The affine ensemble: determinantal point processes associated with the ax+b group

Abstract

We introduce the affine ensemble, a class of determinantal point processes (DPP) in the half-plane C+ associated with the ax + b group, depending on an admissible Hardy function. 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 in C+. As a special case one recovers the DPP related to the weighted Bergman kernel. When the Hardy function is chosen within a finite family whose Fourier transform are Laguerre functions, we obtain the DPP associated to hyperbolic Landau levels, modelled on the eigenspaces associated with the finite spectrum of the Maass Laplacian. The wave functions associated with these DPPs describe the quantum dynamics of a charged particle evolving on the open hyperbolic plane under the action of a constant magnetic field, with the finite number of eigenspaces dependent on the strenght of the magnetic field.

Full text

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
dtn
(e−ttα+n), t > 0.
and the following projective unitary group representation of P SL(2,R) on L2(C+, µ):
(1.4) bτα
naz +b
cz +dF(z) := |cz +d|
cz +d2n+α+1
Faz +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
WψH2C+:= {F∈L2(C+, µ+) : F=Wψf, f ∈H2C+}
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)] = EXψ(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)cWψψ(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)|2ZD(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
2D(0, R)c∩D(ξ(w), R)h∼1
2
arccos(1 −2ξ(w)2)
1−R2.
Thus,
|D(i, R)c∩D(w, R)|h∼1
2
arccos(1 −2w−i
w+i2)
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)|2ZD(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
21
−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
nC+=F∈L2C+, 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|2B−nz−w
w−zB
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 −2w−i
w+i2)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|2α+1
2z−w
w−zα+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|2B−nz−w
w−zB
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
nL2(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−t1
2i(x′−x)+ s′+s
2Lα
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)
nb2
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
22
iα+1 (w−z)n
(w−z)α+n+1 ·P(α,0)
n1−8s′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|2α
2+1
2z−w
w−zα
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∞
0tα/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