processes (DPPs) are random point distributions with negative corre-

lations between points determined by the reproducing kernel of some Hilbert space, usually

called the correlation kernel. Because of the repulsion inherent of the model, DPPs are

convenient to describe physical systems with charged-liked particles, to distribute random

sequences of points in selected regions while avoiding clustering, to promote diversity in

selection algorithms for machine learning [20], or to improve the rate of convergence in

Monte Carlo methods [8]. DPPs have been introduced by Odile Macchi to model fermion

distributions [22].

In this paper we introduce and study some aspects of the aﬃne ensemble, a family of

determinantal point processes on the complex upper half-plane C+, deﬁned in terms of a

representation of the ax +bgroup acting on a vector ψ∈H2(C+), the Hardy space in the

upper half plane. This can be seen as a geometric hyperbolic or algebraic non-unimodular

analogue of the Weyl-Heisenberg ensemble [3, 2], a family of planar euclidean determinantal

point processes associated with the (unimodular) Weyl-Heisenberg group. In terms of the

representation π(z)ψ(t) := s−1

2ψ(s−1(t−x)), the kernel of the aﬃne ensemble is deﬁned for

ψsuch that ∥ψ∥2= 1 and ∥Fψ∥L2(R+,t−1dt)<∞, as a normalizing constant times

(1.1) kψ(z, w) = ⟨π(w)ψ, π(z)ψ⟩H2(C+).

Key words and phrases. Determinantal point processes, hyperbolic half plane, aﬃne group.

This work was supported by the Austrian ministry BMBWF through the WTZ/OeAD-projects SRB

01/2018 ”ANACRES - Analysis and Acoustics Research” and MULT 10/2020 ”Time-Frequency representa-

tions for function spaces - Tireftus and FWF project ‘Operators and Time-Frequency Analysis’ P 31225-N32.

1

2 LU´

IS DANIEL ABREU, PETER BALAZS, AND SMILJANA JAKˇ

SI ´

C

Using the Fourier transform isomorphism F:H2(C+)→L2(0,∞), the kernel (1.1) can be

written in a more convenient form, for z=x+is, w =x′+is′∈C+,

(1.2) kψ(z, w) = (ss′)1

2Z∞

0

e−ix′ξ(Fψ)(s′ξ)e−ixξ(Fψ)(sξ)dξ.

By selecting special functions ψ, a number of processes arise as special cases, automat-

ically inheriting all properties of the aﬃne ensemble. In this paper we will consider only

examples with P SL(2,R) invariance, corresponding to invariance under the linear fractional

transformations of the half plane C+(dilations, rotations and translations). Consider the

mother wavelets chosen from the family {ψα

n}n∈N0,α > 0,

(1.3) (Fψα

n)(ξ) := ξα

2e−ξLα

n(2ξ), ξ > 0,

where Lα

ndenotes the generalized Laguerre polynomials

Lα

n(t) = t−αet

n!d

dtn

(e−ttα+n), t > 0.

and the following projective unitary group representation of P SL(2,R) on L2(C+, µ):

(1.4) bτα

naz +b

cz +dF(z) := |cz +d|

cz +d2n+α+1

Faz +b

cz +d,

where a, b, c, d are real numbers such that det"a b

c d #= 0. Then bτα

nleaves the Hilbert

space with reproducing kernel kψα

n(z, w) invariant. This has been shown in [1] and also that,

essentially, the choice (1.3) is the only leading to spaces invariant under representations of the

form (1.4), if we assume mild reasonable restrictions on ψ(see Theorem 3 in [1]). Moreover,

for ψwithin the family (1.3) and the parameter α= 2(B−n)−1 we obtain reproducing

kernels associated with the eigenspaces of the pure point spectrum of the Maass Laplacian

with weight B[21, 12, 4]:

HB:= s2∂2

∂x2+∂2

∂s2−2iBs ∂

∂x

The last part of this paper will be devoted to these special cases. The pure point spectrum

eigenspaces of the Maass Laplacian HBhave been used in [12, 4] to model the formation

of higher Landau levels in the hyperbolic plane. A physical model, put forward by Alain

Comtet in [12], describes a situation where the number of levels is constrained to be a ﬁnite

number, depending on the strength of the magnetic ﬁeld B, which must exceed a lower bound

for their existence (the magnetic ﬁeld has to be strong enough to capture the particle in a

closed orbit). The connection to analytic wavelets was suggested by the characterization of

hyperbolic Landau coherent states [23] and has been implicit used in [4] and more recently

in [1].

THE AFFINE ENSEMBLE 3

It is reasonable to expect interesting examples arising from other special choices, namely

those leading to the polyanalytic structure discovered by Vasilevski [30] (see [6, 18, 5] for

the special choices leading to polyanalytic spaces) but we will not explore this direction in

the present paper. One of our motivations for this research was the scarceness of examples

on hyperbolic DPPs. Besides the celebrated case studied by Peres and Vir´ag [25], we only

found the higher Landau levels DPP on the disc studied recently by Demni and Lazag in [14],

which strongly inﬂuenced the current paper. The aﬃne ensemble contains an uncountable

number of examples as special cases, of which we only explore a very few.

The paper is organized as follows. The next section contains the required background

on analytic wavelets and hyperbolic geometry. The third section is the core of the paper,

where the aﬃne ensemble is deﬁned and the main results are proved. Section 4 specializes

the results to the class of Maass-Landau ensembles and the calculations are detailed in the

last section of the paper, as an appendix.

2. Background

2.1. The continuous analytic wavelet transform. We will use the basic notation for

H2(C+), the Hardy space in the upper half plane, of analytic functions in C+with the norm

∥f∥H2(C+)= sup

0<s<∞Z∞

−∞

|f(x+is)|2dx < ∞.

To simplify the computations it is often convenient to use the equivalent deﬁnition (since

the Paley-Wiener theorem [15] gives F(H2(C+)) = L2(0,∞))

H2(C+) = f∈L2(R):(Ff)(ξ) = 0 for almost all ξ < 0.

Consider the ax +bgroup (see [11, Chapter 10] for the listed properties) G∼R×R+∼C+

with the multiplication

(x, s)·(x′, s′)=(x+sx′, ss′).

The identiﬁcation G∼C+is done by setting (x, s)∼x+is. The neutral element of the

group is (0,1) ∼iand the inverse element is given by (x, s)−1= (−x

s,1

s)≡ −x

s+i

s. The

ax +bgroup is not unimodular, since the left Haar measure on Gis dxds

s2and the right Haar

measure Gis dxds

s. The left Haar measure of a set Ω ⊆G,

|Ω|=ZΩ

dxds

s2,

coincides, under the identiﬁcation of the ax +bgroup with C+, with the hyperbolic measure

|Ω|=|Ω|h:= ZΩ

s−2dµC+(z),

where dµC+(z) is the Lesbegue measure in C+. We will write

(2.1) dµ+(z) = (Im z)−2dµC+(z).

4 LU´

IS DANIEL ABREU, PETER BALAZS, AND SMILJANA JAKˇ

SI ´

C

For every x∈Rand s∈R+, deﬁne the translation Txby Txf(t) = f(t−x) and the dilation

Dsf(t) = 1

√sf(t/s). Let z=x+is ∈C+and deﬁne the representation, for ψ∈H2(C+),

(2.2) π(z)ψ(t) := TxDsψ(t) = s−1

2ψ(s−1(t−x)).

The theory of general wavelet transforms using group representations requires the admissi-

bility condition [7], Z

G

|⟨ψ, π(z)ψ⟩|2dµ(z)<∞,

to construct square-integrable representations for general groups Gwith left Haar measures

µ. This will lead to an isometric transform thanks to the orthogonality relations (2.6) below.

In the Weyl-Heisenberg representations used in [3, 2], this follows trivially from the square-

integrability of ψonly. But here, in the aﬃne case, we need to take into account that

the ax +bgroup is not unimodular, and the diﬀerent left and right Haar measures of the

representation require a further condition on the integrability of ψ, which will be restricted

to the class of functions such that

(2.3) (ψ∈H2(C+)

0<2π∥Fψ∥2

L2(R+,t−1dt)=Cψ<∞.

Functions satisfying (2.3) are called admissible and the constant Cψis the admissibility

constant. Now, we have an irreducible and unitary representation πof the aﬃne group

on H2(C+) [7], deﬁned in (2.2) for an admissible ψ. By this deﬁnition any admissible

function is automatically in the Hardy space and so the inner product considered for the

wavelet transform or the reproducing kernel is in H2(C+). The continuous analytic wavelet

transform of a function fwith respect to a wavelet ψis deﬁned, for every z=x+is ∈C+,

as

(2.4) Wψf(z) = ⟨f, π(z)ψ⟩H2(C+).

More explicitly,

Wψf(z) = sup

0<s<∞

s−1

2Z∞

−∞

f(t)ψ(s−1(t−x))dt.

Using F(H2(C+)) = L2(0,∞), this can also be written (and we will do it as a rule to simplify

the calculations) as

(2.5) Wψf(z) = s1

2Z∞

0

f(ξ)e−ixξ(Fψ)(sξ)dξ.

As proven recently in [19], Wψf(z) only leads to analytic (Bergman) phase spaces for a very

special choice of ψ, but it is common practice to call it in general continuous analytic wavelet

transform. The orthogonality relations

(2.6) ZC+

Wψ1f1(x, s)Wψ2f2(x, s)dµ+(z)=2π⟨Fψ1,Fψ2⟩L2(R+,t−1dt)⟨f1, f2⟩H2(C+),

THE AFFINE ENSEMBLE 5

are valid for all f1, f2∈H2(C+) and ψ1, ψ2∈H2(C+) admissible. Then, setting ψ1=ψ2=

ψand f1=f2in (2.6), gives

ZC+

|Wψf(x, s)|2dµ+(z) = Cψ∥f∥2

H2(C+)

and the continuous wavelet transform provides an isometric inclusion Wψ:H2(C+)→

L2(C+,dµ+). Setting ψ1=ψ2=ψand f2=π(z)ψin (2.6) then for every f∈H2(C+) one

has

(2.7) Wψf(z) = 1

CψZC+

Wψf(w)⟨π(w)ψ, π(z)ψ⟩dµ+(z), z ∈C+.

Thus, the range of the wavelet transform

WψH2C+:= {F∈L2(C+, µ+) : F=Wψf, f ∈H2C+}

is a reproducing kernel subspace of L2(C+, µ+) with kernel

(2.8) kψ(z, w) = 1

Cψ

⟨π(w)ψ, π(z)ψ⟩H2(C+)=1

Cψ

Wψψ(w−1.z),and kψ(z, z) = ∥ψ∥2

2

Cψ

.

The Fourier transform F:H2(C+)→L2(0,∞) can be used to simplify computations, since

(2.9) ⟨π(w)ψ, π(z)ψ⟩H2(C+)=D\

π(w)ψ, \

π(z)ψEL2(R+,dt)= (ss′)1

2Z∞

0b

ψ(s′ξ)b

ψ(sξ)ei(x−x′)ξdξ.

2.2. Hyperbolic geometry. We will need some elementary facts of hyperbolic geometry.

The hyperbolic metric in C+is deﬁned as [10]

(2.10) d(z1, z2) = log 1 + ϱ(z1, z2)

1−ϱ(z1, z2)= 2 tanh−1(ϱ(z1, z2)) ,

where ϱis the pseudohyperbolic metric in C+,

ϱ(z1, z2) =

z1−z2

z1−z2.

The hyperbolic ball of center z∈C+and radius R < 1 is

D(z, R) = w∈C+:d(w, z)< R.

By direct computation it can be checked that ϱ(z−1, i) = ϱ(i, z) and that D(z−1, R) =

D(z, R).

6 LU´

IS DANIEL ABREU, PETER BALAZS, AND SMILJANA JAKˇ

SI ´

C

3. The affine ensemble

3.1. The aﬃne ensemble. Determinantal Point Processes (we simply list the concepts we

are using; for a complete deﬁnition see [9, Chapter 4]) are deﬁned using an ambient space

Λ, a Radon measure µdeﬁned on Λ, and a reproducing kernel Hilbert space Hcontained in

L2(C+,dµ+). The reproducing kernel of H,K(z, w), is the correlation kernel of the point

process X. The k-point intensities are given by ρk(x1, ..., xk) = det (K(xi, xj))1≤i,j≤k. Given

a set Ω ⊂C+, the 1-point intensity gives the expected number of points to be found in Ω:

E(X(Ω)) = ZΩ

ρ1(z)dµ+(z) = ZΩ

K(z, z)dµ+(z),

The normalization of the kernel Kψ(z, w) in the following deﬁnition is diﬀerent from the

one in (2.8) and is chosen so that Kψ(z, z) = 1. Recall that kψ(z, z) = ∥ψ∥2

2

Cψ.

Deﬁnition 1. The aﬃne ensemble Xψassociated with an admissible function ψis the

Determinantal Point Process with the normalized correlation kernel

(3.1) Kψ(z, w) = kψ(z, w)

kψ(z, z)=Wψψ(w−1·z)

∥ψ∥2

2

=We

ψe

ψ(w−1·z),

where e

ψ=ψ/∥ψ∥2.

We will assume from now on ∥ψ∥2= 1 (if this is not the case, we will use the notation

e

ψ=ψ/∥ψ∥2). Then

Kψ(z, w) = kψ(z, w) = Wψψ(w−1·z), Kψ(z, z) = Wψψ(i)

and, from (2.6),

(3.2) ZC+

|Wψψ(w)|2dµ+(w) = Cψ.

3.2. Variance estimates. We will consider an operator TΩacting on a function fon the

range of Wψ, which smooths out the energy of foutside Ω, by ﬁrst multiplication by 1Ω

and then projecting on the range of Wψ. Using the reproducing kernel property, TΩcan be

written as

(TΩf)(z) = ZΩ

f(w)Kψ(z, w)dµ+(w)(3.3)

=ZΩ

f(z′)ZΩ

Kψ(z, z′)Kψ(z′, w)dµ+(z′)dµ+(w),

providing an extension of TΩto the whole L2(C+,s−2dxds) vanishing in the complement of

the range of Wψ. By deﬁnition of 1-point intensity,

E(Xψ(Ω)) = ZΩ

Kψ(z, z)dµ+(z) = trace (TΩ) = |Ω|h.

THE AFFINE ENSEMBLE 7

while the number variance of Xψ(Ω) is (see [16, pg. 40] for a detailed proof ):

V[Xψ(Ω)] = EXψ(Ω)2−E(Xψ(Ω))2=trace (TΩ)−trace T2

Ω.

Our ﬁrst result gives the asymptotic behavior of the variance and the exact value of the

asymptotic constant. A related formula has been obtained by Shirai for Ginibre ensembles

in higher Landau levels [26]. A new proof of Shirai’s formula has been obtained by Demni

and Lazag [14], using a quite ﬂexible argument based on geometric considerations, which

inspired the following result.

Theorem 1. Let ψadmissible with ∥ψ∥2

H2(C+)= 1. We have the following explicit formula

for the variance of the aﬃne ensemble associated with ψ:

V[Xψ(D(i, R))] = ZC+

|Wψψ(w)|2|D(i, R)c∩D(w, R)|hdµ+(w).

Moreover, as R→1−,

(3.4) V[Xψ(D(i, R))] ∼cψ

1−R2,

where the asymptotic constant cψis given by

(3.5) cψ=1

2ZC+

|Wψψ(w)|2arccos 1−2

w−i

w+i

2!dµ+(w).

Proof. In the context of the concentration operator deﬁned in the beginning of the section,

set Ω = D(i, R). Observe that Kψ(z, z) = Wψψ(z·z−1) = Wψψ(i), that Wψψ(w−1·z) =

⟨π(w)ψ, π(z)ψ⟩H2(C+)=Wψψ(z−1·w), and use the reproducing kernel equation, dµ+as the

left Haar measure on the ax +bgroup, and Fubini, to write:

V[Xψ(D(i, R))]

=ZD(i,R)

Wψψ(i)dµ+(z)−ZD(i,R)×D(i,R)Wψψ(w−1·z)2dµ+(w)dµ+(z)

=ZD(i,R)×C+Wψψ(z−1·w)2dµ+(w)dµ+(z)−ZD(i,R)×D(i,R)Wψψ(z−1·w)2dµ+(w)dµ+(z)

=ZD(i,R)×D(i,R)cWψψ(z−1·w)2dµ+(w)dµ+(z)

=ZC+

1D(i,R)c(z)ZC+

1D(i,R)(w)Wψψ(z−1·w)2dµ+(w)dµ+(z)

=ZC+

1D(i,R)c(z)ZC+

1D(i,R)(z·w)|Wψψ(w)|2dµ+(w)dµ+(z)

=ZC+

1D(i,R)c(z)ZC+

1D(w−1,R)(z)|Wψψ(w)|2dµ+(w)dµ+(z)

=ZC+

|Wψψ(w)|2ZD(i,R)c∩D(w−1,R)

dµ+(z)dµ+(w),

8 LU´

IS DANIEL ABREU, PETER BALAZS, AND SMILJANA JAKˇ

SI ´

C

where 1D(i,R)(z·w)=1D(w−1,R)(z) follows from ϱ(z.w, i) = ϱ(w−1, z). Since D(w−1, R) =

D(w, R), we conclude that

(3.6) V[Xψ(D(i, R))] = ZC+

|Wψψ(w)|2|D(i, R)c∩D(w, R)|hdµ+(w).

To prove (3.4) and to determine the asymptotic constant (3.5), we will need to ﬁnd the area

|D(i, R)c∩D(w−1, R)|hwhen R→1. First move the integrals conformal to the unit disc.

Setting

ξ(z)=z−i

z+i∈D;ξ−1

(w)=w+ 1

i(w−1) ∈C+,

the measures can be related by

(Im z)αdµC+(z) = 2α+1 (1 −ξ(z)2)α

(1 −ξ(z))2α+4 dµD(ξ(z)),

dµDbeing the Lesbegue measure on D. Denoting by D(ξ(w), R) the hyperbolic disc on D

resulting from conformal mapping D(w, R) we have

ZD(i,R)c∩D(w,R)

1

(Im z)2dµC+(z) = 1

2ZD(0,R)c∩D(ξ(w),R)

(1 −ξ(z)2)−2dµD(ξ(z))

and we can use the computation of Theorem 1 in [14], leading to, as R→1−,

|D(i, R)c∩D(w, R)|h=1

2D(0, R)c∩D(ξ(w), R)h∼1

2

arccos(1 −2ξ(w)2)

1−R2.

Thus,

|D(i, R)c∩D(w, R)|h∼1

2

arccos(1 −2w−i

w+i2)

1−R2

It follows that, as R→1−,

V[Xψ(D(i, R))] ∼1

2

1

1−R2ZC+

|Wψψ(w)|2arccos(1 −2

w−i

w+i

2

)dµ+(w).

□

The next result shows with a two-sided inequality that the variance of the aﬃne ensemble

is proportional to |Ω|h. The ﬁrst part of the result is essentially an interpretation of the

results in [13], where the lower inequality is obtained for a large class of sets Ω, assuming

that, for some c > 0,

(3.7) ⟨π(z)ψ, π(w)ψ⟩H(C+)

2≥c

|D(z, R)|2

hZC+

1D(ξ,R)(z)1D(ξ,R)(w)dξ.

For Ω = D(i, R) we provide a proof of an upper bound involving the admissibility constant

Cψ.

THE AFFINE ENSEMBLE 9

Theorem 2. Assuming that (3.7) holds, we have

(3.8) |Ω|h≲V[Xψ(Ω)] ≤ |Ω|h.

If Ω = D(i, R), then

V[Xψ(D(i, R))] ≤Cψ|D(i, R)|h.

Proof. Using the operator TΩwe easily obtain an upper bound for the variance

V[Xψ(Ω)] = trace (TΩ)−trace T2

Ω≤trace (TΩ) = |Ω|h,

since trace (T2

Ω)≤trace (TΩ). The lower inequality follows from [13, Lemma 3.2], where it

is shown that, under the condition (3.7),

trace (TΩ)−trace T2

Ω≳|Ω|h.

Now set Ω = D(i, R). Then, from Theorem 1,

V[Xψ(D(i, R))] = ZC+

|Wψψ(w)|2ZD(i,R)c∩D(w,R)

dµ+(z)dµ+(w)

≤ZC+

|Wψψ(w)|2|D(w, R)|hdµ+(w)

=|D(i, R)|hZC+

|Wψψ(w)|2dµ+(w)

=Cψ|D(i, R)|h,

using |D(w, R)|h=|D(i, R)|hand (3.2). □

Remark 1. A reduced DPP adapted to Ωcan be deﬁned as a Toeplitz smooth restriction of

the aﬃne ensemble to Ωusing the operator (3.3), following a scheme similar to the one used

to deﬁne the ﬁnite Weyl-Heisenberg ensembles in [3]. Denoting by {pΩ

j}the eigenfunctions

of (3.3), one associates with Ωthe reduced ﬁnite dimensional Hilbert space

WNΩ

ψ=Span{pΩ

j}n=1,...,NΩ⊂Wψ,

where NΩ=⌊|Ω|h⌋, the least integer than or equal to |Ω|h. The Ω-reduced aﬃne ensemble is

the ﬁnite dimensional DPP XΩ

ψgenerated by the kernel

Kψ,Ω(z, w) =

NΩ

X

j=0

pΩ

j(z)pΩ

j(w).

10 LU´

IS DANIEL ABREU, PETER BALAZS, AND SMILJANA JAKˇ

SI ´

C

4. The Maass-Landau levels processes

4.1. Bergman spaces. The reproducing kernel of the space Wf

ψα

0(H2(C+)) is the following

weighted Bergman kernel (take n= 0 in Proposition 1 in the Appendix):

Kg

ψα

0

(z, w) = α(4 Im zIm w)α+1

21

−i(z−w)α+1

.

This is the ‘ground level’ case of the structure considered in the next section. For α= 1 it

is a C+weighted version of the DPP studied by Peres and Vir´ag [25].

4.2. Hyperbolic Maass-Landau levels. The Hamiltonian describing the dynamics of a

charged particle moving on the Poincar´e upper half-plane C+under the action of the mag-

netic ﬁeld Bis given by :

HB:= s2∂2

∂x2+∂2

∂s2−2iBs ∂

∂x

The operator HBwas ﬁrst introduced by Maass in number theory [21, 24] and its interpre-

tation as a hyperbolic analogue of the Landau Hamiltonian has been put forward by Comtet

(see [12, 4]). We list here the following important properties of HBas an operator.

(1) HBis an elliptic densely deﬁned operator on the Hilbert space L2(C+, dµ+), with a

unique self-adjoint realization that we denote also by HB.

(2) The spectrum of HBin L2(C+, dµ+) consists of two parts: a continuous part [1/4,+∞[,

corresponding to scattering states and a ﬁnite number of eigenvalues with inﬁnite de-

generacy (hyperbolic Landau levels) of the form

(4.1) ϵB

n:= (B−n) (1 −B+n), n = 0,1,2,· · · ,⌊B−1

2⌋.

The ﬁnite part of the spectrum exists provided 2B > 1. The notation ⌊a⌋stands for

the greatest integer not exceeding a.

(3) For each ﬁxed eigenvalue ϵB

n, we denote by

(4.2) EB

nC+=F∈L2C+, dµ+, HBF=ϵB

nF

the corresponding eigenspace, which has a reproducing kernel given by

Kn,B (z, w)

=(−1)nΓ (2B−n)

n!Γ (2B−2n)4 Im zIm w

|z−w|2B−nz−w

w−zB

F"−2B−n, −n

2B−2n;4 Im zIm w

|z−w|2#,

where Fis the Gauss hypergeometric function:

F"a, b

c;z#=2F1"a, b

c;z#=∞

X

n=0

(a)n(b)n

n!(c)n

zn.

THE AFFINE ENSEMBLE 11

The condition 2B > 1 ensuring the existence of these discrete eigenvalues in (2.) means

that the magnetic ﬁeld has to be strong enough to capture the particle in a closed orbit. If

this condition is not fulﬁlled, the motion will be unbounded and the orbit of the particle

will intercept the upper half-plane boundary whose points stand for ‘points at inﬁnity’ (see

[12, p. 189]). The eigenvalues in (2.) which are below the continuous spectrum have

eigenfunctions called bound states since the particle in such a state cannot leave the system

without additional energy. Then the number of particle layers (Landau levels), ⌊B−1

2⌋,

depends on the strength Bof the magnetic ﬁeld.

To make the identiﬁcation with special cases of the aﬃne ensemble kernel, in the Appendix

we compute the reproducing kernels and admissibility constants of the spaces Wf

ψα

n(H2(C+))

in terms of hypergeometric functions and Jacobi polynomials. According to Proposition 1,

the reproducing kernel of W^

ψ2(B−n)−1

n

=Wf

ψα

n(H2(C+)) is given by

K^

ψ2(B−n)−1

n

(z, w) = Kn,B (z, w) .

Thus, the kernels K^

ψ2(B−n)−1

n

(z, w), are precisely the reproducing kernels of the eigenspaces

associated with the pure point spectrum of the Maass Laplacian. As a result, all properties of

the aﬃne ensemble are automatically translated to the DPP associated with the reproducing

kernels Kψ2(B−n)−1

n(z, w), with asymptotic constant

c^

ψ2(B−n)−1

n

=1

2ZC+W^

ψ2(B−n)−1

n

^

ψ2(B−n)−1

n(w)

2

arccos(1 −2

w−i

w+i

2

)dµ+(w)

=1

2ZC+

|Kn,B(i, w)|2arccos(1 −2

w−i

w+i

2

)dµ+(w)

and admissibility constant

C^

ψ2(B−n)−1

n

=4π

2(B−n)−1.

Then, Theorems 1 and 2 lead to the following result for the Maass-Landau process, the DPP

XB,n generated by the reproducing kernel Kn,B (z, w)of the eigenspace of HBassociated with

the Maass-Landau level eigenvalue ϵB

n:= (B−n) (1 −B+n), for n= 0,1,2,· · · ,⌊B−1

2⌋.

Corollary 1. The variance of XB,n is given by

V[XB,n(D(i, R))] = ZC+

|Kn,B(i, w)|2|D(i, R)c∩D(w, R)|hdµ+(w).

Moreover, when R→1−,

V[XB,n(D(i, R))] ∼cn,B

1−R2,

12 LU´

IS DANIEL ABREU, PETER BALAZS, AND SMILJANA JAKˇ

SI ´

C

where cn,B =1

2RC+|Kn,B(i, w)|2arccos(1 −2w−i

w+i2)dµ+(w). Finally, the variance of XB,n

satisﬁes the non-asymptotic bound

V[XB,n(D(i, R))] ≤4π

2(B−n)−1|D(i, R)|h.

5. Appendix

5.1. Reproducing kernels of special aﬃne ensembles. Let α > −1. For n= 0,1,2, . . .

we deﬁne the normalized functions f

ψα

nsuch that

f

ψα

n

H2(C+)= 1:

(Ff

ψα

n)(t) = s2α+1n!

Γ(n+α+ 1)tα

2e−tLα

n(2t), t > 0.

Proposition 1. The reproducing kernel of Wf

ψα

nis given by

Kf

ψα

n(z, w) = (−1)nΓ(n+1+α)

n!Γ(1 + α)4 Im zIm w

|z−w|2α+1

2z−w

w−zα+1

2+n

×F"n+α+ 1,−n

1 + α;4 Im zIm w

|z−w|2#,

where F=2F1denotes the hypergeometric function. Setting α= 2(B−n)−1we obtain

K^

ψ2(B−n)−1

n

(z, w)

=(−1)nΓ(2B−n)

n!Γ(2B−2n)4 Im zIm w

|z−w|2B−nz−w

w−zB

F"2B−n, −n

2B−2n;4 Im zIm w

|z−w|2#

Proof. For simpliﬁcation write z=x+is, w =x′+is′∈C+. Formula (2.9) gives:

Kf

ψα

n(z, w) = ⟨π(w)f

ψα

n, π(z)f

ψα

n⟩H2(C+)=\

π(w)f

ψα

n,\

π(z)f

ψα

nL2(R+)

=2α+1n!

Γ(n+α+ 1)s1

2s′1

2Z∞

0

e−ix′t(ts′)α

2e−s′tLα

n(2s′t)eixt(ts)α

2e−stLα

n(2st)dt

=n!

Γ(n+α+ 1)sα

2+1

2s′α

2+1

2Z∞

0

tαe−t1

2i(x′−x)+ s′+s

2Lα

n(s′t)Lα

n(st)dt.

To compute the integral we will use the following integral formula [17, p. 810, 7.414 (13)]:

Z∞

0

e−t(k+a1+a2

2)tαLα

n(a1t)Lα

n(a2t)dt =Γ(1 + α+n)

b1+α+n

0

bn

2

n!P(α,0)

nb2

1

b0b2,

where

b0=k+a1+a2

2, b2=k−a1+a2

2, b2

1=b0b2+ 2a1a2,Re α > −1,Re b0>0

THE AFFINE ENSEMBLE 13

and P(α,β)

ndenotes the Jacobi polynomial. Setting

b0=1

2i(x′−x) + s′+s=1

2i(x′−x−is′−is) = 1

2i(x′−is′−(x+is)) = 1

2i(w−z),

b2=1

2i(x′−x)−s′−s=1

2i(x′−x+is′+is) = 1

2i(x′+is′−(x−is)) = 1

2i(w−¯z)

and

b2

1=1

4(−|z−¯w|2+ 8s′s).

gives

Kf

ψα

n(z, w)=(ss′)α

2+1

22

iα+1 (w−z)n

(w−z)α+n+1 ·P(α,0)

n1−8s′s

|z−w|2,

where the Jacobi polynomial is deﬁned as

(5.1) P(α,β)

n(x) = Γ(n+1+α)

n!Γ(1 + α)F"n+α+β+ 1,−n

1 + α;1−x

2#.

Thus,

Kf

ψα

n(z, w)

= (4ss′)α

2+1

21

iα+1

Γ(n+α+ 1)

n!Γ(α+ 1)

(w−z)n

(w−z)α+n+1 F"n+1+α, −n

1 + α;4s′s

|z−w|2#.

Next, notice that

1

(|z−w|2)α

2+1

2

(w−z)n(z−w)α

2+1

2(z−w)α

2+1

2

(w−z)α+n+1

=1

(|z−w|2)α

2+1

2

(−1)n(z−w)n+α

2+1

2(−1)α

2+1

2(w−z)α

2+1

2

(w−z)α+n+1 .

Hence,

Kf

ψα

n(z, w)

=(−1)nΓ(n+α+ 1)

n!Γ(α+ 1) 4s′s

|z−w|2α

2+1

2z−w

w−zα

2+1

2+n

F"n+1+α, −n

1 + α;4s′s

|z−w|2#.

□

5.2. Norms and admissible constants. The orthogonality relation

(5.2) Z+∞

0

Lα

n(t)Lα

m(t)tαe−tdt =Γ(n+α+ 1)

n!δn,m

written as Z+∞

0

tα

2e−tLα

n(2t)tα

2e−tLα

m(2t)dt =Γ(n+α+ 1)

2α+1n!δn,m

14 LU´

IS DANIEL ABREU, PETER BALAZS, AND SMILJANA JAKˇ

SI ´

C

shows that Ff

ψα

n∈L2(R+). Hence, we observe that f

ψα

n∈H2(C+) and

f

ψα

n

2

H2(C+)=

Ff

ψα

n

2

L2(R+)= 1.

Next, we calculate the admissibility constant Cf

ψα

n. The formula [27, (10)] gives

Z∞

0

tα−1e−tLα

n(2t)2dt =Γ(n+α+ 1)

2αn!α

Cf

ψα

n= 2π∥F f

ψα

n∥L2(R+,t−1)=π2α+2n!

Γ(n+α+ 1) Z∞

0tα/2e−tLα

n(2t)2dt

t=4π

α.

Consequently,

C^

ψ2(B−n)−1

n

=2

2(B−n)−1.

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Email address:abreuluisdaniel@gmail.com

NuHAG, Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, A-

1090, Vienna, Austria

Email address:peter.balazs@oeaw.ac.at

Acoustics Research Institute, Vienna 1040, Austria

Email address:smiljana.jaksic@sfb.bg.ac.rs

Faculty of Forestry, University of Belgrade, Kneza Vi seslava 1, 11000, Belgrade, Serbia