text: FOURIER INTERPOLATION ON THE REAL LINE 11 and the series converges absolutely. Our next task is to show that this identity holds ∈ for all τ H. ∈ { −} ∈ R Proposition 2. For any ε +, and x the function F (τ; x) admits an analytic ε continuation to H. Moreover, the analytic continuation satisﬁes the functional equations (cid:16) (cid:17) − F (τ; x) F (τ + 2; x) = 0, ε ε (24) − − − 1 − − − F (τ; x) + ε( iτ) 1/2F ; x = eiπτx2 + ε( iτ) 1/2eiπ( 1/τ)x2. ε ε τ Proof. To prove the theorem, it is enough to show that there exists an analytic contin- D uation to some open set Ω containing the boundary of , on which the equations (24) hold. Indeed, we can then choose Ω such that D ⊂ ⊂ D ∪ D ∪ D ∪ − D Ω S T 2 T 2 ∪ Since ∈ gΩ = H, we can construct a continuation by repeatedly using (24). Since H g Γ θ is simply-connected, this gives a well-deﬁned analytic function on H. The ﬁrst functional equation in (24) is clearly satisﬁed, since the integral that deﬁnes F { ∈ ∀ ∈ Z | − | }ε automatically deﬁnes a 2-periodic function on the open set τ H : k , τ 2k > 1 ± that contains the vertical lines Im(τ) = 1. Hence, we only need to deal with the second functional equation. We can get an analytic { ∈ | | (cid:54) } continuation of F to some neighborhood of z H : z = 1, z = i by changing the ε contour of integration in (22). First, we rewrite the integral as (cid:90) (cid:90) (cid:90) i (cid:90) 1 2F (τ, x) = K (τ, z)eiπx2zdz + K (τ, z)eiπx2zdz ε ε ε − 1 i (cid:90) i − i − − − (25) = K (τ, z)eiπx2zdz K (τ, 1/z)eiπx2( 1/z)z 2dz ε ε − − 1 1 i − − − = K (τ, z)(eiπx2z + ε( iz) 1/2 eiπx2( 1/z))dz, ε − 1 where we have used the transformation property (20). Note, that if τ belongs to D ∪ D D ∪ D S , then the only poles of K (τ, z) (as a function of z) inside S are at z = τ − ε − and z = 1/τ. Let γ be the circle arc from 1 to i, and let γ be a simple smooth path − 1 D 2 F from 1 to i that lies inside S and strictly below γ . Denote by the region enclosed 1 F between γ and γ . We will now build a continuation of F to and show that it satisﬁes 1 2 (cid:90) ε the functional equation. We deﬁne a continuation by the contour integral 1 − − − F˜ (τ; x) = K (τ, z)(eiπx2z + ε( iz) 1/2eiπx2( 1/z)) dz. ε ε 2 γ2 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: FOURIER

text: INTERPOLATION

text: ON

text: THE

text: REAL

text: LINE

text: and

text: the

text: series

text: converges

text: absolutely.

text: Our

text: next

text: task

text: is

text: to

text: show

text: that

text: this

text: identity

text: holds

text: for

text: all

text: τ

text: H.

text: R

text: Proposition

text: For

text: any

text: ε

text: and

text: x

text: the

text: function

text: F

text: (τ;

text: x)

text: admits

text: an

text: analytic

text: ε

text: continuation

text: to

text: H.

text: Moreover,

text: the

text: analytic

text: continuation

text: satisﬁes

text: the

text: functional

text: equations

text: F

text: (τ;

text: x)

text: F

text: (τ

text: x)

text: ε

text: ε

text: F

text: (τ;

text: x)

text: ε(

text: iτ)

text: x

text: ε(

text: iτ)

text: ε

text: ε

text: τ

text: Proof.

text: To

text: prove

text: the

text: theorem,

text: it

text: is

text: enough

text: to

text: show

text: that

text: there

text: exists

text: an

text: analytic

text: contin-

text: D

text: uation

text: to

text: some

text: open

text: set

text: Ω

text: containing

text: the

text: boundary

text: of

text: on

text: which

text: the

text: equations

text: hold.

text: Indeed,

text: we

text: can

text: then

text: choose

text: Ω

text: such

text: that

text: D

text: D

text: D

text: D

text: D

text: Ω

text: S

text: T

text: T

text: Since

text: gΩ

text: H,

text: we

text: can

text: construct

text: a

text: continuation

text: by

text: repeatedly

text: using

text: Since

text: H

text: g

text: Γ

text: θ

text: is

text: simply-connected,

text: this

text: gives

text: a

text: well-deﬁned

text: analytic

text: function

text: on

text: H.

text: The

text: ﬁrst

text: functional

text: equation

text: in

text: is

text: clearly

text: satisﬁed,

text: since

text: the

text: integral

text: that

text: deﬁnes

text: F

text: Z

text: }ε

text: automatically

text: deﬁnes

text: a

text: function

text: on

text: the

text: open

text: set

text: τ

text: H

text: k

text: τ

text: that

text: contains

text: the

text: vertical

text: lines

text: Im(τ)

text: Hence,

text: we

text: only

text: need

text: to

text: deal

text: with

text: the

text: second

text: functional

text: equation.

text: We

text: can

text: get

text: an

text: analytic

text: continuation

text: of

text: F

text: to

text: some

text: neighborhood

text: of

text: z

text: H

text: z

text: z

text: i

text: by

text: changing

text: the

text: ε

text: contour

text: of

text: integration

text: in

text: First,

text: we

text: rewrite

text: the

text: integral

text: as

text: i

text: (τ,

text: x)

text: K

text: (τ,

text: K

text: (τ,

text: ε

text: ε

text: ε

text: i

text: i

text: i

text: K

text: (τ,

text: K

text: (τ,

text: ε

text: ε

text: i

text: K

text: (τ,

text: ε(

text: iz)

text: ε

text: where

text: we

text: have

text: used

text: the

text: transformation

text: property

text: Note,

text: that

text: if

text: τ

text: belongs

text: to

text: D

text: D

text: D

text: D

text: S

text: then

text: the

text: only

text: poles

text: of

text: K

text: (τ,

text: z)

text: (as

text: a

text: function

text: of

text: z)

text: inside

text: S

text: are

text: at

text: z

text: τ

text: ε

text: and

text: z

text: Let

text: γ

text: be

text: the

text: circle

text: arc

text: from

text: to

text: i,

text: and

text: let

text: γ

text: be

text: a

text: simple

text: smooth

text: path

text: D

text: F

text: from

text: to

text: i

text: that

text: lies

text: inside

text: S

text: and

text: strictly

text: below

text: γ

text: Denote

text: by

text: the

text: region

text: enclosed

text: F

text: between

text: γ

text: and

text: γ

text: We

text: will

text: now

text: build

text: a

text: continuation

text: of

text: F

text: to

text: and

text: show

text: that

text: it

text: satisﬁes

text: ε

text: the

text: functional

text: equation.

text: We

text: deﬁne

text: a

text: continuation

text: by

text: the

text: contour

text: integral

text: F˜

text: (τ;

text: x)

text: K

text: (τ,

text: ε(

text: iz)

text: dz.

text: ε

text: ε

text: γ2

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 12 RADCHENKO AND VIAZOVSKA D i γ Sτ 1 τ F γ 2 − 1 1 Figure 2. Fundamental domain for Γ and the contour of integration. θ Clearly, F = F˜ f(cid:16)or τ wit(cid:17)h big enough ima(cid:90)ginary part, so F˜ indeed deﬁnes an analytic ε Fε ∈ F continuation to . For τ we compute − − √ ε − 1 − 1 (cid:90) εK (√1/τ, z) εei√πx2( 1/z) F˜ (τ; x) + − F ; x = F˜ (τ; x) −ε − (eiπx2z + − )dz ε ε ε iτ τ 2 iτ iz γ1 (cid:90) − 1 − − − = F˜ (τ; x) K (τ, z)(eiπx2z + ε( iz) 1/2eiπx2( 1/z))dz ε ε 2 γ1 1 (cid:88) (cid:0) − − − (cid:1) = K (τ, z)(eiπx2z + ε( iz) 1/2eiπx2( 1/z))dz ε 2 F ∂ − − − = iπ Res K (τ, z)(eiπx2z + ε( iz) 1/2eiπx2( 1/z)) z=x ε ∈F x − − − = eiπx2τ + ε( iτ) 1/2eiπx2( 1/τ), which is precisely the functional equation that we needed. Similar computation works for the arc from i to 1. The only thing that is left is to check that F has no pole at τ = i. ε − − For ε = 1 this follows from the second functional equation, while for ε = 1 both 2λ(z) 1 − − − and eiπzr2 + ε( iz) 1/2eiπ( 1/z)r2 vanish at z = i, so that they cancel the double pole at i − (cid:3) coming from J(i) J(z), and hence the integral (25) converges at τ = i. ∈ As an immediate corollary, we obtain that formula (23) is valid for all τ H. This already implies that for all δ > 0 we have bε (x) = O((1 + δ)n). In the next section we n prove a much stronger estimate. Note that the only properties of K that were used in the proof are the modularity in τ ε ∈ and in z, as well as the fact that the only poles are at z Γ τ, and that the residue at θ z = τ is equal to 1/(iπ). 5. Growth estimate The main result of this section is the following. ∈ { −} Theorem 4. For any ε +, the numbers bε (x) satisfy n | | bε (x) = O(n2) n uniformly in x. ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: paragraph

text: RADCHENKO

text: AND

text: VIAZOVSKA

text: D

text: i

text: γ

text: Sτ

text: τ

text: F

text: γ

text: Figure

text: Fundamental

label: caption

text: domain

text: for

text: Γ

text: and

text: the

text: contour

text: of

text: integration.

text: θ

text: Clearly,

text: F

text: F˜

text: τ

text: big

text: enough

text: part,

text: so

text: F˜

text: indeed

text: deﬁnes

text: an

text: analytic

text: ε

text: Fε

text: F

text: continuation

text: to

text: For

text: τ

text: we

text: compute

text: ε

text: εK

text: z)

text: F˜

text: (τ;

text: x)

text: F

text: x

text: F˜

text: (τ;

text: x)

text: −ε

text: )dz

text: ε

text: ε

text: ε

text: iτ

text: τ

text: iτ

text: iz

text: γ1

text: F˜

text: (τ;

text: x)

text: K

text: (τ,

text: ε(

text: iz)

text: ε

text: ε

text: γ1

text: K

text: (τ,

text: ε(

text: iz)

text: ε

text: F

text: iπ

text: Res

text: K

text: (τ,

text: ε(

text: iz)

text: z=x

text: ε

text: ∈F

text: x

text: ε(

text: iτ)

text: which

text: is

text: precisely

text: the

text: functional

text: equation

text: that

text: we

text: needed.

text: Similar

text: computation

text: works

text: for

text: the

text: arc

text: from

text: i

text: to

text: The

text: only

text: thing

text: that

text: is

text: left

text: is

text: to

text: check

text: that

text: F

text: has

text: no

text: pole

text: at

text: τ

text: i.

text: ε

text: For

text: ε

text: this

text: follows

text: from

text: the

text: second

text: functional

text: equation,

text: while

text: for

text: ε

text: both

text: and

text: ε(

text: iz)

text: vanish

text: at

text: z

text: i,

text: so

text: that

text: they

text: cancel

text: the

text: double

text: pole

text: at

text: i

text: coming

text: from

text: J(i)

text: J(z),

text: and

text: hence

text: the

text: integral

text: converges

text: at

text: τ

text: i.

text: As

text: an

text: immediate

text: corollary,

text: we

text: obtain

text: that

text: formula

text: is

text: valid

text: for

text: all

text: τ

text: H.

text: This

text: already

text: implies

text: that

text: for

text: all

text: δ

text: we

text: have

text: bε

text: (x)

text: δ)n).

text: In

text: the

text: next

text: section

text: we

text: n

text: prove

text: a

text: much

text: stronger

text: estimate.

text: Note

text: that

text: the

text: only

text: properties

text: of

text: K

text: that

text: were

text: used

text: in

text: the

text: proof

text: are

text: the

text: modularity

text: in

text: τ

text: ε

text: and

text: in

text: z,

text: as

text: well

text: as

text: the

text: fact

text: that

text: the

text: only

text: poles

text: are

text: at

text: z

text: Γ

text: τ,

text: and

text: that

text: the

text: residue

text: at

text: θ

text: z

text: τ

text: is

text: equal

text: to

text: Growth

text: estimate

text: The

text: main

text: result

text: of

text: this

text: section

text: is

text: the

text: following.

text: Theorem

text: For

text: any

text: ε

text: the

text: numbers

text: bε

text: (x)

text: satisfy

text: n

text: bε

text: (x)

text: n

text: uniformly

text: in

text: x.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: FOURIER INTERPOLATION ON THE REAL LINE 7 (cid:28) We freely use the standard big O notation. In addition, we also use Vinogradov’s “ ” sign (cid:28) ⇔ f g f = O (g). ε,δ,... ε,δ,... Notationally, we prefer to use “O” for sequences and additive remainders, while for most (cid:28) inequalities with implied constants we use “ ”. 3. Weakly holomorphic modular forms on Γ of weight 3/2 θ We begin by constructing a basis for a certain space of weakly holomorphic modular { } { − } forms of weight 3/2. Namely, let g+(z) ≥ and g (z) ≥ be two collections of holo- n n 0 n n 1 morphic functions on the upper half-plane H that satisfy the transformation properties gε(z + 2) = gε(z), n n (15) − − − ( iz) 3/2gε( 1/z) = εgε(z), n n as well as the following behavior at the cusps − → ∞ g+(z) = q n/2 + O(q1/2), z i , n − − → ∞ (16) g (z) = q n/2 + O(1), z i , n → → ∞ gε(1 + i/t) 0, t . n The reason behind these conditions will be made clear in the next section. We make the following ansatz: − g+(z) = θ3(z)P +(J 1(z)), n n (17) − − − − g (z) = θ3(z)(1 2λ(z))P (J 1(z)), n n ± ∈ Q − where P [x] are monic polynomials of degree n and P (0) = 0. The polynomials ± n n − P are uniquely determined by the ﬁrst two conditions in (16), since J 1 has q-expansion n − ± starting with q 1/2+24+O(q1/2), and thus the coeﬃcients of P can be found by inverting n an upper-triangular matrix. The transformation properties (15) follow from the properties of J(z) and λ(z). The ﬁrst few of these functions are − · − · − g+ = θ3, g = θ3 (1 2λ) (J 1), 0 1 · − − − · − · − − − g+ = θ3 (J 1 30), g = θ3 (1 2λ) (J 2 22J 1), 1 2 · − − − − · − · − − − − g+ = θ3 (J 2 54J 1 + 192), g = θ3 (1 2λ) (J 3 46J 2 + 252J 1). 2 3 ± The polynomials P are analogues of Faber polynomials, see [3]. The main diﬀerence n here is that the group Γ has two cusps, so some additional normalization is needed. In θ { ±} the next theorem we give closed form expressions for generating functions of g . n { } { − } Theorem 3. T(cid:88)he generating functions for g+(z) ≥ and g (z) ≥ are given by n n 0 n n 1 ∞ − θ(τ)(1 2λ(τ))θ3(z)J(z) (cid:88) g+(z)eiπnτ = − =: K (τ, z), n J(z) J(τ) + n=0 (18) ∞ − − θ(τ)J(τ)θ3(z)(1 2λ(z)) g (z)eiπnτ = − =: K−(τ, z). n J(z) J(τ) n=1 ##LTLine## ##LTLine##

text: FOURIER

text: INTERPOLATION

text: ON

text: THE

text: REAL

text: LINE

text: We

text: freely

text: use

text: the

text: standard

text: big

text: O

text: notation.

text: In

text: addition,

text: we

text: also

text: use

text: Vinogradov’s

text: sign

text: f

text: g

text: f

text: O

text: (g).

text: ε,δ,...

text: ε,δ,...

text: Notationally,

text: we

text: prefer

text: to

text: use

text: “O”

text: for

text: sequences

text: and

text: additive

text: remainders,

text: while

text: for

text: most

text: inequalities

text: with

text: implied

text: constants

text: we

text: use

label: section

text: Weakly

text: holomorphic

text: modular

text: forms

text: on

text: Γ

text: of

text: weight

text: θ

text: We

text: begin

text: by

text: constructing

text: a

text: basis

text: for

text: a

text: certain

text: space

text: of

text: weakly

text: holomorphic

text: modular

text: forms

text: of

text: weight

text: Namely,

text: let

text: g+(z)

text: and

text: g

text: (z)

text: be

text: two

text: collections

text: of

text: holo-

text: n

text: n

text: n

text: n

text: morphic

text: functions

text: on

text: the

text: upper

text: half-plane

text: H

text: that

text: satisfy

text: the

text: transformation

text: properties

text: gε(z

text: gε(z),

text: n

text: n

text: iz)

text: εgε(z),

text: n

text: n

text: as

text: well

text: as

text: the

text: following

text: behavior

text: at

text: the

text: cusps

text: g+(z)

text: q

text: z

text: i

text: n

text: g

text: (z)

text: q

text: z

text: i

text: n

text: i/t)

text: t

text: n

text: The

text: reason

text: behind

text: these

text: conditions

text: will

text: be

text: made

text: clear

text: in

text: the

text: next

text: section.

text: We

text: make

text: the

text: following

text: ansatz:

text: g+(z)

text: +(J

text: n

text: n

text: g

text: (z)

text: (J

text: n

text: n

text: Q

text: where

text: P

text: [x]

text: are

text: monic

text: polynomials

text: of

text: degree

text: n

text: and

text: P

text: The

text: polynomials

text: n

text: n

text: P

text: are

text: uniquely

text: determined

text: by

text: the

text: ﬁrst

text: two

text: conditions

text: in

text: since

text: J

text: has

text: q-expansion

text: n

text: starting

text: with

text: q

text: and

text: thus

text: the

text: coeﬃcients

text: of

text: P

text: can

text: be

text: found

text: by

text: inverting

text: n

text: an

text: upper-triangular

text: matrix.

text: The

text: transformation

text: properties

text: follow

text: from

text: the

text: properties

text: of

text: J(z)

text: and

text: λ(z).

text: The

text: ﬁrst

text: few

text: of

text: these

text: functions

text: are

text: g+

text: g

text: θ3

text: (J

text: g+

text: θ3

text: (J

text: g

text: θ3

text: (J

text: g+

text: θ3

text: (J

text: g

text: θ3

text: (J

text: The

text: polynomials

text: P

text: are

text: analogues

text: of

text: Faber

text: polynomials,

text: see

text: The

text: main

text: diﬀerence

text: n

text: here

text: is

text: that

text: the

text: group

text: Γ

text: has

text: two

text: cusps,

text: so

text: some

text: additional

text: normalization

text: is

text: needed.

text: In

text: θ

text: the

text: next

text: theorem

text: we

text: give

text: closed

text: form

text: expressions

text: for

text: generating

text: functions

text: of

text: g

text: n

text: Theorem

text: generating

text: functions

text: for

text: g+(z)

text: and

text: g

text: (z)

text: are

text: given

text: by

text: n

text: n

text: n

text: n

text: g+(z)eiπnτ

text: K

text: (τ,

text: z),

text: n

text: J(z)

text: J(τ)

text: g

text: (z)eiπnτ

text: K−(τ,

text: z).

text: n

text: J(z)

text: J(τ)

text: ##LTLine##

text: ##LTLine##


text: TRUNCATED CONTROL VARIATES FOR WEAK APPROXIMATION SCHEMES 15 then, due to (1.5) and (7.1), we get for (cid:88)l = j(cid:16), . . . , J − 1, (cid:17) ≤ − ρr,k (1 + A ∆)ρr,k + C ∆(ρr,k + ρr,s ) + (d 1)B ∆ρr,s . l+1,2,1 2 l,2,1 1,1 l,2,1 l,2,1 2 l,2,1 (cid:54) s:s=k (cid:88) Further, denote d ρr := ρr,s , l+1,n,1 l+1,n,1 s=1 then we get ≤ − − ρr (1 + A ∆)ρr + 2(d 1)C ∆ρr + (d 1)2B ∆ρr . l+1,2,1 2 l,2,1 1,1 l,2,1 2 l,2,1 This gives us ≤ ρr (1 + κ ∆)ρr l+1,2,1 1 l,2,1 for some constant κ > 0, leading to 1 ≤ − − − (7.2) ρr (1 + κ ∆)l j 1ρr , l = j + 1, . . . , J 1, l,2,1 1 (cid:34)j+1,2,1 (cid:35) (cid:18) (cid:19) (cid:88) where d E ∂ 2 ρr = Φs (x, ξ ) , j+1,2,1 ∂xr ∆ j+1 which is bounded due to (1.5). Together ws=i1th (7.2) we obtain the boundedness of {ρr : J ∈ N} and J,2,1 hence the bound(cid:12)edness of (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:88) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) d ∂ ≤ E(cid:118) ∂ ∂ ∂xr qj(x) (cid:88) (cid:117)(cid:117) ∂x(cid:34)s(cid:18)f(GJ,j(x))∂xr G(cid:19)sJ,j(cid:35)(x) s=1 (cid:116) ≤ (cid:118)(cid:117)(cid:117)d (cid:88)E (cid:34)(cid:18)∂∂xs f(GJ,j(x)) (cid:19)2 (cid:35)ρrJ,,s2,1 (cid:113) (cid:116)s=1 ≤ d d E ∂ f(G (x)) 2 ρr,s ≤ const ρr ∂xs J,j J,2,1 J,2,1 s=1 ∈ { } (cid:80) (cid:80) for all r 1, . . . , d , since f is assumed to be continuously diﬀerentiable with bounded partial derivatives. (cid:34) (cid:35) ≤ − Let us proceed with (cid:18)the case K = 2(cid:19). We have, due to ( d a )n dn 1 d an, k=1 k k=1 k E  ∂ 4  Gk (x) ∂xr (cid:88)l+1,(cid:0)j (cid:1)   ≤ E − − − γ4 + 4γ3γ + 6(d 1)γ2γ2 + 4(d 1)2γ γ3 + (d 1)3γ4 k k s k s k s s (cid:54) s:s=k and thus, due to 4a3b ≤ 3a4 + b4 a(cid:88)nd 2(cid:16)a2b2 ≤ a4 + b4, ≤ − ρr,k (1 + A ∆)ρr,k + C ∆(3ρr,k + ρr,s ) + 3(d 1)C ∆(ρr,k + ρr,s )(cid:17) l+1,4,1 4 l,4,1 3,1 l,4,1 l,4,1 2,2 l,4,1 l,4,1 (cid:54) s:s=k − − +(d 1)2C ∆(ρr,k + 3ρr,s ) + (d 1)3B ∆ρr,s . 1,3 l,4,1 l,4,1 4 l,4,1 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: TRUNCATED

text: CONTROL

text: VARIATES

text: FOR

text: WEAK

text: APPROXIMATION

text: SCHEMES

text: then,

text: due

text: to

text: and

text: we

text: get

text: for

label: equation

text: J

text: ρr,k

text: A

text: ∆)ρr,k

text: C

text: ∆(ρr,k

text: ρr,s

text: (d

text: ∆ρr,s

font: default

text: s:s=k

text: Further,

text: denote

text: d

text: ρr

text: ρr,s

text: then

text: we

text: get

text: ρr

text: A

text: ∆)ρr

text: ∆ρr

text: (d

text: ∆ρr

text: This

text: gives

text: us

text: ρr

text: κ

text: ∆)ρr

text: for

text: some

text: constant

text: κ

text: leading

text: to

text: ρr

text: κ

text: ∆)l

text: j

text: l

text: j

text: J

text: where

text: d

text: E

text: ρr

text: Φs

text: (x,

text: ξ

text: ∂xr

text: which

text: is

text: bounded

text: due

text: to

text: Together

text: we

text: obtain

text: the

text: boundedness

text: of

text: {ρr

text: J

text: N}

text: and

text: hence

text: the

text: of

text: d

text: ∂xr

text: qj(x)

text: f(GJ,j(x))

text: d

text: d

text: E

text: f(G

text: (x))

text: ρr,s

text: const

text: ρr

text: ∂xs

text: J,j

text: for

text: all

text: r

text: d

text: since

text: f

text: is

text: assumed

text: to

text: be

text: continuously

text: diﬀerentiable

text: with

text: bounded

text: partial

text: derivatives.

text: Let

text: us

text: proceed

text: with

text: case

text: K

text: We

text: have,

text: due

text: to

text: d

text: a

text: )n

text: dn

text: d

text: an,

text: k

text: k

text: E

text: Gk

text: (x)

text: ∂xr

text: E

text: γ4

text: 4γ3γ

text: γ3

text: (d

text: k

text: k

text: s

text: k

text: s

text: k

text: s

text: s

text: s:s=k

text: and

text: thus,

text: due

text: to

text: ρr,k

text: A

text: ∆)ρr,k

text: C

text: ρr,s

text: ∆(ρr,k

text: ρr,s

text: s:s=k

text: +(d

text: ∆(ρr,k

text: (d

text: ∆ρr,s

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: TRUNCATED CONTROL VARIATES FOR WEAK APPROXIMATION SCHEMES 17 (cid:88) Further, denote d ρr,o = ρr,o,s , l+1,n,2 l+1,n,2 s=1 − then we get for l = j + 1, . . . , J 1, (cid:18) (cid:19) ≤ (cid:0)− (cid:1) − (cid:0) (cid:1) ρr,o (1 + A ∆)ρr,o + 2(d 1)C ∆ρr,o + (d 1)2B ∆ρr,o l+1,2,2 2 l,2,2 1,1 l,2,2 2 l,2,2 1 1 +d3C ∆ ρr,o + ρr + ρo + d4B ∆ ρr + ρo . 1,1 l,2,2 2 l,4,1 l,4,1 2 2 l,4,1 l,4,1 This gives us ≤ ρr,o (1 + κ ∆)ρr,o + κ ∆, l+1,2,2 3 l,2,2 4 for some constants κ , κ > 0, leading to 3 4 ≤ − − − ρr,o (1 + κ ∆)l j 1ρr,o + κ , l = j + 1, . . . , J 1, l,2,2 3 (cid:34)j+1,2,2 5 (cid:35) (cid:18) (cid:19) (cid:88) where κ > 0 and 5 ρr,o = d E ∂2 Φs (x, ξ ) 2 . j+1,2,2 ∂xr∂xo ∆ j+1 s=1 Thus, we obtain the boundedness of (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:88) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) ∂2 ≤ d E ∂ ∂2 ∂xr∂xo qj(x) (cid:88) ∂x(cid:12)(cid:12)(cid:12)s f(GJ,j(x))∂xr∂xo GsJ,j(x) (cid:12)(cid:12)(cid:12) s=1 (cid:12) (cid:12) +(cid:88) (cid:118)(cid:117)(cid:117)d E(cid:34)(cid:18)∂x∂s∂2xu f(GJ,j(x(cid:19)))(cid:35)∂∂xr GsJ,j(x)∂∂xo GuJ,j(x) s,u(cid:116)=1 ≤ d (cid:88)E(cid:118)(cid:117)(cid:117) ∂(cid:34)∂x(cid:18)s f(GJ,j(x)) 2 ρrJ,,o2(cid:19),,s2 (cid:35)(cid:113) s=1 (cid:116) + d E ∂2 f(G (x)) 2 4 ρr,s ρo,u ∂xs∂xu J,j J,4,1 J,4,1 s,u=1 ∈ { } for all r, o 1, . . . , d , since f is assumed to be twice continuously diﬀerentiable with bounded partial derivatives up to order 2. Let us proceed w(cid:34)it(cid:18)h the ﬁnal case(cid:19)K(cid:35) = 3. We have E  ∂ 6 Gk (x) ∂xr (cid:88)l+1,(cid:0)j  ≤ E − − − γ6 + 6γ5γ + 15(d 1)γ4γ2 + 20(d 1)2γ3γ3 + 15(d 1)3γ2γ4 k k s k s k s k s (cid:54) (cid:1)(cid:3) s:s=k − − +6(d 1)4γ γ5 + (d 1)5γ6 k s s ≤ ≤ ≤ and thus, due to 6a5b 5a6 + b6, 3a4b2 2a6 + b6 and 2a3b3 a6 + b6, ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: TRUNCATED

text: CONTROL

text: VARIATES

text: FOR

text: WEAK

text: APPROXIMATION

text: SCHEMES

text: Further,

text: denote

text: d

text: ρr,o

text: ρr,o,s

text: then

text: we

text: get

text: for

text: l

text: j

text: J

text: ρr,o

text: A

text: ∆)ρr,o

text: ∆ρr,o

text: (d

text: ∆ρr,o

text: ρr,o

text: ρr

text: ρo

text: ρr

text: ρo

text: This

text: gives

text: us

text: ρr,o

text: κ

text: ∆)ρr,o

text: κ

text: for

text: some

text: constants

text: κ

text: κ

text: leading

text: to

text: ρr,o

text: κ

text: ∆)l

text: j

text: κ

text: l

text: j

text: J

text: where

text: κ

text: and

text: ρr,o

text: d

text: E

text: ∂2

text: Φs

text: (x,

text: ξ

text: ∂xr∂xo

text: Thus,

text: we

text: obtain

text: the

text: boundedness

text: of

text: ∂2

text: d

text: E

text: ∂2

text: ∂xr∂xo

text: qj(x)

text: f(GJ,j(x))∂xr∂xo

text: GsJ,j(x)

text: GsJ,j(x)∂∂xo

text: GuJ,j(x)

text: d

text: f(GJ,j(x))

text: d

text: E

text: ∂2

text: f(G

text: (x))

text: ρr,s

text: ρo,u

text: ∂xs∂xu

text: J,j

text: for

text: all

text: r,

text: o

text: d

text: since

text: f

text: is

text: assumed

text: to

text: be

text: twice

text: continuously

text: diﬀerentiable

text: with

text: bounded

text: partial

text: derivatives

text: up

text: to

text: order

text: Let

text: us

text: proceed

text: the

text: ﬁnal

text: We

text: have

text: E

text: Gk

text: (x)

text: ∂xr

text: E

text: γ6

text: 6γ5γ

text: k

text: k

text: s

text: k

text: s

text: k

text: s

text: k

text: s

text: s:s=k

text: γ5

text: (d

text: k

text: s

text: s

text: and

text: thus,

text: due

text: to

text: and

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 18 DENIS BELOMESTNY, STEFAN HÄFNER, AND MIKHAIL URUSOV (cid:88) (cid:16) ≤ ρr,k (1 + A ∆)ρr,k l+1,6,1 6 l,6,1 − + C ∆(5ρr,k + ρr,s ) + 5(d 1)C ∆(2ρr,k + ρr,s ) 5,1 l,6,1 l,6,1 4,2 l,6,1 l,6,1 (cid:54) s:s=k − − + 10(d 1)2C ∆(ρr,k + ρr,s ) + 5(d 1)3C ∆((cid:17)ρr,k + 2ρr,s ) 3,3 l,6,1 l,6,1 2,4 l,6,1 l,6,1 − − +(d 1)4C ∆(ρr,k + 5ρr,s ) + (d 1)5B ∆ρr,s . 1,5 l,6,1 l,6,1 6 l,6,1 This gives us ≤ − − ρr (1 + A ∆)ρr + 6(d 1)C ∆ρr + 15(d 1)2C ∆ρr l+1,6,1 6 l,6,1 5,1 l,6,1 4,2 l,6,1 − − − +20(d 1)3C ∆ρr + 15(d 1)4C ∆ρr + 6(d 1)5C ∆ρr 3,3 l,6,1 2,4 l,6,1 1,5 l,6,1 − +(d 1)6B ∆ρr . 6 l,6,1 Hence, we obtain ≤ ρr (1 + κ ∆)ρr l+1,6,1 6 l,6,1 for some constant κ > 0, leading to 6 ≤ − − − ρrl,6,1 (1 + κ6∆)(cid:88)l j 1ρ(cid:34)rj(cid:18)+1,6,1, l = j + 1,(cid:19). .(cid:35). , J 1, where d E ∂ 6 ρr = Φs (x, ξ ) j+1,6,1 ∂xr ∆ j+1 s=1 Moreover, we h(cid:34)a(cid:18)ve (cid:19) (cid:35) E  ∂ 8 Gk (x) ∂xr (cid:88)l+1,(cid:0)j  ≤ E − − − γ8 + 8γ7γ + 28(d 1)γ6γ2 + 56(d 1)2γ5γ3 + 70(d 1)3γ4γ4 (cid:1)(cid:3) k k s k s k s k s (cid:54) s:s=k − − − − +56(d 1)4γ3γ5 + 28(d 1)5γ2γ6 + 8(d 1)6γ γ7 + (d 1)7γ8 k s k s k s s ≤ ≤ ≤ ≤ and thus, due to 8a7b 7a8 + b8, 4a6b2 3a8 + b8, 8a5b3 5a8 + 3b8 and 2a4b4 a8 + b8, (cid:88) (cid:16) ≤ ρr,k (1 + A ∆)ρr,k l+1,8,1 8 l,8,1 − + C ∆(7ρr,k + ρr,s ) + 7(d 1)C ∆(3ρr,k + ρr,s ) 7,1 l,8,1 l,8,1 6,2 l,8,1 l,8,1 (cid:54) s:s=k − − + 7(d 1)2C ∆(5ρr,k + 3ρr,s ) + 35(d 1)3C ∆(ρr,k + ρr,s ) 5,3 l,8,1 l,8,1 4,4 l,8,1 l,8,1 − − + 7(d 1)4C ∆(3ρr,k + 5ρr,s ) + 7(d 1)5C ∆(cid:17)(ρr,k + 3ρr,s ) 3,5 l,8,1 l,8,1 2,6 l,8,1 l,8,1 − − +(d 1)6C ∆(ρr,k + 7ρr,s ) + (d 1)7B ∆ρr,s . 1,7 l,8,1 l,8,1 8 l,8,1 This gives us ##LTLine## ##LTLine##

text: DENIS

text: BELOMESTNY,

text: STEFAN

text: HÄFNER,

text: AND

text: MIKHAIL

text: URUSOV

text: ρr,k

text: A

text: ∆)ρr,k

text: C

text: ρr,s

text: ρr,s

text: s:s=k

text: ∆(ρr,k

text: ρr,s

text: +(d

text: ∆(ρr,k

text: (d

text: ∆ρr,s

text: This

text: gives

text: us

text: ρr

text: A

text: ∆)ρr

text: ∆ρr

text: ∆ρr

text: ∆ρr

text: ∆ρr

text: ∆ρr

text: +(d

text: ∆ρr

text: Hence,

text: we

text: obtain

text: ρr

text: κ

text: ∆)ρr

text: for

text: some

text: constant

text: κ

text: leading

text: to

text: j

text: l

text: j

text: J

text: where

text: d

text: E

text: ρr

text: Φs

text: (x,

text: ξ

text: ∂xr

text: Moreover,

text: we

text: E

text: Gk

text: (x)

text: ∂xr

text: E

text: γ8

text: 8γ7γ

text: k

text: k

text: s

text: k

text: s

text: k

text: s

text: k

text: s

text: s:s=k

text: γ7

text: (d

text: k

text: s

text: k

text: s

text: k

text: s

text: s

text: and

text: thus,

text: due

text: to

text: and

text: ρr,k

text: A

text: ∆)ρr,k

text: C

text: ρr,s

text: ρr,s

text: s:s=k

text: ∆(ρr,k

text: ρr,s

text: +(d

text: ∆(ρr,k

text: (d

text: ∆ρr,s

text: This

text: gives

text: us

text: ##LTLine##

text: ##LTLine##


text: 6 DENIS BELOMESTNY, STEFAN HÄFNER, AND MIKHAIL URUSOV and the system A { ∈ P I × P I ∪ (cid:54) ∅} = (U , U ) ( ) ( ) : U U = , 1 2 1 2 1 2 P I I ⊆ I ∈ { } where ( ) denotes the set of all subsets of a set . For any U1 1 and o 1, 2 U1, we write o as o = (o ) ∈ . Below we use the convention that a product over the empty set is always one. For rkr∈UN1 , H : R → R stands for the (normalized) k-th Hermite polynomial, i.e. 0 k − (√1)k x2 dk −x2 ∈ R Hk(x) := k! e 2 dxk e 2 , x . ≡ − We remark that, in particular, H 1, H (x) = x and H (x) = √1 (x2 1). 0 1 2 2 As in Subsection , we summarise important representations from [1] below. Proposition 2.4. It holds (cid:88) (cid:88) (cid:88) (cid:89) (cid:89) J (2.9) f(X ) = Ef(X ) + a (X − ) H (ξr) V kl, ∆,T ∆,T j,o,U1,U2 ∆,(j 1)∆ or j j ∈A ∈{ } ∈ ∈ j=1 (U1,U2) o 1,2 U1 r U1 (k,l) U2 R → R (cid:12)  where the coeﬃcients a : d can be computed by the (cid:12)formula j,o,U1,U2 (cid:89) (cid:89) (cid:12)  (cid:12)  (cid:12) (cid:12) (2.10) a (x) = E f(X ) H (ξr) V kl X − = x . j,o,U1,U2 ∆,T or j j ∆,(j 1)∆ ∈ ∈ r U1 (k,l) U2 ∈ { } Moreover, we have for each j 1, . . . , J , (cid:88) (cid:88) q − (x) =E[q (X )|X − = x] (cid:80) j 1 j ∆,j∆ ∆,(j 1)∆ 1 1 = m(m−1) 6m √ √ 4 mi=1 I(yi=0)qj(Φ∆(x, y, z)), 2 2 (y1,...,ym)∈{− 3,0, 3}m (zuv)1≤u<v≤m∈{−1,1}m(m2−1) ∈ A ∈ { } and, for all (U1, U2) , o 1, 2 U1, it holds (cid:88) (cid:88) 1 1 (2.11) aj,o,U1,U2(x) = m(m−1) 6m (cid:89)√ √ (cid:89) 2 2 (cid:80) (y1,...,ym)∈{− 3,0, 3}m (zuv)1≤u<v≤m∈{−1,1}m(m2−1) 4 mi=1 I(yi=0) Hor(yr) zkl qj(Φ∆(x, y, z)), ∈ ∈ r U1 (k,l) U2 × − − where y = (y1, . . . , ym), z = (zuv) is the m m-matrix with zvu = zuv, u < v, zuu = 1 and ≡ q f. J Using Theorem 2.4, we obtain the following result (see Proposition 3.6 in [1]), which provides a bound for the discretisation error and a perfect control variate for the discretised quantity. Proposition 2.5. Assume, that µ and σ in (0.1) are Lipschitz continuous with components R → R µi, σi,r : d , i = 1, . . . , d, r = 1, . . . , m, being 6 times continuously diﬀerentiable with their R → R partial derivatives of order up to 6 having polynomial growth. Let f : d be 6 times continuously diﬀerentiable with partial derivatives of order up to 6 having polynomial growth. Provided that (2.8) ∈ N E| | holds and that, for suﬃciently large p , the expectations X 2p are uniformly bounded in J ∆,j∆ and j = 0, . . . , J, we have for this “simpliﬁed second order weak Taylor(cid:104)scheme” (cid:105) |E − | ≤ [f(X ) f(X )] c∆2, T ∆,T − where the constant c does not depend on ∆. Moreover, we have Var f(X ) M(2) = 0 for the ∆,T ∆,T control variate ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: DENIS

text: BELOMESTNY,

text: STEFAN

text: HÄFNER,

text: AND

text: MIKHAIL

text: URUSOV

text: and

text: the

text: system

text: A

text: P

text: I

text: P

text: I

text: (U

text: U

text: U

text: U

text: P

text: I

text: I

text: I

text: where

text: denotes

text: the

text: set

text: of

text: all

text: subsets

text: of

text: a

text: set

text: For

text: any

text: and

text: o

text: we

text: write

text: o

text: as

text: o

text: (o

text: Below

text: we

text: use

text: the

text: convention

text: that

text: a

text: product

text: over

text: the

text: empty

text: set

text: is

text: always

text: one.

text: For

text: H

text: R

text: R

text: stands

text: for

text: the

text: (normalized)

text: k-th

text: Hermite

text: polynomial,

text: i.e.

text: k

text: dk

text: R

text: Hk(x)

text: k!

text: e

text: dxk

text: e

text: x

text: We

text: remark

text: that,

text: in

text: particular,

text: H

text: H

text: (x)

text: x

text: and

text: H

text: (x)

text: √1

text: As

text: in

text: Subsection

text: we

text: summarise

text: important

text: representations

text: from

text: below.

text: Proposition

text: It

text: holds

text: J

text: f(X

text: Ef(X

text: a

text: (X

text: H

text: (ξr)

text: V

text: kl,

text: ∆,T

text: ∆,T

text: ∆,(j

text: or

text: j

text: j

text: ∈A

text: o

text: r

text: (k,l)

text: R

text: R

text: where

text: the

text: coeﬃcients

text: a

text: d

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text: be

text: computed

text: by

text: the

text: a

text: (x)

text: E

text: f(X

text: H

text: (ξr)

text: V

text: kl

text: X

text: x

text: ∆,T

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text: j

text: j

text: ∆,(j

text: r

text: (k,l)

text: Moreover,

text: we

text: have

text: for

text: each

text: j

text: J

text: q

text: (x)

text: =E[q

text: (X

text: )|X

text: x]

text: j

text: j

text: ∆,j∆

text: ∆,(j

text: y,

text: z)),

text: A

text: and,

text: for

text: all

text: o

text: it

text: holds

text: Hor(yr)

text: zkl

text: qj(Φ∆(x,

text: y,

text: z)),

text: r

text: (k,l)

text: where

text: y

text: ym),

text: z

text: (zuv)

text: is

text: the

text: m

text: m-matrix

text: with

text: zvu

text: zuv,

text: u

text: v,

text: zuu

text: and

text: q

text: f.

text: J

text: Using

text: Theorem

text: we

text: obtain

text: the

text: following

text: result

text: (see

text: Proposition

text: in

text: which

text: provides

text: a

text: bound

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text: the

text: discretisation

text: error

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text: perfect

text: control

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text: discretised

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text: Proposition

text: Assume,

text: that

text: µ

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text: their

text: R

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text: derivatives

text: of

text: order

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text: polynomial

text: growth.

text: Let

text: f

text: d

text: be

text: times

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text: diﬀerentiable

text: with

text: partial

text: derivatives

text: of

text: order

text: up

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text: polynomial

text: growth.

text: Provided

text: that

text: N

text: E|

text: holds

text: and

text: that,

text: for

text: suﬃciently

text: large

text: p

text: the

text: expectations

text: X

text: are

text: uniformly

text: bounded

text: in

text: J

text: ∆,j∆

text: and

text: j

text: J,

text: we

text: have

text: for

text: this

text: “simpliﬁed

text: second

text: order

text: weak

text: |E

text: [f(X

text: f(X

text: T

text: ∆,T

text: where

text: the

text: constant

text: c

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text: depend

text: on

text: Moreover,

text: we

text: have

text: Var

text: f(X

text: for

text: the

text: ∆,T

text: ∆,T

text: control

text: variate

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: Self-Interference in Full-Duplex Multi-User MIMO Channels Arman Shojaeifard, Member, IEEE, Kai-Kit Wong, Fellow, IEEE, Marco Di Renzo, Senior Member, IEEE, Gan Zheng, Senior Member, IEEE, Khairi Ashour Hamdi, Senior Member, IEEE, Jie Tang, Member, IEEE Abstract We consider a multi-user multiple-input multiple-output (MIMO) setup where full-duplex (FD) multi-antenna nodes apply linear beamformers to simultaneously transmit and receive multiple streams over Rician fading channels. The exact ﬁrst and second positive moments of the residual self-interference (SI), involving the squared norm of a sum of non-identically distributed random variables, are derived in closed-form. The method of moments is hence invoked to provide a Gamma approximation for the residual SI distribution. The proposed theorem holds under arbitrary linear precoder/decoder design, number of antennas and streams, and SI cancellation capability. Index Terms Multiple-Input Multiple-Output (MIMO), Full-Duplex (FD), Self-Interference (SI), Rician Fading Model. I. INTRODUCTION To date, wireless systems, have been designed under a separation of the transmit/receive functions, a.k.a., half-duplex (HD) mode. This is typically achieved via orthogonal radio frequency (RF) partitioning, e.g., in time-division duplex (TDD) and frequency-division duplex (FDD) systems. The motivation behind this trend has been to avoid the overwhelming self-interference (SI) arising from the bi-directional operation. The rapid increase in trafﬁc, under limited RF resources, however, has driven the incentive for an overhaul in the wireless system design. A candidate solution involves transceiving information over the same RF resources, i.e., full-duplex (FD) mode [1]. There is a growing surge of interest in this topic, as in theory, FD has the potential to double the spectral efﬁciency compared to HD. Several point-to-point prototypes of FD radio nodes have been demonstrated in practice recently [2]. Despite pioneering efforts, FD operation in the context of multi-user multiple-input multiple-output (MIMO) setups is not well understood. A long-standing limitation is in the modeling of the residual SI using the Rayleigh distribution [3]. This approach, whilst improving on the common perfect SI cancellation assumption, differs from measurements showing the residual SI channel undergoes Rician fading [1]. The residual SI distribution over Rician fading channels is thus required for facilitating performance analysis and optimization. We consider a FD multi-antenna node communicating with multiple FD radios over the same RF resources. The Rician fading model is employed to capture the residual SI under arbitrary cancellation through tuning of the distribution parameters by design or measurements. With linear processing, it is not feasible to derive the statistics of the residual SI channel power gain directly. Here, exact closed-form expressions for the residual SI ﬁrst and second positive moments are developed. We then exploit the method of moments in order to obtain an explicit Gamma approximation for the distribution of the residual SI over FD multi-user MIMO Rician fading channels. The validity of the theoretical ﬁndings is conﬁrmed through simulations of the entire RF transmit/receive processing chain. Notation: is a matrix with ( )-th entry { } ; is a vector with -th element { } ; , †, and are the transpose, X n,m X n,m x k x k T + Hermitian-transpose, and pseudo-inverse; E{ } is the expected value; V{ } is the variance; P is the probability density function . . (.) (pdf); | | is the modulus; (cid:107) (cid:107) is the norm; and is the zeroth-order Bessel function of the ﬁrst kind, respectively. . . I0(.) II. SYSTEM MODEL Consider cells, where in each cell , ∈ L { }, a FD multi-antenna node communicates with respect to multiple L l l = 1,...,L l0 FD radios , ∈ K { }. Let and respectively denote the number of transmit and receive antennas at the lk k = 1,...,K M N FD multi-antenna nodes. The FD radios have two-antennas, one for transmission, and the other for reception. An application example is FD multi-antenna base stations communicating in the downlink/uplink with multiple FD mobile terminals. Assuming ≤ , channel assignment is not required. K min(N,M) A. Shojaeifard and K.-K Wong are with the Department of Electronic and Electrical Engineering, University College London, London, United Kingdom (e-mail: a.shojaeifard@ucl.ac.uk; kai-kit.wong@ucl.ac.uk). M. Di Renzo is with the Laboratoire des Signaux et Syste`mes, CNRS, CentraleSupe´lec, Univ Paris Sud, Universite´ Paris-Saclay, Gif-sur-Yvette, France (e-mail: marco.direnzo@l2s.centralesupelec.fr). G. Zheng is with the Wolfson School of Mechanical, Electrical and Manufacturing Engineering, Loughborough University, Loughborough, United Kingdom (e-mail: g.zheng@lboro.ac.uk). K. A. Hamdi is with the School of Electrical and Electronic Engineering, University of Manchester, Manchester, United Kingdom (e-mail: k.hamdi@ manchester.ac.uk). J. Tang is with the School of Electronic and Information Engineering, South China University of Technology, Guangzhou, China (e-mail: eej- tang@scut.edu.cn). This work was supported by the Engineering and Physical Sciences Research Council (EPSRC) under grants EP/N008219/1 and EP/N007840/1.

text: Self-Interference

label: title

text: in

text: Full-Duplex

text: Multi-User

text: MIMO

text: Channels

text: Arman

label: author

text: Shojaeifard,

text: Member,

text: IEEE,

text: Kai-Kit

text: Wong,

text: Fellow,

text: IEEE,

text: Marco

text: Di

text: Renzo,

text: Senior

text: Member,

text: IEEE,

text: Gan

text: Zheng,

text: Senior

text: Member,

text: IEEE,

text: Khairi

text: Ashour

text: Hamdi,

text: Senior

text: Member,

text: IEEE,

text: Jie

text: Tang,

text: Member,

text: IEEE

text: Abstract

text: We

label: abstract

text: consider

text: a

text: multi-user

text: multiple-input

text: multiple-output

text: (MIMO)

text: setup

text: where

text: full-duplex

text: (FD)

text: multi-antenna

text: nodes

text: apply

text: linear

text: beamformers

text: to

text: simultaneously

text: transmit

text: and

text: receive

text: multiple

text: streams

text: over

text: Rician

text: fading

text: channels.

text: The

text: exact

text: ﬁrst

text: and

text: second

text: positive

text: moments

text: of

text: the

text: residual

text: self-interference

text: (SI),

text: involving

text: the

text: squared

text: norm

text: of

text: a

text: sum

text: of

text: non-identically

text: distributed

text: random

text: variables,

text: are

text: derived

text: in

text: closed-form.

text: The

text: method

text: of

text: moments

text: is

text: hence

text: invoked

text: to

text: provide

text: a

text: Gamma

text: approximation

text: for

text: the

text: residual

text: SI

text: distribution.

text: The

text: proposed

text: theorem

text: holds

text: under

text: arbitrary

text: linear

text: precoder/decoder

text: design,

text: number

text: of

text: antennas

text: and

text: streams,

text: and

text: SI

text: cancellation

text: capability.

text: Index

text: Terms

text: Multiple-Input

text: Multiple-Output

text: (MIMO),

text: Full-Duplex

text: (FD),

text: Self-Interference

text: (SI),

text: Rician

text: Fading

text: Model.

text: I.

text: INTRODUCTION

text: To

text: date,

text: wireless

text: systems,

text: have

text: been

text: designed

text: under

text: a

text: separation

text: of

text: the

text: transmit/receive

text: functions,

text: a.k.a.,

text: half-duplex

text: (HD)

text: mode.

text: This

text: is

text: typically

text: achieved

text: via

text: orthogonal

text: radio

text: frequency

text: (RF)

text: partitioning,

text: e.g.,

text: in

text: time-division

text: duplex

text: (TDD)

text: and

text: frequency-division

text: duplex

text: (FDD)

text: systems.

text: The

text: motivation

text: behind

text: this

text: trend

text: has

text: been

text: to

text: avoid

text: the

text: overwhelming

text: self-interference

text: (SI)

text: arising

text: from

text: the

text: bi-directional

text: operation.

text: The

text: rapid

text: increase

text: in

text: trafﬁc,

text: under

text: limited

text: RF

text: resources,

text: however,

text: has

text: driven

text: the

text: incentive

text: for

text: an

text: overhaul

text: in

text: the

text: wireless

text: system

text: design.

text: A

text: candidate

text: solution

text: involves

text: transceiving

text: information

text: over

text: the

text: same

text: RF

text: resources,

text: i.e.,

text: full-duplex

text: (FD)

text: mode

text: There

text: is

text: a

text: growing

text: surge

text: of

text: interest

text: in

text: this

text: topic,

text: as

text: in

text: theory,

text: FD

text: has

text: the

text: potential

text: to

text: double

text: the

text: spectral

text: efﬁciency

text: compared

text: to

text: HD.

text: Several

text: point-to-point

text: prototypes

text: of

text: FD

text: radio

text: nodes

text: have

text: been

text: demonstrated

text: in

text: practice

text: recently

text: Despite

text: pioneering

text: efforts,

text: FD

text: operation

text: in

text: the

text: context

text: of

text: multi-user

text: multiple-input

text: multiple-output

text: (MIMO)

text: setups

text: is

text: not

text: well

text: understood.

text: A

text: long-standing

text: limitation

text: is

text: in

text: the

text: modeling

text: of

text: the

text: residual

text: SI

text: using

text: the

text: Rayleigh

text: distribution

text: This

text: approach,

text: whilst

text: improving

text: on

text: the

text: common

text: perfect

text: SI

text: cancellation

text: assumption,

text: differs

text: from

text: measurements

text: showing

text: the

text: residual

text: SI

text: channel

text: undergoes

text: Rician

text: fading

text: The

text: residual

text: SI

text: distribution

text: over

text: Rician

text: fading

text: channels

text: is

text: thus

text: required

text: for

text: facilitating

text: performance

text: analysis

text: and

text: optimization.

text: We

text: consider

text: a

text: FD

text: multi-antenna

text: node

text: communicating

text: with

text: multiple

text: FD

text: radios

text: over

text: the

text: same

text: RF

text: resources.

text: The

text: Rician

text: fading

text: model

text: is

text: employed

text: to

text: capture

text: the

text: residual

text: SI

text: under

text: arbitrary

text: cancellation

text: through

text: tuning

text: of

text: the

text: distribution

text: parameters

text: by

text: design

text: or

text: measurements.

text: With

text: linear

text: processing,

text: it

text: is

text: not

text: feasible

text: to

text: derive

text: the

text: statistics

text: of

text: the

text: residual

text: SI

text: channel

text: power

text: gain

text: directly.

text: Here,

text: exact

text: closed-form

text: expressions

text: for

text: the

text: residual

text: SI

text: ﬁrst

text: and

text: second

text: positive

text: moments

text: are

text: developed.

text: We

text: then

text: exploit

text: the

text: method

text: of

text: moments

text: in

text: order

text: to

text: obtain

text: an

text: explicit

text: Gamma

text: approximation

text: for

text: the

text: distribution

text: of

text: the

text: residual

text: SI

text: over

text: FD

text: multi-user

text: MIMO

text: Rician

text: fading

text: channels.

text: The

text: validity

text: of

text: the

text: theoretical

text: ﬁndings

text: is

text: conﬁrmed

text: through

text: simulations

text: of

text: the

text: entire

text: RF

text: transmit/receive

text: processing

text: chain.

text: Notation:

text: is

text: a

text: matrix

text: with

text: )-th

text: entry

text: is

text: a

text: vector

text: with

text: -th

text: element

text: and

text: are

text: the

text: transpose,

text: X

text: n,m

text: X

text: n,m

text: x

text: k

text: x

text: k

text: T

text: Hermitian-transpose,

text: and

text: pseudo-inverse;

text: E{

text: is

text: the

text: expected

text: value;

text: V{

text: is

text: the

text: variance;

text: P

text: is

text: the

text: probability

text: density

text: function

text: (pdf);

text: is

text: the

text: modulus;

text: is

text: the

text: norm;

text: and

text: is

text: the

text: zeroth-order

text: Bessel

text: function

text: of

text: the

text: ﬁrst

text: kind,

text: respectively.

text: II.

text: SYSTEM

text: MODEL

text: Consider

text: cells,

text: where

text: in

text: each

text: cell

text: L

text: a

text: FD

text: multi-antenna

text: node

text: communicates

text: with

text: respect

text: to

text: multiple

text: L

text: l

text: l

text: FD

text: radios

text: K

text: Let

text: and

text: respectively

text: denote

text: the

text: number

text: of

text: transmit

text: and

text: receive

text: antennas

text: at

text: the

text: lk

text: k

text: M

text: N

text: FD

text: multi-antenna

text: nodes.

text: The

text: FD

text: radios

text: have

text: two-antennas,

text: one

text: for

text: transmission,

text: and

text: the

text: other

text: for

text: reception.

text: An

text: application

text: example

text: is

text: FD

text: multi-antenna

text: base

text: stations

text: communicating

text: in

text: the

text: downlink/uplink

text: with

text: multiple

text: FD

text: mobile

text: terminals.

text: Assuming

text: channel

text: assignment

text: is

text: not

text: required.

text: K

text: min(N,M)

text: A.

text: Shojaeifard

text: and

text: K.-K

text: Wong

text: are

text: with

text: the

text: Department

text: of

text: Electronic

text: and

text: Electrical

text: Engineering,

text: University

text: College

text: London,

text: London,

text: United

text: Kingdom

text: (e-mail:

text: a.shojaeifard@ucl.ac.uk;

text: kai-kit.wong@ucl.ac.uk).

text: M.

text: Di

text: Renzo

text: is

text: with

text: the

text: Laboratoire

text: des

text: Signaux

text: et

text: Syste`mes,

text: CNRS,

text: CentraleSupe´lec,

text: Univ

text: Paris

text: Sud,

text: Universite´

text: Paris-Saclay,

text: Gif-sur-Yvette,

text: France

text: (e-mail:

text: G.

text: Zheng

text: is

text: with

text: the

text: Wolfson

text: School

text: of

text: Mechanical,

text: Electrical

text: and

text: Manufacturing

text: Engineering,

text: Loughborough

text: University,

text: Loughborough,

text: United

text: Kingdom

text: (e-mail:

text: g.zheng@lboro.ac.uk).

text: K.

text: A.

text: Hamdi

text: is

text: with

text: the

text: School

text: of

text: Electrical

text: and

text: Electronic

text: Engineering,

text: University

text: of

text: Manchester,

text: Manchester,

text: United

text: Kingdom

text: (e-mail:

text: k.hamdi@

text: manchester.ac.uk).

text: J.

text: Tang

text: is

text: with

text: the

text: School

text: of

text: Electronic

text: and

text: Information

text: Engineering,

text: South

text: China

text: University

text: of

text: Technology,

text: Guangzhou,

text: China

text: (e-mail:

text: eej-

text: tang@scut.edu.cn).

text: This

text: work

text: was

text: supported

text: by

text: the

text: Engineering

text: and

text: Physical

text: Sciences

text: Research

text: Council

text: (EPSRC)

text: under

text: grants

text: and


text: INFINITE FAMILIES OF SIMPLE HOLOMORPHIC ETA QUOTIENTS 3 Corollary 1. For any integer n > 3, the eta quotient f is irreducible. p,n Also, from Corollary 1 in [2], we recall that given an irreducibile holomor- phic eta quotient f of a prime power level, all the rescalings of f by positive integers are irreducible. Thus we obtain: Corollary 2 (Irreducibility criterion for holomorphic eta quotients). ∈ N Given a holomorphic eta quotient g, if there exist n, d and a prime p, such that g is the rescaling of f by d, then g is irreducible. Here f is p,n p,n as deﬁned in (2.1). With a good amount of numerical evidence, we conjecture that Conjecture 1. For any integer n > 3 and for any odd prime p, there are no simple holomorphic eta quotients of level pn and of weight greater that of f . p,n 3. Notations and the basic facts N ∈ N D By we denote the set of positive integers. For N , by we denote ∈ ZD (cid:89) N the set of divisors of N. For X N , we deﬁne the eta quotient ηX by (3.1) ηX := ηXd, d ∈D d N ∈ D where X is the value of X at d whereas η denotes the rescaling of d N d η by d. Clearly, the level of ηX divides N. In other words, ηX transforms like (cid:88) ZD → Z a modular form on Γ0(N). We deﬁne the summatory function σ : N by (3.2) σ(X) := X . d ∈D d N ∈ ZD Since η is of weight 1/2, the weight of ηX is σ(X)/2 for all X N . Recall that an eta quotient f on Γ (N) is holomorphic if it does not have 0 P Q any poles at the cusps of Γ (N). Under the action of Γ (N) on 1( ) by 0 ∈ Z 0 M¨obius transformation, for a, b with gcd(a, b) = 1, we have ∼ (cid:48) (3.3) [a : b] [a : gcd(N, b)] Γ0(N) (cid:48) ∈ Z P Q for some a which is coprime to gcd(N, b) (see [10]). We identify 1( ) Q ∪ {∞} (cid:54) with via the canonical bijection that maps [α : λ] to α/λ if λ = 0 ∞ ∈ Q ∪ {∞} and to if λ = 0. For s and a weakly holomorphic modular form f on Γ (N), the order of f at the cusp s of Γ (N) is the exponent of 0 0 q1/ws occurring with the ﬁrst nonzero coeﬃcient in the q-expansion of f at the cusp s, where w is the width of the cusp s (see [10], [23]). The following (cid:110)s (cid:12) (cid:111) is a minimal set of representatives of the cusps of Γ (N) (see [10], [20]): (cid:12) 0 S a ∈ Q ∈ D ∈ Z ∼ (3.4) := t , a , gcd(a, t) = 1 / , N N t ##LTLine##

text: INFINITE

text: FAMILIES

text: OF

text: SIMPLE

text: HOLOMORPHIC

text: ETA

text: QUOTIENTS

text: Corollary

text: For

text: any

text: integer

text: n

text: the

text: eta

text: quotient

text: f

text: is

text: irreducible.

text: p,n

text: Also,

text: from

text: Corollary

text: in

text: we

text: recall

text: that

text: given

text: an

text: irreducibile

text: holomor-

text: phic

text: eta

text: quotient

text: f

text: of

text: a

text: prime

text: power

text: level,

text: all

text: the

text: rescalings

text: of

text: f

text: by

text: positive

text: integers

text: are

text: irreducible.

text: Thus

text: we

text: obtain:

text: Corollary

text: (Irreducibility

text: criterion

text: for

text: holomorphic

text: eta

text: quotients).

text: N

text: Given

text: a

text: holomorphic

text: eta

text: quotient

text: g,

text: if

text: there

text: exist

text: n,

text: d

text: and

text: a

text: prime

text: p,

text: such

text: that

text: g

text: is

text: the

text: rescaling

text: of

text: f

text: by

text: d,

text: then

text: g

text: is

text: irreducible.

text: Here

text: f

text: is

text: p,n

text: p,n

text: as

text: deﬁned

text: in

text: With

text: a

text: good

text: amount

text: of

text: numerical

text: evidence,

text: we

text: conjecture

text: that

text: Conjecture

text: For

text: any

text: integer

text: n

text: and

text: for

text: any

text: odd

text: prime

text: p,

text: there

text: are

text: no

text: simple

text: holomorphic

text: eta

text: quotients

text: of

text: level

text: pn

text: and

text: of

text: weight

text: greater

text: that

text: of

text: f

text: p,n

text: Notations

text: and

text: the

text: basic

text: facts

text: N

text: N

text: D

text: By

text: we

text: denote

text: the

text: set

text: of

text: positive

text: integers.

text: For

text: N

text: by

text: we

text: denote

text: ZD

text: N

text: the

text: set

text: of

text: divisors

text: of

text: N.

text: For

text: X

text: N

text: we

text: deﬁne

text: the

text: eta

text: quotient

text: ηX

text: by

text: ηX

text: ηXd,

text: d

text: ∈D

text: d

text: N

text: D

text: where

text: X

text: is

text: the

text: value

text: of

text: X

text: at

text: d

text: whereas

text: η

text: denotes

text: the

text: rescaling

text: of

text: d

text: N

text: d

text: η

text: by

text: d.

text: Clearly,

text: the

text: level

text: of

text: ηX

text: divides

text: N.

text: In

text: other

text: words,

text: ηX

text: transforms

text: like

text: ZD

text: Z

text: a

text: modular

text: form

text: on

text: We

text: deﬁne

text: the

text: summatory

text: function

text: σ

text: N

text: by

text: σ(X)

text: X

text: d

text: ∈D

text: d

text: N

text: ZD

text: Since

text: η

text: is

text: of

text: weight

text: the

text: weight

text: of

text: ηX

text: is

text: for

text: all

text: X

text: N

text: Recall

text: that

text: an

text: eta

text: quotient

text: f

text: on

text: Γ

text: (N)

text: is

text: holomorphic

text: if

text: it

text: does

text: not

text: have

text: P

text: Q

text: any

text: poles

text: at

text: the

text: cusps

text: of

text: Γ

text: (N).

text: Under

text: the

text: action

text: of

text: Γ

text: (N)

text: on

text: by

text: Z

text: M¨obius

text: transformation,

text: for

text: a,

text: b

text: with

text: gcd(a,

text: b)

text: we

text: have

text: [a

text: b]

text: [a

text: gcd(N,

text: b)]

text: Z

text: P

text: Q

text: for

text: some

text: a

text: which

text: is

text: coprime

text: to

text: gcd(N,

text: b)

text: (see

text: We

text: identify

text: Q

text: with

text: via

text: the

text: canonical

text: bijection

text: that

text: maps

text: [α

text: λ]

text: to

text: α/λ

text: if

text: λ

text: Q

text: and

text: to

text: if

text: λ

text: For

text: s

text: and

text: a

text: weakly

text: holomorphic

text: modular

text: form

text: f

text: on

text: Γ

text: (N),

text: the

text: order

text: of

text: f

text: at

text: the

text: cusp

text: s

text: of

text: Γ

text: (N)

text: is

text: the

text: exponent

text: of

text: occurring

text: with

text: the

text: ﬁrst

text: nonzero

text: coeﬃcient

text: in

text: the

text: q-expansion

text: of

text: f

text: at

text: the

text: cusp

text: s,

text: where

text: w

text: is

text: the

text: width

text: of

text: the

text: cusp

text: s

text: (see

text: The

text: following

text: is

text: a

text: minimal

text: set

text: of

text: representatives

text: of

text: the

text: cusps

text: of

text: Γ

text: (N)

text: (see

text: S

text: a

text: Q

text: D

text: Z

text: t

text: a

text: gcd(a,

text: t)

text: N

text: N

text: t

text: ##LTLine##


text: INFINITE FAMILIES OF SIMPLE HOLOMORPHIC ETA QUOTIENTS 5 (cid:88) · · k ψ(N) (3.11) ϕ(gcd(t, N/t)) ord (f ; Γ (N)) = . 1/t 0 24 | t N ∈ Z Since ord (f ; Γ (N)) 1 (see (3.5)), from (3.11) it follows that of any 1/t 0 24 particular weight, there are only ﬁnitely many holomorphic eta quotients on Γ (N). More precisely, the number of holomorphic eta quotients of weight 0 (cid:88) k/2 on Γ (N) is at most the number of solutions of the following equation 0 · · (3.12) ϕ(gcd(t, N/t)) x = k ψ(N) t ∈D t N in nonnegative integers x . Ot ZD → ZD We deﬁ∈neZtDhe order map N : N 214 N of level N as the map which sends X N to the ordered set of orders of the eta quotient ηX at the { } ∈ ZD ×D cusps 1/t t∈DN of Γ0(N). Also, we deﬁne the order matrix AN N N of level N by · (3.13) A (t, d) := 24 ord (η ; Γ (N)) N 1/t d 0 ∈ D   for all t, d . For example, for a prime power pn, we have N    − − · · ·   pn pn 1 pn 2 p 1     − − − · · ·  pn 2 pn 1 pn 2 p 1     − − − · · ·  pn 4 pn 3 pn 2 p 1    (3.14) Apn =  . . . · · · . .  .  . . . . .  . . . . . · · · − − 1 p p2 pn 1 pn 2 · · · − 1 p p2 pn 1 pn By linearity of the order map, we have O 1 · (3.15) (X) = A X . N N 24 ∈ N (cid:48) ∈ ZD − (cid:48) For r D, if Y, Y Nr is ≥such(cid:48) that Y Y is nonne∈gaZtDive at each element of r , then we write Y Y . In particular, for X N , the eta N ≥ quotient ηX is holomorphic if and only if A X 0. N From (3.13) and (3.5), we note that A (t, d) is multiplicative in N, t and N (cid:79) d. Hence, it follows that (3.16) A = A , N pn (cid:107) pn N p prime ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: INFINITE

text: FAMILIES

text: OF

text: SIMPLE

text: HOLOMORPHIC

text: ETA

text: QUOTIENTS

text: k

text: ψ(N)

text: ϕ(gcd(t,

text: N/t))

text: ord

text: (f

text: Γ

text: (N))

text: t

text: N

text: Z

text: Since

text: ord

text: (f

text: Γ

text: (N))

text: (see

text: from

text: it

text: follows

text: that

text: of

text: any

text: particular

text: weight,

text: there

text: are

text: only

text: ﬁnitely

text: many

text: holomorphic

text: eta

text: quotients

text: on

text: Γ

text: (N).

text: More

text: precisely,

text: the

text: number

text: of

text: holomorphic

text: eta

text: quotients

text: of

text: weight

text: on

text: Γ

text: (N)

text: is

text: at

text: most

text: the

text: number

text: of

text: solutions

text: of

text: the

text: following

text: equation

text: ϕ(gcd(t,

text: N/t))

text: x

text: k

text: ψ(N)

text: t

text: ∈D

text: t

text: N

text: in

text: nonnegative

text: integers

text: x

text: Ot

text: ZD

text: ZD

text: We

text: deﬁ∈neZtDhe

text: order

text: map

text: N

text: N

text: N

text: of

text: level

text: N

text: as

text: the

text: map

text: which

text: sends

text: X

text: N

text: to

text: the

text: ordered

text: set

text: of

text: orders

text: of

text: the

text: eta

text: quotient

text: ηX

text: at

text: the

text: ZD

text: ×D

text: cusps

text: t∈DN

text: of

text: Also,

text: we

text: deﬁne

text: the

text: order

text: matrix

text: AN

text: N

text: N

text: of

text: level

text: N

text: by

text: A

text: (t,

text: d)

text: ord

text: (η

text: Γ

text: (N))

text: N

text: d

text: D

text: for

text: all

text: t,

text: d

text: For

text: example,

text: for

text: a

text: prime

text: power

text: pn,

text: we

text: have

text: N

text: pn

text: pn

text: pn

text: p

text: pn

text: pn

text: pn

text: p

text: pn

text: pn

text: pn

text: p

text: Apn

text: p

text: pn

text: pn

text: p

text: pn

text: pn

text: By

text: linearity

text: of

text: the

text: order

text: map,

text: we

text: have

text: O

text: (X)

text: A

text: X

text: N

text: N

text: N

text: ZD

text: For

text: r

text: D,

text: if

text: Y,

text: Y

text: Nr

text: is

text: that

text: Y

text: Y

text: is

text: nonne∈gaZtDive

text: at

text: each

text: element

text: of

text: r

text: then

text: we

text: write

text: Y

text: Y

text: In

text: particular,

text: for

text: X

text: N

text: the

text: eta

text: N

text: quotient

text: ηX

text: is

text: holomorphic

text: if

text: and

text: only

text: if

text: A

text: X

text: N

text: From

text: and

text: we

text: note

text: that

text: A

text: (t,

text: d)

text: is

text: multiplicative

text: in

text: N,

text: t

text: and

text: N

text: d.

text: Hence,

text: it

text: follows

text: that

text: A

text: A

text: N

text: pn

text: pn

text: N

text: p

text: prime

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: | (cid:105) method since the average value of the Hamiltonian in the trial state is , m E (m) = V (m) 1 1 according to Eq.(22). Using the identity − ∂ − ∆ = log(∆ 1) (29) m ∂m2 m (cid:18) (cid:19) the sum of one-loop graphs in Fig.1 can be written as − ∂ − 1 V (m) = 1 I (m) = I (m) m2I (m) (30) 1L ∂m2 1 1 2 0 (cid:90) (cid:90) where the diverging integrals , generalize Eq.(10) and are deﬁned as I (m) I (m) 1 0 1 d4p − − d4p 1 I (m) = log( p2 + m2), I (m) = i − (31) 1 2i (2π)4 0 (2π)4 p2 + m2 so that ∂I (m) 1 1 (32) = I (m). ∂m2 2 0 We recognize as the standard one-loop eﬀective potential of Weinberg and Coleman for I (m) 1 → a massive scalar particle in the limit . This term contains the quantum ﬂuctuations at ϕ 0 one-loop. The second term is a correction coming from the counterterm and arises because the exact Lagrangian was massless. The calculation of the GEP also requires the two-loop graph in Fig.1 that is ﬁrst-order in . A lazy way to evaluate it is by substituting the vertex λ − im2 in the crossed one-loop graph with the seagull one-loop self energy graph iΣ that 1L reads[12] λ (33) Σ = I (m) 1L 0 2 and adding a symmetry factor. The resulting two-loop term is 1/2 λ V (m) = [I (m)]2. (34) 2L 0 8 The GEP follows as the sum V + V 1L 2L − 1 λ V (m) = I (m) m2I (m) + [I (m)]2. (35) GEP 1 0 0 2 8 . . . . + + + + + Figure 2: Pictorial display of the right hand side of Eq.(28). 11 ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: method

text: since

text: the

text: average

text: value

text: of

text: the

text: Hamiltonian

text: in

text: the

text: trial

text: state

text: is

text: m

text: E

text: (m)

text: V

text: (m)

text: according

text: to

text: Using

text: the

text: identity

text: log(∆

text: m

text: m

text: the

text: sum

text: of

text: one-loop

text: graphs

text: in

text: can

text: be

text: written

text: as

text: V

text: (m)

text: I

text: (m)

text: I

text: (m)

text: (m)

text: where

text: the

text: diverging

text: integrals

text: generalize

text: and

text: are

text: deﬁned

text: as

text: I

text: (m)

text: I

text: (m)

text: I

text: (m)

text: log(

text: I

text: (m)

text: i

text: so

text: that

text: ∂I

text: (m)

text: I

text: (m).

text: We

text: recognize

text: as

text: the

text: standard

text: one-loop

text: eﬀective

text: potential

text: of

text: Weinberg

text: and

text: Coleman

text: for

text: I

text: (m)

text: a

text: massive

text: scalar

text: particle

text: in

text: the

text: limit

text: This

text: term

text: contains

text: the

text: quantum

text: ﬂuctuations

text: at

text: ϕ

text: one-loop.

text: The

text: second

text: term

text: is

text: a

text: correction

text: coming

text: from

text: the

text: counterterm

text: and

text: arises

text: because

text: the

text: exact

text: Lagrangian

text: was

text: massless.

text: The

text: calculation

text: of

text: the

text: GEP

text: also

text: requires

text: the

text: two-loop

text: graph

text: in

text: that

text: is

text: ﬁrst-order

text: in

text: A

text: lazy

text: way

text: to

text: evaluate

text: it

text: is

text: by

text: substituting

text: the

text: vertex

text: λ

text: in

text: the

text: crossed

text: one-loop

text: graph

text: with

text: the

text: seagull

text: one-loop

text: self

text: energy

text: graph

text: iΣ

text: that

text: λ

text: Σ

text: I

text: (m)

text: and

text: adding

text: a

text: symmetry

text: factor.

text: The

text: resulting

text: two-loop

text: term

text: is

text: λ

text: V

text: (m)

text: [I

text: The

text: GEP

text: follows

text: as

text: the

text: sum

text: V

text: V

text: λ

text: V

text: (m)

text: I

text: (m)

text: (m)

text: [I

text: GEP

text: Figure

text: Pictorial

text: display

text: of

text: the

text: right

text: hand

text: side

text: of

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: where the adimensional potential (cid:2) is (cid:3) U(α, x) − − U(α, x) = x2 α(log x)2 2 log x 1 (43) and α is the eﬀective coupling α = λ/(16π2). The behavior of the potential is shown in Fig.3. For any coupling the point U(α, x) α is a relative minimum while the potential has a relative maximum at . The x = 0 x = 1/e − absolute minimum is at x = exp(2/α) where U(α, x ) = x2 < 0. The two stationary 0 0 0 points and are the points where the ﬁrst or second factor in Eq.(36) is zero, x = 1/e x = x 0 respectively. Thus the absolute minimum m2/Λ2 = x = exp(2/α) is the solution of the gap 0 equation, Eq.(37). However, since the original theory has no scale, the quantitative value of remains arbitrary as it depends on the unknown scale . We can only predict that, since m Λ the GEP provides a genuine variational approximation for the vacuum energy, the massless vacuum must be unstable towards the vacuum of a massive scalar particle with an exact ≤ eﬀective potential . V (m) V (m) < 0 exact GEP In the chiral limit, the GEP can be easily extended to more complex theories by just adding up the graphs of Fig.1. It is instructive to see how Eq.(17) can be recovered by the graphs for scalar electrodynamics. In the next section the GEP is evaluated for the U(1) chiral limit of QCD. IV. QCD IN THE CHIRAL LIMIT The full Lagrangian of QCD, including (cid:88)N ma(cid:104)ssless chiral (cid:105)Quarks, can be written as f N f (cid:54) − (cid:54) L = L + Ψ¯ i∂ gA Tˆ Ψ (44) QCD Y M i a a i i=1 where L is the full SU(N) Yang-Mills Lagrangian, including a covariant gauge-ﬁxing Y M term and the ghost terms arising from the Faddeev-Popov determinant. The generators of (cid:104) (cid:105) satisfy the algebra SU(N) ˆ ˆ ˆ (45) T , T = if T a b abc c with the structure constants normalized according to (46) f f = Nδ . abc dbc ad In a background ﬁeld A˜µ the eﬀective action Γ(A˜) is the sum of 1PI graphs that can be a 14

text: where

text: the

text: adimensional

text: potential

text: is

text: U(α,

text: x)

text: U(α,

text: x)

text: α(log

text: log

text: x

text: and

text: α

text: is

text: the

text: eﬀective

text: coupling

text: α

text: The

text: behavior

text: of

text: the

text: potential

text: is

text: shown

text: in

text: For

text: any

text: coupling

text: the

text: point

text: U(α,

text: x)

text: α

text: is

text: a

text: relative

text: minimum

text: while

text: the

text: potential

text: has

text: a

text: relative

text: maximum

text: at

text: The

text: x

text: x

text: absolute

text: minimum

text: is

text: at

text: x

text: where

text: U(α,

text: x

text: The

text: two

text: stationary

text: points

text: and

text: are

text: the

text: points

text: where

text: the

text: ﬁrst

text: or

text: second

text: factor

text: in

text: is

text: zero,

text: x

text: x

text: x

text: respectively.

text: Thus

text: the

text: absolute

text: minimum

text: x

text: is

text: the

text: solution

text: of

text: the

text: gap

text: equation,

text: However,

text: since

text: the

text: original

text: theory

text: has

text: no

text: scale,

text: the

text: quantitative

text: value

text: of

text: remains

text: arbitrary

text: as

text: it

text: depends

text: on

text: the

text: unknown

text: scale

text: We

text: can

text: only

text: predict

text: that,

text: since

text: m

text: Λ

text: the

text: GEP

text: provides

text: a

text: genuine

text: variational

text: approximation

text: for

text: the

text: vacuum

text: energy,

text: the

text: massless

text: vacuum

text: must

text: be

text: unstable

text: towards

text: the

text: vacuum

text: of

text: a

text: massive

text: scalar

text: particle

text: with

text: an

text: exact

text: eﬀective

text: potential

text: V

text: (m)

text: V

text: (m)

text: exact

text: GEP

text: In

text: the

text: chiral

text: limit,

text: the

text: GEP

text: can

text: be

text: easily

text: extended

text: to

text: more

text: complex

text: theories

text: by

text: just

text: adding

text: up

text: the

text: graphs

text: of

text: It

text: is

text: instructive

text: to

text: see

text: how

text: can

text: be

text: recovered

text: by

text: the

text: graphs

text: for

text: scalar

text: electrodynamics.

text: In

text: the

text: next

text: section

text: the

text: GEP

text: is

text: evaluated

text: for

text: the

text: chiral

text: limit

text: of

text: QCD.

text: IV.

text: QCD

text: IN

text: THE

text: CHIRAL

text: LIMIT

text: The

text: full

text: Lagrangian

text: of

text: QCD,

text: including

text: chiral

text: can

text: be

text: written

text: as

text: f

text: N

text: f

text: L

text: L

text: Ψ¯

text: i∂

text: gA

text: Tˆ

text: Ψ

text: QCD

text: Y

text: M

text: i

text: a

text: a

text: i

text: where

text: L

text: is

text: the

text: full

text: SU(N)

text: Yang-Mills

text: Lagrangian,

text: including

text: a

text: covariant

text: gauge-ﬁxing

text: Y

text: M

text: term

text: and

text: the

text: ghost

text: terms

text: arising

text: from

text: the

text: Faddeev-Popov

text: determinant.

text: The

text: generators

text: of

text: satisfy

text: the

text: algebra

text: SU(N)

text: ˆ

text: ˆ

text: ˆ

text: T

text: T

text: if

text: T

text: a

text: b

text: abc

text: c

text: with

text: the

text: structure

text: constants

text: normalized

text: according

text: to

text: f

text: f

text: Nδ

text: abc

text: dbc

text: ad

text: In

text: a

text: background

text: ﬁeld

text: A˜µ

text: the

text: eﬀective

text: action

text: Γ(A˜)

text: is

text: the

text: sum

text: of

text: graphs

text: that

text: can

text: be

text: a


text: their physical value at the minimum of the free energy . That point may be found by ϕ min requiring that ∂V eff (14) = 0 ∂ϕ2 where as usual the partial derivative is allowed as far as the gap equations (12),(13) are satisﬁed[17]. The condition (14) combined with the gap equation (12) yields the very simple result Ω2 ϕ2 = . (15) min 8λ B However, we notice that here the mass must be found by solution of the coupled gap Ω equations. Thus Eqs.(15), (12) and (13) must be regarded as a set of coupled equations and must be solved together in order to ﬁnd the physical values for the correlation lengths and the order parameter. (cid:18) (cid:19) Insertion of Eq.(15) into Eq.(13) yields a simple relation for the GL parameter κ GL λ 2 1 κ2 = = κ (16) GL ξ 0 I (Ω) 0 1 + ϕ2 min where κ = e2 /(8λ ) is the mean-ﬁeld GL parameter which does not depend on temperature. 0 B B Eq.(16) shows that the GL parameter is predicted to be temperature dependent through the non trivial dependence of and . At low temperature, where the order parameter Ω ϕ min is large, the deviation from the mean-ﬁeld value is negligible. Conversely, close to ϕ κ min 0 the critical point, where the order parameter is vanishing, the correction factor in Eq.(16) becomes very important[3, 4]. → It is instructive to look at the eﬀective potential in the limit of the unbroken- ϕ 0 symmetry phase and at the chiral point where the original Lagrangian is scaleless. m = 0 B In that limit Eq.(9) reads (cid:2) (cid:3) − 1 V [0] = [I (Ω) + 2I (∆)] Ω2I (Ω) + 2∆2I (∆) eff 1 1 0 0 2 + 3λ [I (Ω)]2 + e2 I (Ω)I (∆) (17) B 0 B 0 0 and is a function of the mass parameters. Its minimum might fall at a ﬁnite set of masses , yielding a generation of mass from a scaleless Lagrangian. That property turns out ∆ Ω 0 0 6 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: their

text: physical

text: value

text: at

text: the

text: minimum

text: of

text: the

text: free

text: energy

text: That

text: point

text: may

text: be

text: found

text: by

text: ϕ

text: min

text: requiring

text: that

text: ∂V

text: eff

text: ∂ϕ2

text: where

text: as

text: usual

text: the

text: partial

text: derivative

text: is

text: allowed

text: as

text: far

text: as

text: the

text: gap

text: equations

text: are

text: The

text: condition

text: combined

text: with

text: the

text: gap

text: equation

text: yields

text: the

text: very

text: simple

text: result

text: Ω2

text: ϕ2

text: min

text: 8λ

text: B

text: However,

text: we

text: notice

text: that

text: here

text: the

text: mass

text: must

text: be

text: found

text: by

text: solution

text: of

text: the

text: coupled

text: gap

text: Ω

text: equations.

text: Thus

text: and

text: must

text: be

text: regarded

text: as

text: a

text: set

text: of

text: coupled

text: equations

text: and

text: must

text: be

text: solved

text: together

text: in

text: order

text: to

text: ﬁnd

text: the

text: physical

text: values

text: for

text: the

text: correlation

text: lengths

text: and

text: the

text: order

text: parameter.

text: Insertion

text: of

text: into

text: yields

text: a

text: simple

text: relation

text: for

text: the

text: GL

text: parameter

text: κ

text: GL

text: λ

text: κ2

text: κ

text: GL

text: ξ

text: I

text: (Ω)

text: ϕ2

text: min

text: where

text: κ

text: is

text: the

text: mean-ﬁeld

text: GL

text: parameter

text: which

text: does

text: not

text: depend

text: on

text: temperature.

text: B

text: B

text: shows

text: that

text: the

text: GL

text: parameter

text: is

text: predicted

text: to

text: be

text: temperature

text: dependent

text: through

text: the

text: non

text: trivial

text: dependence

text: of

text: and

text: At

text: low

text: temperature,

text: where

text: the

text: order

text: parameter

text: Ω

text: ϕ

text: min

text: is

text: large,

text: the

text: deviation

text: from

text: the

text: mean-ﬁeld

text: value

text: is

text: negligible.

text: Conversely,

text: close

text: to

text: ϕ

text: κ

text: min

text: the

text: critical

text: point,

text: where

text: the

text: order

text: parameter

text: is

text: vanishing,

text: the

text: correction

text: factor

text: in

text: becomes

text: very

text: It

text: is

text: instructive

text: to

text: look

text: at

text: the

text: eﬀective

text: potential

text: in

text: the

text: limit

text: of

text: the

text: unbroken-

text: ϕ

text: symmetry

text: phase

text: and

text: at

text: the

text: chiral

text: point

text: where

text: the

text: original

text: Lagrangian

text: is

text: scaleless.

text: m

text: B

text: In

text: that

text: limit

text: reads

text: V

text: [I

text: (Ω)

text: (Ω)

text: eff

text: 3λ

text: [I

text: I

text: (Ω)I

text: B

text: B

text: and

text: is

text: a

text: function

text: of

text: the

text: mass

text: parameters.

text: Its

text: minimum

text: might

text: fall

text: at

text: a

text: ﬁnite

text: set

text: of

text: masses

text: yielding

text: a

text: generation

text: of

text: mass

text: from

text: a

text: scaleless

text: Lagrangian.

text: That

text: property

text: turns

text: out

text: Ω

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: (cid:104) (cid:105) potential is evaluated for any value of the average and the best also depends on ϕ = φ m that average. If the symmetry is not broken, then the minimum of the eﬀective potential is at where is a function of the trial mass, to be ﬁxed by the stationary condition ϕ = 0 V (m) 1 Eq.(23). We assume that the gauge symmetry is not broken in QCD so that at V (m) ϕ = 0 1 is the eﬀective potential we are interested in. The variational nature of the method ensures | (cid:105) that the true vacuum energy is smaller than the minimum of . At the minimum, V (m) m 1 provides an approximation for the vacuum and is given by the vacuum of a free massive (cid:54) scalar particle with mass equal to the optimized mass parameter . Of course, m = m B | (cid:105) the optimal state is just a ﬁrst approximation and the actual vacuum is much richer. m However, we expect that a perturbative expansion around that approximate vacuum would be the best choice for the Lagrangian L, prompting towards an expansion with an interaction L and a free part L that depend on m and can be optimized by a clever choice of the int 0 parameter . Diﬀerent strategies have been proposed for the optimization, ranging from the m stationary condition of the GEP, Eq.(23), to Stevenson’s principle of minimal sensitivity[28]. A method based on the minimal variance has been recently proposed for QCD and other gauge theories[10–14]. In all those approaches, the underlying idea is that an optimal choice of could minimize the eﬀect of higher orders in the expansion. Since the total Lagrangian m does not depend on , the physical observables are expected to be stationary at the optimal m , thus suggesting the use of stationary conditions for determining the free parameter. As m a matter of fact, if all graphs were summed up exactly, then the dependence on would m cancel in the ﬁnal result, so that the strength of that dependence measures the weight of the neglected graphs at any order. Leaving aside the problem of the best choice of , we observe that at the calculation m ϕ = 0 of the ﬁrst-order eﬀective potential is quite straightforward and follows from the ﬁrst- V (m) 1 (cid:90) order expansion of the eﬀective action Γ(ϕ) eiΓ(ϕ) = D eiS0(φ+ϕ)+iSint(φ+ϕ) (24) φ 1PI where the functional integral is the sum of all one-particle irreducible (1PI) graphs and − S = S + S is the action. The eﬀective potential then follows as V (m) = Γ(0)/V 0 int 1 4 where V is a total space-time volume. Moreover, being interested in the chiral limit, let set 4 in the interaction Eq.(20) and study a massless scalar theory. m = 0 B The vertices of the theory can be read from L in Eq.(20) where we set m = 0 and int B 9

text: potential

text: is

text: evaluated

text: for

text: any

text: value

text: of

text: the

text: average

text: and

text: the

text: best

text: also

text: depends

text: on

text: ϕ

text: φ

text: m

text: that

text: average.

text: If

text: the

text: symmetry

text: is

text: not

text: broken,

text: then

text: the

text: minimum

text: of

text: the

text: eﬀective

text: potential

text: is

text: at

text: where

text: is

text: a

text: function

text: of

text: the

text: trial

text: mass,

text: to

text: be

text: ﬁxed

text: by

text: the

text: stationary

text: condition

text: ϕ

text: V

text: (m)

text: We

text: assume

text: that

text: the

text: gauge

text: symmetry

text: is

text: not

text: broken

text: in

text: QCD

text: so

text: that

text: at

text: V

text: (m)

text: ϕ

text: is

text: the

text: eﬀective

text: potential

text: we

text: are

text: interested

text: in.

text: The

text: variational

text: nature

text: of

text: the

text: method

text: ensures

text: that

text: the

text: true

text: vacuum

text: energy

text: is

text: smaller

text: than

text: the

text: minimum

text: of

text: At

text: the

text: minimum,

text: V

text: (m)

text: m

text: provides

text: an

text: approximation

text: for

text: the

text: vacuum

text: and

text: is

text: given

text: by

text: the

text: vacuum

text: of

text: a

text: free

text: massive

text: scalar

text: particle

text: with

text: mass

text: equal

text: to

text: the

text: optimized

text: mass

text: parameter

text: Of

text: course,

text: m

text: m

text: B

text: the

text: optimal

text: state

text: is

text: just

text: a

text: ﬁrst

text: approximation

text: and

text: the

text: actual

text: vacuum

text: is

text: much

text: richer.

text: m

text: However,

text: we

text: expect

text: that

text: a

text: perturbative

text: expansion

text: around

text: that

text: approximate

text: vacuum

text: would

text: be

text: the

text: best

text: choice

text: for

text: the

text: Lagrangian

text: L,

text: prompting

text: towards

text: an

text: expansion

text: with

text: an

text: interaction

text: L

text: and

text: a

text: free

text: part

text: L

text: that

text: depend

text: on

text: m

text: and

text: can

text: be

text: optimized

text: by

text: a

text: clever

text: choice

text: of

text: the

text: int

text: parameter

text: Diﬀerent

text: strategies

text: have

text: been

text: proposed

text: for

text: the

text: optimization,

text: ranging

text: from

text: the

text: m

text: stationary

text: condition

text: of

text: the

text: GEP,

text: to

text: Stevenson’s

text: principle

text: of

text: minimal

text: A

text: method

text: based

text: on

text: the

text: minimal

text: variance

text: has

text: been

text: recently

text: proposed

text: for

text: QCD

text: and

text: other

text: gauge

text: In

text: all

text: those

text: approaches,

text: the

text: underlying

text: idea

text: is

text: that

text: an

text: optimal

text: choice

text: of

text: could

text: minimize

text: the

text: eﬀect

text: of

text: higher

text: orders

text: in

text: the

text: expansion.

text: Since

text: the

text: total

text: Lagrangian

text: m

text: does

text: not

text: depend

text: on

text: the

text: physical

text: observables

text: are

text: expected

text: to

text: be

text: stationary

text: at

text: the

text: optimal

text: m

text: thus

text: suggesting

text: the

text: use

text: of

text: stationary

text: conditions

text: for

text: determining

text: the

text: free

text: parameter.

text: As

text: m

text: a

text: matter

text: of

text: fact,

text: if

text: all

text: graphs

text: were

text: summed

text: up

text: exactly,

text: then

text: the

text: dependence

text: on

text: would

text: m

text: cancel

text: in

text: the

text: ﬁnal

text: result,

text: so

text: that

text: the

text: strength

text: of

text: that

text: dependence

text: measures

text: the

text: weight

text: of

text: the

text: neglected

text: graphs

text: at

text: any

text: order.

text: Leaving

text: aside

text: the

text: problem

text: of

text: the

text: best

text: choice

text: of

text: we

text: observe

text: that

text: at

text: the

text: calculation

text: m

text: ϕ

text: of

text: the

text: ﬁrst-order

text: eﬀective

text: potential

text: is

text: quite

text: straightforward

text: and

text: follows

text: from

text: the

text: ﬁrst-

text: V

text: (m)

text: order

text: expansion

text: of

text: the

text: eﬀective

text: action

text: Γ(ϕ)

text: eiΓ(ϕ)

text: D

text: φ

text: where

text: the

text: functional

text: integral

text: is

text: the

text: sum

text: of

text: all

text: one-particle

text: irreducible

text: graphs

text: and

text: S

text: S

text: S

text: is

text: the

text: action.

text: The

text: eﬀective

text: potential

text: then

text: follows

text: as

text: V

text: (m)

text: int

text: where

text: V

text: is

text: a

text: total

text: space-time

text: volume.

text: Moreover,

text: being

text: interested

text: in

text: the

text: chiral

text: limit,

text: let

text: set

text: in

text: the

text: interaction

text: and

text: study

text: a

text: massless

text: scalar

text: theory.

text: m

text: B

text: The

text: vertices

text: of

text: the

text: theory

text: can

text: be

text: read

text: from

text: L

text: in

text: where

text: we

text: set

text: m

text: and

text: int

text: B


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text: Royal

text: Society

text: Open

text: Science,

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text: Kwak,

text: C.

text: Lee,

text: H.

text: Park,

text: and

text: S.

text: Moon,

text: What

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text: a

text: social

text: network

text: or

text: a

text: news

text: media?,

text: in

text: Proceedings

text: of

text: the

text: international

text: conference

text: on

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text: wide

text: web,

text: ACM,

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text: and

text: N.

text: Cristianini,

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text: pp.

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text: in

text: top

text: ﬁve

text: of

text: all

text: referendums,

text: accessed

text: B.

text: Pang

text: and

text: L.

text: Lee,

text: Seeing

text: stars:

text: Exploiting

text: class

text: relationships

text: for

text: sentiment

text: categorization

text: with

text: respect

text: to

text: rating

text: scales,

text: in

text: Proceedings

text: of

text: the

text: annual

text: meeting

text: on

text: association

text: for

text: computational

text: linguistics,

text: Association

text: for

text: Computational

text: Linguistics,

text: pp.

text: B.

text: Pang

text: and

text: L.

text: Lee,

text: Opinion

text: mining

text: and

text: sentiment

text: analysis,

text: Found.

text: Trends

text: Inf.

text: Retr.,

text: pp.

text: RTE

text: news,

text: Marriage

text: referendum

text: leading

text: Irish

text: Twitter

text: topic

text: for

text: (accessed

text: M.

text: Thelwall,

text: Heart

text: and

text: soul:

text: sentiment

text: strength

text: detection

text: in

text: the

text: social

text: web

text: with

text: sentistrength,

text: Pro-

text: ceedings

text: of

text: the

text: CyberEmotions,

text: pp.

text: A.

text: Tumasjan,

text: T.

text: O.

text: Sprenger,

text: P.

text: G.

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