text: 3 References 56 I. INTRODUCTION It is widely acknowledged that for quantum ﬁeld theories to be fundamental and predictive up to highest energies, their short-distance behaviour should be controlled by an ultraviolet (UV) ﬁxed point under the renormalisation group [1, 2]. In particle physics, it is known since long that ultraviolet ﬁxed points can be free, such as in asymptotic freedom of non-abelian gauge theories [3, 4]. Ultraviolet ﬁxed points can also be interacting, such as in asymptotic safety [5]. In the absence of gravity, general theorems for asymptotic safety [6] and proofs of existence in four dimensional theories are available, covering simple [7], semi-simple [8], and supersymmetric gauge theories [9]. It has also been speculated that a quantum theory for gravity with matter may exist with the help of an interacting UV ﬁxed point [10–16]. In gravity, and much unlike in non-Abelian gauge theories, proofs or theorems for asymptotic safety are presently not at hand. This is largely due to the dimensional nature of Newton’s cou- pling whose canonical mass dimension must be compensated by large anomalous dimensions and strong quantum eﬀects. Still, at strong coupling, the asymptotic safety conjecture can be tested systematically with the help of a bootstrap search strategy [17]. By now, an extensive body of evidence for asymptotic safety has grown over the years using the renormalisation group within increasingly sophisticated approximations [17–80]. Invariably, the ﬂuctuations of the metric ﬁeld are found to be strong, strong enough for gravity to become anti-screening [10–12]. Canonical power counting continues to be a “good” ordering principle [5, 17]: canonically relevant couplings remain relevant at interacting ﬁxed points, canonically irrelevant couplings remain irrelevant, and the spectrum becomes near-Gaussian with increasing mass dimension [17, 55]. It would then seem that asymptotically safe gravity becomes “as Gaussian as it gets”. Much of the above picture has been established for f(R) quantum gravity and high-order poly- nomial approximations thereof. In this work, we extend investigations of the asymptotic safety conjecture towards actions beyond Ricci scalars. We use the method of functional renormalisation to derive renormalisation group ﬂow equation for gravity including at strong coupling. Our primary goal is to generalise the successful ﬂow equations for f(R) type approximations to much more gen- eral classes of actions involving Ricci and Riemann tensor invariants, or products and combinations thereof. A main challenge for such a programme is that explicit ﬂow equations for general actions of the form f(R, R , R , · · · ) are not available. The main new idea is the observation that for µν µνρσ certain subsets of actions, ﬂow equations can nevertheless be derived and analysed. For want of example, we consider the UV ﬁxed point of quantum gravity based on actions of the form f(R, R ) = F (R Rµν) + R · Z(R Rµν) . (1.1) µν µν µν where F and Z are unspeciﬁed functions, ultimately determined by the UV ﬁxed point itself. For small curvature, the action reduces to the standard Einstein-Hilbert action. For large curvature, higher order terms modify the action and the dynamics of the theory. The virtue of actions of the form (1.1) is that, when Taylor-expanded in curvature invariants, only a single term arises per canonical mass dimension. In other words, evaluating the renormalisation group equations on spaces with constant curvature is suﬃcient to ﬁnd the running for all polynomial couplings of the theory, as well as the non-perturbative renormalisation group ﬂow of the functions F and Z. By and large, these ideas are similar to what had been done previously for f(R) type ac- tions. Generalising those for actions involving more general curvature invariants will enable us to investigate the impact on the gravitational UV ﬁxed point of curvature invariants beyond Ricci scalars. Unlike f(R) theories, our approximations also incorporate the full kinematics of fourth order gravity alongside higher order curvature invariants. As a ﬁrst application of our results,

text: References

text: I.

text: INTRODUCTION

text: It

text: is

text: widely

text: acknowledged

text: that

text: for

text: quantum

text: ﬁeld

text: theories

text: to

text: be

text: fundamental

text: and

text: predictive

text: up

text: to

text: highest

text: energies,

text: their

text: short-distance

text: behaviour

text: should

text: be

text: controlled

text: by

text: an

text: ultraviolet

text: (UV)

text: ﬁxed

text: point

text: under

text: the

text: renormalisation

text: group

text: In

text: particle

text: physics,

text: it

text: is

text: known

text: since

text: long

text: that

text: ultraviolet

text: ﬁxed

text: points

text: can

text: be

text: free,

text: such

text: as

text: in

text: asymptotic

text: freedom

text: of

text: non-abelian

text: gauge

text: theories

text: Ultraviolet

text: ﬁxed

text: points

text: can

text: also

text: be

text: interacting,

text: such

text: as

text: in

text: asymptotic

text: safety

text: In

text: the

text: absence

text: of

text: gravity,

text: general

text: theorems

text: for

text: asymptotic

text: safety

text: and

text: proofs

text: of

text: existence

text: in

text: four

text: dimensional

text: theories

text: are

text: available,

text: covering

text: simple

text: semi-simple

text: and

text: supersymmetric

text: gauge

text: theories

text: It

text: has

text: also

text: been

text: speculated

text: that

text: a

text: quantum

text: theory

text: for

text: gravity

text: with

text: matter

text: may

text: exist

text: with

text: the

text: help

text: of

text: an

text: interacting

text: UV

text: ﬁxed

text: point

text: In

text: gravity,

text: and

text: much

text: unlike

text: in

text: non-Abelian

text: gauge

text: theories,

text: proofs

text: or

text: theorems

text: for

text: asymptotic

text: safety

text: are

text: presently

text: not

text: at

text: hand.

text: This

text: is

text: largely

text: due

text: to

text: the

text: dimensional

text: nature

text: of

text: Newton’s

text: cou-

text: pling

text: whose

text: canonical

text: mass

text: dimension

text: must

text: be

text: compensated

text: by

text: large

text: anomalous

text: dimensions

text: and

text: strong

text: quantum

text: eﬀects.

text: Still,

text: at

text: strong

text: coupling,

text: the

text: asymptotic

text: safety

text: conjecture

text: can

text: be

text: tested

text: systematically

text: with

text: the

text: help

text: of

text: a

text: bootstrap

text: search

text: strategy

text: By

text: now,

text: an

text: extensive

text: body

text: of

text: evidence

text: for

text: asymptotic

text: safety

text: has

text: grown

text: over

text: the

text: years

text: using

text: the

text: renormalisation

text: group

text: within

text: increasingly

text: sophisticated

text: approximations

text: Invariably,

text: the

text: ﬂuctuations

text: of

text: the

text: metric

text: ﬁeld

text: are

text: found

text: to

text: be

text: strong,

text: strong

text: enough

text: for

text: gravity

text: to

text: become

text: anti-screening

text: Canonical

text: power

text: counting

text: continues

text: to

text: be

text: a

text: “good”

text: ordering

text: principle

text: canonically

text: relevant

text: couplings

text: remain

text: relevant

text: at

text: interacting

text: ﬁxed

text: points,

text: canonically

text: irrelevant

text: couplings

text: remain

text: irrelevant,

text: and

text: the

text: spectrum

text: becomes

text: near-Gaussian

text: with

text: increasing

text: mass

text: dimension

text: It

text: would

text: then

text: seem

text: that

text: asymptotically

text: safe

text: gravity

text: becomes

text: “as

text: Gaussian

text: as

text: it

text: gets”.

text: Much

text: of

text: the

text: above

text: picture

text: has

text: been

text: established

text: for

text: f(R)

text: quantum

text: gravity

text: and

text: high-order

text: poly-

text: nomial

text: approximations

text: thereof.

text: In

text: this

text: work,

text: we

text: extend

text: investigations

text: of

text: the

text: asymptotic

text: safety

text: conjecture

text: towards

text: actions

text: beyond

text: Ricci

text: scalars.

text: We

text: use

text: the

text: method

text: of

text: functional

text: renormalisation

text: to

text: derive

text: renormalisation

text: group

text: ﬂow

text: equation

text: for

text: gravity

text: including

text: at

text: strong

text: coupling.

text: Our

text: primary

text: goal

text: is

text: to

text: generalise

text: the

text: successful

text: ﬂow

text: equations

text: for

text: f(R)

text: type

text: approximations

text: to

text: much

text: more

text: gen-

text: eral

text: classes

text: of

text: actions

text: involving

text: Ricci

text: and

text: Riemann

text: tensor

text: invariants,

text: or

text: products

text: and

text: combinations

text: thereof.

text: A

text: main

text: challenge

text: for

text: such

text: a

text: programme

text: is

text: that

text: explicit

text: ﬂow

text: equations

text: for

text: general

text: actions

text: of

text: the

text: form

text: f(R,

text: R

text: R

text: are

text: not

text: available.

text: The

text: main

text: new

text: idea

text: is

text: the

text: observation

text: that

text: for

text: µν

text: µνρσ

text: certain

text: subsets

text: of

text: actions,

text: ﬂow

text: equations

text: can

text: nevertheless

text: be

text: derived

text: and

text: analysed.

text: For

text: want

text: of

text: example,

text: we

text: consider

text: the

text: UV

text: ﬁxed

text: point

text: of

text: quantum

text: gravity

text: based

text: on

text: actions

text: of

text: the

text: form

text: f(R,

text: R

text: F

text: (R

text: Rµν)

text: R

text: Z(R

text: Rµν)

text: µν

text: µν

text: µν

text: where

text: F

text: and

text: Z

text: are

text: unspeciﬁed

text: functions,

text: ultimately

text: determined

text: by

text: the

text: UV

text: ﬁxed

text: point

text: itself.

text: For

text: small

text: curvature,

text: the

text: action

text: reduces

text: to

text: the

text: standard

text: Einstein-Hilbert

text: action.

text: For

text: large

text: curvature,

text: higher

text: order

text: terms

text: modify

text: the

text: action

text: and

text: the

text: dynamics

text: of

text: the

text: theory.

text: The

text: virtue

text: of

text: actions

text: of

text: the

text: form

text: is

text: that,

text: when

text: Taylor-expanded

text: in

text: curvature

text: invariants,

text: only

text: a

text: single

text: term

text: arises

text: per

text: canonical

text: mass

text: dimension.

text: In

text: other

text: words,

text: evaluating

text: the

text: renormalisation

text: group

text: equations

text: on

text: spaces

text: with

text: constant

text: curvature

text: is

text: suﬃcient

text: to

text: ﬁnd

text: the

text: running

text: for

text: all

text: polynomial

text: couplings

text: of

text: the

text: theory,

text: as

text: well

text: as

text: the

text: non-perturbative

text: renormalisation

text: group

text: ﬂow

text: of

text: the

text: functions

text: F

text: and

text: Z.

text: By

text: and

text: large,

text: these

text: ideas

text: are

text: similar

text: to

text: what

text: had

text: been

text: done

text: previously

text: for

text: f(R)

text: type

text: ac-

text: tions.

text: Generalising

text: those

text: for

text: actions

text: involving

text: more

text: general

text: curvature

text: invariants

text: will

text: enable

text: us

text: to

text: investigate

text: the

text: impact

text: on

text: the

text: gravitational

text: UV

text: ﬁxed

text: point

text: of

text: curvature

text: invariants

text: beyond

text: Ricci

text: scalars.

text: Unlike

text: f(R)

text: theories,

text: our

text: approximations

text: also

text: incorporate

text: the

text: full

text: kinematics

text: of

text: fourth

text: order

text: gravity

text: alongside

text: higher

text: order

text: curvature

text: invariants.

text: As

text: a

text: ﬁrst

text: application

text: of

text: our

text: results,


text: 28 ﬁxed point couplings scaling exponents N (cid:48) (cid:48) λ∗ g∗ g∗ · λ∗ θ θ θ 2 3 2 0.129268 0.984172 0.127222 2.38241 3 0.170801 1.46679 0.250528 1.62684 21.233 4 0.18978224 1.189653 0.2257749 2.45042 1.1075 −8.27329 5 0.19532118 1.203080 0.2349870 2.41473 0.99548 −5.31158 6 0.19594 1.20458 0.236025 2.40791 0.97504 −5.05665 7 0.19563165 1.203838 0.2355087 2.40905 0.98114 −5.00970 8 0.19562751 1.203828 0.2355018 2.40904 0.98111 −5.00233 9 0.19562851 1.203830 0.2355035 2.40902 0.98103 −5.00244 10 0.19562502 1.203822 0.2354976 2.40903 0.98118 −5.00295 11 0.19562516 1.203822 0.2354979 2.40903 0.98117 −5.00273 12 0.19562515 1.203822 0.2354979 2.40903 0.98117 −5.00278 13 0.19562509 1.203822 0.2354978 2.40903 0.98117 −5.00281 14 0.19562511 1.203822 0.2354978 2.40903 0.98117 −5.00280 21 0.19562511 1.203822 0.2354978 2.40903 0.98117 −5.00280 Table 4. The ﬁxed point values for the dimensionless Newton coupling g , the dimensionless cosmological ∗ constant λ , the universal product λ·g, and the ﬁrst four scaling exponents to various orders in the expansion. ∗ Within the accuracy of the displayed digits, the values no longer vary between approximation orders N = 14 and N = 21. operators rapidly stabilises the ﬁxed point values. We estimate the asymptotic values of the ﬁrst few couplings by taking the average over the four best approximation orders to ﬁnd − (cid:104)λ (cid:105) = 0.3250067185 ±2 · 10 10 0 − (cid:104)λ (cid:105) = −0.8306876250 ±3 · 10 10 1 − (cid:104)λ (cid:105) = 0.1915686687 ±2 · 10 10 2 − (cid:104)λ (cid:105) = −0.6291586052 ±9 · 10 10 3 − (5.14) (cid:104)λ (cid:105) = −0.229072717 ±6 · 10 9 4 − (cid:104)λ (cid:105) = −0.00337078 ±2 · 10 8 5 − (cid:104)λ (cid:105) = 0.11247802 ±8 · 10 8 6 − (cid:104)λ (cid:105) = 0.0061012 ±2 · 10 7 . 7 The indicated error corresponds to a standard deviation in the data from the mentioned average. The fast convergence can also be read oﬀ from Fig. 2, showing that the accuracy in the couplings increase by roughly a decimal place for approximation order N → N + 2. This result is quite the opposite of what has been observed previously in f(R) models of gravity. There, the coupling proportional to the square of the Ricci scalar leads to a strong impact on the ﬁxed point coordinate. Also, the R2 coupling itself showed a slow convergence with increasing approximation order. A brief remark on accuracy and error estimates is in order. The error estimates in (5.14) refer to the accuracy with which the relevant ﬁxed point equations have been solved. In principle, results at low polynomial orders cannot be taken for granted due to the non-perturbative nature of the graviton [17]. However, the high numerical accuracy and the fast convergence make sure that the ﬁxed point is a reliable approximation of the full, non-perturbative, ﬁxed point of (4.14), (4.15). On the other hand, our error estimate in (5.14) does not account for proper systematic errors. For strategies to estimate the latter using functional renormalisation we refer to [91].

label: paragraph

text: ﬁxed

label: table

text: point

text: couplings

text: scaling

text: exponents

text: N

text: λ∗

text: g∗

text: g∗

text: λ∗

text: θ

text: θ

text: θ

text: Table

text: The

text: ﬁxed

text: point

text: values

text: for

text: the

text: dimensionless

text: Newton

text: coupling

text: g

text: the

text: dimensionless

text: cosmological

text: constant

text: λ

text: the

text: universal

text: product

text: λ·g,

text: and

text: the

text: ﬁrst

text: four

text: scaling

text: exponents

text: to

text: various

text: orders

text: in

text: the

text: expansion.

text: Within

text: the

text: accuracy

text: of

text: the

text: displayed

text: digits,

text: the

text: values

text: no

text: longer

text: vary

text: between

text: approximation

text: orders

text: N

text: and

text: N

text: operators

text: rapidly

text: stabilises

text: the

text: ﬁxed

text: point

text: values.

text: We

text: estimate

text: the

text: asymptotic

text: values

text: of

text: the

text: ﬁrst

text: few

text: couplings

text: by

text: taking

text: the

text: average

text: over

text: the

text: four

text: best

text: approximation

text: orders

text: to

text: ﬁnd

text: ±2

text: ±3

text: ±2

text: ±9

text: ±6

text: ±2

text: ±8

text: ±2

text: The

text: indicated

text: error

text: corresponds

text: to

text: a

text: standard

text: deviation

text: in

text: the

text: data

text: from

text: the

text: mentioned

text: average.

text: The

text: fast

text: convergence

text: can

text: also

text: be

text: read

text: oﬀ

text: from

text: Fig.

text: showing

text: that

text: the

text: accuracy

text: in

text: the

text: couplings

text: increase

text: by

text: roughly

text: a

text: decimal

text: place

text: for

text: approximation

text: order

text: N

text: N

text: This

text: result

text: is

text: quite

text: the

text: opposite

text: of

text: what

text: has

text: been

text: observed

text: previously

text: in

text: f(R)

text: models

text: of

text: gravity.

text: There,

text: the

text: coupling

text: proportional

text: to

text: the

text: square

text: of

text: the

text: Ricci

text: scalar

text: leads

text: to

text: a

text: strong

text: impact

text: on

text: the

text: ﬁxed

text: point

text: coordinate.

text: Also,

text: the

text: coupling

text: itself

text: showed

text: a

text: slow

text: convergence

text: with

text: increasing

text: approximation

text: order.

text: A

text: brief

text: remark

text: on

text: accuracy

text: and

text: error

text: estimates

text: is

text: in

text: order.

text: The

text: error

text: estimates

text: in

text: refer

text: to

text: the

text: accuracy

text: with

text: which

text: the

text: relevant

text: ﬁxed

text: point

text: equations

text: have

text: been

text: solved.

text: In

text: principle,

text: results

text: at

text: low

text: polynomial

text: orders

text: cannot

text: be

text: taken

text: for

text: granted

text: due

text: to

text: the

text: non-perturbative

text: nature

text: of

text: the

text: graviton

text: However,

text: the

text: high

text: numerical

text: accuracy

text: and

text: the

text: fast

text: convergence

text: make

text: sure

text: that

text: the

text: ﬁxed

text: point

text: is

text: a

text: reliable

text: approximation

text: of

text: the

text: full,

text: non-perturbative,

text: ﬁxed

text: point

text: of

text: On

text: the

text: other

text: hand,

text: our

text: error

text: estimate

text: in

text: does

text: not

text: account

text: for

text: proper

text: systematic

text: errors.

text: For

text: strategies

text: to

text: estimate

text: the

text: latter

text: using

text: functional

text: renormalisation

text: we

text: refer

text: to


text: 29 D. Universal coupling ratios Fixed point couplings are non-universal. Still, some universal quantities of interest are given by speciﬁc products of couplings which remain invariant under global re-scalings of the metric ﬁeld g → (cid:96) g . (5.15) µν µν Under (5.15), the couplings scale as − λ → (cid:96)4 2n λ . (5.16) n n Note that the classically dimensionless coupling λ remains invariant under the rescaling (5.15). 2 All other couplings scale non-trivially. Consequently, various products of couplings can be formed which stay invariant under (5.15). Such invariants may serve as a measure for the relative strength of the gravitational interactions [101]. For couplings including up to λ RR Rµν we may construct 3 µν three independent invariants with values − (cid:104)λ /λ2(cid:105) = 0.4709956080 ± 5 · 10 10 0 1 − (cid:104)λ λ2(cid:105) = 0.1286508384 ± 4 · 10 10 (5.17) 0 3 − (cid:104)λ λ (cid:105) = 0.5226342675 ± 6 · 10 10 . 1 3 Tab. 4 show the convergence of couplings, universal coupling ratios, and the leading scaling exponents. We postpone a detailed discussion of universality until Sect. . E. Beyond polynomial expansions In order to go beyond the polynomial approximations we can integrate the ﬁxed point condition numerically. In a ﬁrst step, we consider even and odd parts of the ﬁxed point condition in R (5.6), leading to (cid:0) (cid:48)(cid:1) 48π 4f − R2f = I [f, z](R) + I [f, z](−R) (5.18) 0 0 (cid:0) (cid:48)(cid:1) 48π R 2 z − R2 z = I [f, z](R) − I [f, z](−R) (5.19) 0 0 These two equations can then be taken as diﬀerential equations for the functions f(x) and z(x) which depend on R through the variable x = R2 . Taking the initial conditions from our polynomial 4 solutions we can solve these two equations to give f and z. Since the equations depend on up to the third derivatives of these functions we must give three initial conditions for both f and z. Resolving them for the highest derivatives, the coupled diﬀerential equations take the form (cid:48)(cid:48) df = H [f, z; R] (5.20) f dR (cid:48)(cid:48) dz = H [f, z; R] (5.21) z dR where the functions H and H depend on f and z up to including their second derivatives (the f z explicit expressions are rather lenghty and not given here). From the structure of the ﬁxed point (cid:48)(cid:48)(cid:48) (cid:48)(cid:48)(cid:48) equation (C3) we observe that both f and z are proportional to the polynomial (C24), which vanishes at ##LTLine## ##LTLine## ##LTLine##

text: D.

label: section

text: Universal

text: coupling

text: ratios

text: Fixed

text: point

text: couplings

text: are

text: non-universal.

text: Still,

text: some

text: universal

text: quantities

text: of

text: interest

text: are

text: given

text: by

text: speciﬁc

text: products

text: of

text: couplings

text: which

text: remain

text: invariant

text: under

text: global

text: re-scalings

text: of

text: the

text: metric

text: ﬁeld

text: g

label: equation

text: g

text: µν

text: µν

text: Under

text: the

text: couplings

text: scale

text: as

text: λ

text: λ

text: n

text: n

text: Note

text: that

text: the

text: classically

text: dimensionless

text: coupling

text: λ

text: remains

text: invariant

text: under

text: the

text: rescaling

text: All

text: other

text: couplings

text: scale

text: non-trivially.

text: Consequently,

text: various

text: products

text: of

text: couplings

text: can

text: be

text: formed

text: which

text: stay

text: invariant

text: under

text: Such

text: invariants

text: may

text: serve

text: as

text: a

text: measure

text: for

text: the

text: relative

text: strength

text: of

text: the

text: gravitational

text: interactions

text: For

text: couplings

text: including

text: up

text: to

text: λ

text: RR

text: Rµν

text: we

text: may

text: construct

text: µν

text: three

text: independent

text: invariants

text: with

text: values

text: λ

text: Tab.

text: show

text: the

text: convergence

text: of

text: couplings,

text: universal

text: coupling

text: ratios,

text: and

text: the

text: leading

text: scaling

text: exponents.

text: We

text: postpone

text: a

text: detailed

text: discussion

text: of

text: universality

text: until

text: Sect.

text: E.

text: Beyond

text: polynomial

text: expansions

text: In

text: order

text: to

text: go

text: beyond

text: the

text: polynomial

text: approximations

text: we

text: can

text: integrate

text: the

text: ﬁxed

text: point

text: condition

text: numerically.

text: In

text: a

text: ﬁrst

text: step,

text: we

text: consider

text: even

text: and

text: odd

text: parts

text: of

text: the

text: ﬁxed

text: point

text: condition

text: in

text: R

text: leading

text: to

text: I

text: [f,

text: z](R)

text: I

text: [f,

text: z](−R)

text: R

text: z

text: z

text: I

text: [f,

text: z](R)

text: I

text: [f,

text: z](−R)

text: These

text: two

text: equations

text: can

text: then

text: be

text: taken

text: as

text: diﬀerential

text: equations

text: for

text: the

text: functions

text: f(x)

text: and

text: z(x)

text: which

text: depend

text: on

text: R

text: through

text: the

text: variable

text: x

text: Taking

text: the

text: initial

text: conditions

text: from

text: our

text: polynomial

text: solutions

text: we

text: can

text: solve

text: these

text: two

text: equations

text: to

text: give

text: f

text: and

text: z.

text: Since

text: the

text: equations

text: depend

text: on

text: up

text: to

text: the

text: third

text: derivatives

text: of

text: these

text: functions

text: we

text: must

text: give

text: three

text: initial

text: conditions

text: for

text: both

text: f

text: and

text: z.

text: Resolving

text: them

text: for

text: the

text: highest

text: derivatives,

text: the

text: coupled

text: diﬀerential

text: equations

text: take

text: the

text: form

text: df

text: H

text: [f,

text: z;

text: R]

text: f

text: dR

text: dz

text: H

text: [f,

text: z;

text: R]

text: z

text: dR

text: where

text: the

text: functions

text: H

text: and

text: H

text: depend

text: on

text: f

text: and

text: z

text: up

text: to

text: including

text: their

text: second

text: derivatives

text: (the

text: f

text: z

text: explicit

text: expressions

text: are

text: rather

text: lenghty

text: and

text: not

text: given

text: here).

text: From

text: the

text: structure

text: of

text: the

text: ﬁxed

text: point

text: equation

text: we

text: observe

text: that

text: both

text: f

text: and

text: z

text: are

text: proportional

text: to

text: the

text: polynomial

text: which

text: vanishes

text: at

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 32 0 3 5 9 11 13 7 2 - m=0 4 - 6 - H L 8   - log 1 R✓e0 J(HmNN) L -10 log11001 - R✓m0e Jm(Nm mNamxa)x 5 10 15 20 N Figure 6. The rate of convergence of the real part of universal eigenvalues θ(cid:48) (m = 0, 2, 3, 5, · · · 13) with m increasing approximation order N towards their best values at N = 21 (see main text). max (cid:48) (cid:48) (cid:48) (cid:48) (cid:48) approximation order. In Fig. 6, we show the fast convergence of {θ , θ , θ , θ , · · · , θ }, again as 0 2 3 5 15 functions of the polynomial approximation order. The values of the critical exponents can be found in Tab. 4. It is noteworthy that the values are quite stable from order to order, except for the ﬁrst occurrence of the perturbatively marginal coupling λ with eigenvalue θ at order N = 3. The reason for this has been given in [55]: the (cid:82)2√ 2 invariant gR Rµν is classically marginal meaning that the RG ﬂow for the corresponding µν coupling does not have a term linear in the coupling. Consequently, higher order interactions are needed to help the classically marginal coupling to develop a ﬁxed point of its own. In other words, the operators with irrelevant scaling dimension are necessary to help the relevant and marginal invariants to develop stable ﬁxed points. This structural pattern is generic for ﬁxed points of quantum ﬁelds theories within polynomial or vertex expansions. We estimate the asymptotic values of scaling exponents by taking an average over the best four approximation orders. We ﬁnd (cid:48) − (cid:104)θ (cid:105) = 2.409028030 ± 5 · 10 9 0(cid:48) − (cid:104)θ (cid:105) = 0.98117249 ± 1 · 10 8 2(cid:48) − (6.2) (cid:104)θ (cid:105) = −5.0028024 ± 2 · 10 7 3(cid:48) − (cid:104)θ (cid:105) = −9.921004 ± 2 · 10 6 5 as real parts of the ﬁrst few exponents. Except for the eigenvalue θ , all others relate to complex 2 conjugate pairs and appear with the degeneracy two. For the corresponding imaginary parts, we have ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

font: default

text: H

text: L

text: log

text: J(HmNN)

text: L

text: Jm(Nm

text: mNamxa)x

text: N

text: Figure

text: The

label: caption

text: rate

text: of

text: convergence

text: of

text: the

text: real

text: part

text: of

text: universal

text: eigenvalues

text: (m

text: with

text: m

text: increasing

text: approximation

text: order

text: N

text: towards

text: their

text: best

text: values

text: at

text: N

text: (see

text: main

text: text).

text: max

text: approximation

text: order.

text: In

text: Fig.

text: we

text: show

text: the

text: fast

text: convergence

text: of

text: {θ

text: θ

text: θ

text: θ

text: θ

text: again

text: as

text: functions

text: of

text: the

text: polynomial

text: approximation

text: order.

text: The

text: values

text: of

text: the

text: critical

text: exponents

text: can

text: be

text: found

text: in

text: Tab.

text: It

text: is

text: noteworthy

text: that

text: the

text: values

text: are

text: quite

text: stable

text: from

text: order

text: to

text: order,

text: except

text: for

text: the

text: ﬁrst

text: occurrence

text: of

text: the

text: perturbatively

text: marginal

text: coupling

text: λ

text: with

text: eigenvalue

text: θ

text: at

text: order

text: N

text: The

text: reason

text: for

text: this

text: has

text: been

text: given

text: in

text: the

text: invariant

text: gR

text: Rµν

text: is

text: classically

text: marginal

text: meaning

text: that

text: the

text: RG

text: ﬂow

text: for

text: the

text: corresponding

text: µν

text: coupling

text: does

text: not

text: have

text: a

text: term

text: linear

text: in

text: the

text: coupling.

text: Consequently,

text: higher

text: order

text: interactions

text: are

text: needed

text: to

text: help

text: the

text: classically

text: marginal

text: coupling

text: to

text: develop

text: a

text: ﬁxed

text: point

text: of

text: its

text: own.

text: In

text: other

text: words,

text: the

text: operators

text: with

text: irrelevant

text: scaling

text: dimension

text: are

text: necessary

text: to

text: help

text: the

text: relevant

text: and

text: marginal

text: invariants

text: to

text: develop

text: stable

text: ﬁxed

text: points.

text: This

text: structural

text: pattern

text: is

text: generic

text: for

text: ﬁxed

text: points

text: of

text: quantum

text: ﬁelds

text: theories

text: within

text: polynomial

text: or

text: vertex

text: expansions.

text: We

text: estimate

text: the

text: asymptotic

text: values

text: of

text: scaling

text: exponents

text: by

text: taking

text: an

text: average

text: over

text: the

text: best

text: four

text: approximation

text: orders.

text: We

text: ﬁnd

text: as

text: real

text: parts

text: of

text: the

text: ﬁrst

text: few

text: exponents.

text: Except

text: for

text: the

text: eigenvalue

text: θ

text: all

text: others

text: relate

text: to

text: complex

text: conjugate

text: pairs

text: and

text: appear

text: with

text: the

text: degeneracy

text: two.

text: For

text: the

text: corresponding

text: imaginary

text: parts,

text: we

text: have

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: ##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: ##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: ##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: ##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: ##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: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 48 to the second lowest eigenvalue of −∇2 − −2 R. Thus we have for the trace of a symmetric tensor d 1 (cid:16) (cid:17) (cid:20) (cid:21) Tr(2) (cid:104)et∇2(cid:105) =Tr(2T) (cid:104)et∇2(cid:105) + Tr(1T) et ∇2+d(dd+−11)R + Tr(0) (cid:104)et∇2(cid:105) + Tr(0) (cid:104)et(∇2+d−21R)(cid:105) (B8) − etd−21R − (d + 1)etd−11R − d(d + 1)etd(d2−1)R. 2 Again we can relate the spectrum of the constrained transverse traceless tensor to that of rank-2 tensors, vectors, and scalars. In order to clarify how the contributions from the exponents play a role in our calculation we (cid:80)∞ expand the exponential in powers of R such as c Rm. Taking into account that the volume − m=0 m of the sphere goes like V ∼ R d/2 and that the heat kernel coeﬃcients go like b ∼ Rn/2 we ﬁnd n that (cid:90) √ − ddx g bn ∼ Rn2d . (B9) Since ultimately we are interested in comparing powers of R, the exponentials contribute when 2m = n − d, and the coeﬃcients b receive contributions only when n ≥ d. Another way to n − see where the excluded modes enter is from the expansion e tz = 1 − tz + 1t2z2 + .... In order (cid:82) ∞ − 2 for the parameter t to be included in Q [W ] = dt t iW˜ (t) we see directly from (3.5) that the i (cid:80)∞ 0 corresponding power m of the expansion c Rm is such that 2m = n − d. m=0 m Heat-kernel coeﬃcients Here we summarise the trace of the heat kernel coeﬃcients tr b = b | for the ﬁelds that we s n n s will be interested in after taking into account their constraints. We write 0 for the scalars, 1 for the vectors and 2 for the tensors. For the scalars we have b | = 1 (B10) 0 0 1 b | = R (B11) 2 0 6 (cid:0) (cid:1) 5d2 − 7d + 6 R2 b | = (B12) 4 0 360(d − 1)d (cid:0) (cid:1) 35d4 − 112d3 + 187d2 − 110d + 96 R3 b | = (B13) 6 0 45360(d − 1)2d2 (cid:0) (cid:1) 175d6 − 945d5 + 2389d4 − 3111d3 + 3304d2 − 516d + 2160 R4 b | = . (B14) 8 0 5443200(d − 1)3d3 For the transverse vector ﬁelds we have ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: to

text: the

text: second

text: lowest

text: eigenvalue

text: of

text: −∇2

text: −2

text: R.

text: Thus

text: we

text: have

text: for

text: the

text: trace

text: of

text: a

text: symmetric

text: tensor

text: d

text: et

text: (d

text: d(d

text: Again

text: we

text: can

text: relate

text: the

text: spectrum

text: of

text: the

text: constrained

text: transverse

text: traceless

text: tensor

text: to

text: that

text: of

text: tensors,

text: vectors,

text: and

text: scalars.

text: In

text: order

text: to

text: clarify

text: how

text: the

text: contributions

text: from

text: the

text: exponents

text: play

text: a

text: role

text: in

text: our

text: calculation

text: we

text: expand

text: the

text: exponential

text: in

text: powers

text: of

text: R

text: such

text: as

text: c

text: Rm.

text: Taking

text: into

text: account

text: that

text: the

text: volume

text: m

text: of

text: the

text: sphere

text: goes

text: like

text: V

text: R

text: and

text: that

text: the

text: heat

text: kernel

text: coeﬃcients

text: go

text: like

text: b

text: we

text: ﬁnd

text: n

text: that

text: ddx

text: g

text: bn

text: Since

text: ultimately

text: we

text: are

text: interested

text: in

text: comparing

text: powers

text: of

text: R,

text: the

text: exponentials

text: contribute

text: when

text: n

text: d,

text: and

text: the

text: coeﬃcients

text: b

text: receive

text: contributions

text: only

text: when

text: n

text: d.

text: Another

text: way

text: to

text: n

text: see

text: where

text: the

text: excluded

text: modes

text: enter

text: is

text: from

text: the

text: expansion

text: e

text: tz

text: tz

text: In

text: order

text: for

text: the

text: parameter

text: t

text: to

text: be

text: included

text: in

text: Q

text: [W

text: dt

text: t

text: iW˜

text: (t)

text: we

text: see

text: directly

text: from

text: that

text: the

text: i

text: corresponding

text: power

text: m

text: of

text: the

text: expansion

text: c

text: Rm

text: is

text: such

text: that

text: n

text: d.

text: m

text: Heat-kernel

text: coeﬃcients

text: Here

text: we

text: summarise

text: the

text: trace

text: of

text: the

text: heat

text: kernel

text: coeﬃcients

text: tr

text: b

text: b

text: for

text: the

text: ﬁelds

text: that

text: we

text: s

text: n

text: n

text: s

text: will

text: be

text: interested

text: in

text: after

text: taking

text: into

text: account

text: their

text: constraints.

text: We

text: write

text: for

text: the

text: scalars,

text: for

text: the

text: vectors

text: and

text: for

text: the

text: tensors.

text: For

text: the

text: scalars

text: we

text: have

text: b

text: b

text: R

text: b

text: b

text: b

text: For

text: the

text: transverse

text: vector

text: ﬁelds

text: we

text: have

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: 52 (cid:48) (cid:48) DT [f, z] = 36f + (24R + 36)z − (7R2 − 6R + 36)(f + R z ) (C9) (cid:48) (cid:48) DS[f, z] = 8f + 12z + (−2R2 − 8R + 24)f + (2R3 − 32R2 + 60R)z (C10) (cid:48)(cid:48) (cid:48)(cid:48) +R2(R − 3)2(f + R z ) DV = 4 − R (C11) c DS = 3 − R (C12) c 607 P V = R2 − 24R − 144 (C13) c 15 511 P S = R2 − 12R − 36 (C14) c 30 311 P Tz0 = R3 − 4R2 − 1080R + 2880 (C15) 0 63 R3 P Tf1 = − − 116R2 + 1800R − 4320 (C16) 0 3 371 P Tz1 = R5 + R4 + 122R3 + 840R2 − 3240R (C17) 0 108 731 R5 P T2 = − R6 + + 29R4 − 300R3 + 540R2 (C18) 0 1512 6 37 116 P Sz0 = R3 + R2 + 72R + 192 (C19) 0 189 15 116 248 P Sf1 = − R3 − R2 + 96R + 576 (C20) 0 45 15 1111 29 170 P Sz1 = R5 − R4 − R3 + 40R2 + 1080R (C21) 0 2268 15 3 1333 29 62 P Sf2 = R6 + R5 + R4 − 16R3 + 36R2 (C22) 0 4536 9 15 27991 406 943 P Sz2 = R7 + R6 + R5 − 16R4 − 126R3 (C23) 0 45360 45 30 181 29 91 P S3 = R8 + R7 + R6 − 27R4 (C24) 0 3360 30 20 311 P T = R3 − R2 − 180R + 360 (C25) 1 126 37 29 P S = R3 + R2 + 12R + 24 (C26) 1 378 15 731 R3 P T = R4 − − 29R2 + 300R − 540 (C27) 2 1512 6 127 58 62 P S = − R4 − R3 − R2 + 16R + 72 (C28) 2 1620 45 15 731 R4 P T = R5 − − 29R3 + 300R2 − 540R (C29) 3 1512 6 1333 232 67 P S = − R5 − R4 − R3 + 16R2 + 180R (C30) 3 4536 45 3 181 29 91 P S = − R6 − R5 − R4 + 27R2 (C31) 4 3360 30 20 181 29 91 P S = − R7 − R6 − R5 + 27R3 (C32) 5 3360 30 20 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: DT

text: [f,

text: z]

text: R

text: z

text: DS[f,

text: z]

text: R

text: z

text: DV

text: R

text: c

text: DS

text: R

text: c

text: P

text: V

text: c

text: P

text: S

text: c

text: P

text: P

text: P

text: P

text: P

text: P

text: P

text: P

text: P

text: P

text: P

text: T

text: P

text: S

text: P

text: T

text: P

text: S

text: P

text: T

text: P

text: S

text: P

text: S

text: P

text: S

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: ##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: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 7 equation 2 ∇¯ ∇¯ σ + ∇¯ ∇¯ σ = g¯ ∇¯ 2σ (2.15) µ ν ν µ µν d and similarly from the modes of the ξµ ﬁeld that obey the Killing equation ∇¯ ξ + ∇¯ ξ = 0. (2.16) µ ν ν µ These modes, when evaluated on the sphere, correspond to the lowest two modes of σ and the lowest mode of ξµ respectively and therefore they should be excluded from the trace evaluation. For details of the heat kernel coeﬃcients of the constrained ﬁelds as well as for the exclusion of lowest modes see appendix . C. Gauge ﬁxing and ghosts Because of the diﬀeomorphism invariance of the metric ﬁeld the eﬀective action has to be supplemented with a gauge ﬁxing term so that only the physical modes of the ﬁeld are taken into account. It is convenient to choose a gauge ﬁxing term of the form (cid:90) 1 √ S = ddx g¯ g¯µνF F (2.17) gf µ ν 2α with the gauge ﬁxing condition F = 0. Here we will choose F to be given by µ µ (cid:18) (cid:19) √ 1 + δ F = 2κ ∇¯ νh − ∇¯ h , (2.18) µ µν µ d − with κ = (32πG ) 1/2 and h = hµνg¯ being the trace of the ﬂuctuation. The gauge ﬁxing N µν condition that was originally used in [18] corresponds to δ = d − 1. Here and from now on bared 2 geometrical quantities, such as ∇¯ above, mean that they are constructed by the background metric g¯ . Then, by substituting F into the gauge ﬁxing action we get µν µ (cid:34) (cid:35) (cid:90) (cid:18) (cid:19) κ2 √ 1 + δ 2 1 + δ S = ddx g¯ ∇¯ ρh ∇¯ λhµ − h∇¯ 2h + 2 h∇¯ ∇¯ ρhµ . (2.19) gf α µρ λ d d µ ρ For the above gauge ﬁxing, the corresponding ghost action is (cid:90) √ √ S = − 2 ddx g¯ C¯µM Cν, (2.20) gh µν with the Faddeev-Popov operator given by 1 + δ M = ∇¯ ρg¯ ∇ + ∇¯ ρg¯ ∇ − 2 ∇¯ g¯ρσg¯ ∇ . (2.21) µν µν ρ ρν µ µ ρν σ d ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: equation

text: σ

text: σ

text: g¯

text: 2σ

text: µ

text: ν

text: ν

text: µ

text: µν

text: d

text: and

text: similarly

text: from

text: the

text: modes

text: of

text: the

text: ξµ

text: ﬁeld

text: that

text: obey

text: the

text: Killing

text: equation

text: ξ

text: ξ

text: µ

text: ν

text: ν

text: µ

text: These

text: modes,

text: when

text: evaluated

text: on

text: the

text: sphere,

text: correspond

text: to

text: the

text: lowest

text: two

text: modes

text: of

text: σ

text: and

text: the

text: lowest

text: mode

text: of

text: ξµ

text: respectively

text: and

text: therefore

text: they

text: should

text: be

text: excluded

text: from

text: the

text: trace

text: evaluation.

text: For

text: details

text: of

text: the

text: heat

text: kernel

text: coeﬃcients

text: of

text: the

text: constrained

text: ﬁelds

text: as

text: well

text: as

text: for

text: the

text: exclusion

text: of

text: lowest

text: modes

text: see

text: appendix

text: C.

text: Gauge

text: ﬁxing

text: and

text: ghosts

text: Because

text: of

text: the

text: diﬀeomorphism

text: invariance

text: of

text: the

text: metric

text: ﬁeld

text: the

text: eﬀective

text: action

text: has

text: to

text: be

text: supplemented

text: with

text: a

text: gauge

text: ﬁxing

text: term

text: so

text: that

text: only

text: the

text: physical

text: modes

text: of

text: the

text: ﬁeld

text: are

text: taken

text: into

text: account.

text: It

text: is

text: convenient

text: to

text: choose

text: a

text: gauge

text: ﬁxing

text: term

text: of

text: the

text: form

text: S

text: ddx

text: g¯

text: g¯µνF

text: F

text: gf

text: µ

text: ν

text: 2α

text: with

text: the

text: gauge

text: ﬁxing

text: condition

text: F

text: Here

text: we

text: will

text: choose

text: F

text: to

text: be

text: given

text: by

text: µ

text: µ

text: δ

text: F

text: 2κ

text: νh

text: h

text: µ

text: µν

text: µ

text: d

text: with

text: κ

text: and

text: h

text: hµνg¯

text: being

text: the

text: trace

text: of

text: the

text: ﬂuctuation.

text: The

text: gauge

text: ﬁxing

text: N

text: µν

text: condition

text: that

text: was

text: originally

text: used

text: in

text: corresponds

text: to

text: δ

text: d

text: Here

text: and

text: from

text: now

text: on

text: bared

text: geometrical

text: quantities,

text: such

text: as

text: above,

text: mean

text: that

text: they

text: are

text: constructed

text: by

text: the

text: background

text: metric

text: g¯

text: Then,

text: by

text: substituting

text: F

text: into

text: the

text: gauge

text: ﬁxing

text: action

text: we

text: get

text: µν

text: µ

text: κ2

text: δ

text: δ

text: S

text: ddx

text: g¯

text: ρh

text: λhµ

text: h∇¯

text: h∇¯

text: ρhµ

text: gf

text: α

text: µρ

text: λ

text: d

text: d

text: µ

text: ρ

text: For

text: the

text: above

text: gauge

text: ﬁxing,

text: the

text: corresponding

text: ghost

text: action

text: is

text: S

text: ddx

text: g¯

text: C¯µM

text: Cν,

text: gh

text: µν

text: with

text: the

text: Faddeev-Popov

text: operator

text: given

text: by

text: δ

text: M

text: ρg¯

text: ρg¯

text: g¯ρσg¯

text: µν

text: µν

text: ρ

text: ρν

text: µ

text: µ

text: ρν

text: σ

text: d

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: R. M. Adin, D. Kalmanovich and E. Nevo On the cubical g-vector R(cid:98) (cid:99) Denote by e the i-th unit vector in d/2 . Stacked cubical polytopes show that the ray spanned by e i 1 C C is in , and neighborly cubical polytopes show that the ray spanned by e(cid:98) (cid:99) is in . Our main result is d d/2 d C that all the rays spanned by the vectors e are in . Thus i d A ⊆ C Theorem 1. . d d The conjecture of Adin and Babson–Billera–Chan is A C Conjecture 2. = . d d An analogue of Theorem 1 was previously known for the much wider class of PL cubical spheres [BBC97, ≤ Theorem 5.7]. Also, Conjecture 2 holds for d 5, by combining the constructions above with Steve Klee’s ≥ result [Kle11, Prop.3.7] asserting that gc(Q) 0 for any cubical polytope Q of dimension 2k + 1. k − − Sanyal and Ziegler [SZ10] showed how to construct, from any simplicial (d 2)-polytope P on n 1 vertices and a total order v < v < . . . < v − on its vertices, a cubical d-polytope Q = Q(P, <) on 1 2 n 1 R 2n vertices; it is the projection of a deformed n-cube in n onto the last d coordinates. Further, they showed that if P is k-neighborly then the k-skeleton of Q is isomorphic to the k-skeleton of the n-cube. − − We apply their construction to the case where P is the k-neighborly k-stacked (d 2)-polytope on n 1 n ≤ ≤ (cid:98) − (cid:99) vertices constructed by McMullen and Walkup [MW71], with 1 k d 2 , and with a suitable total 2 order <. Analyzing the cubical g-vectors of the resulting polytopes Q = Q(P , <), as n tends to inﬁnity, n n gives Theorem 1. See Theorem 12 and Corollary 13 for the exact values and asymptotic behavior of the cubical g-vectors. The generalized lower bound theorem for simplicial polytopes (GLBT), conjectured by McMullen and ≤ (cid:98) (cid:99) Walkup [MW71] and proved by Murai and Nevo [MN13], asserts that for 1 k < d , a simplicial d- 2 polytope P is k-stacked if and only if g (P) = 0. The polytopes Q demonstrate that the natural cubical k+1 n analogue of the GLBT does not hold: while gc (Q ) = 0, Q is not (k + 1)-stacked according to the k+2 n n deﬁnition of Babson, Billera and Chan [BBC97]. The paper is organized as follows. Basic deﬁnitions and notation are are given in section . In section we deﬁne our variant of the McMullen–Walkup polytopes. A sketch of the Sanyal–Ziegler construction is given in section . In section we construct the polytopes Q(k, d, n) = Q mentioned above and analyze n their cubical g-vector. In the ﬁnal section we show that a cubical GLBT fails, by showing that Q is not n (k + 1)-stacked. Acknowledgments The authors thank Michael Joswig and Isabella Novik for helpful comments. This work was partially supported by the National Science Foundation under Grant No. DMS-1440140, while the last author was in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Fall 2017 semester. Research of Kalmanovich and Nevo partially supported by Israel Science Foundation grant ISF-1695/15 and by grant 2528/16 of the ISF-NRF Singapore joint research program. 2 Preliminaries The purpose of this section is mainly to set the notation that we will use throughout the paper. For undeﬁned terminology we refer the reader to [Zie95]. A d-dimensional polytope P is the convex hull 2 ##LTLine## ##LTLine## ##LTLine##

text: R.

text: M.

text: Adin,

text: D.

text: Kalmanovich

text: and

text: E.

text: Nevo

text: On

text: the

text: cubical

text: g-vector

text: Denote

text: by

text: e

text: the

text: i-th

text: unit

text: vector

text: in

text: Stacked

text: cubical

text: polytopes

text: show

text: that

text: the

text: ray

text: spanned

text: by

text: e

text: i

text: C

text: C

text: is

text: in

text: and

text: neighborly

text: cubical

text: polytopes

text: show

text: that

text: the

text: ray

text: spanned

text: by

text: is

text: in

text: Our

text: main

text: result

text: is

text: d

text: d

text: C

text: that

text: all

text: the

text: rays

text: spanned

text: by

text: the

text: vectors

text: e

text: are

text: in

text: Thus

text: i

text: d

text: A

text: C

text: Theorem

text: d

text: d

text: The

text: conjecture

text: of

text: Adin

text: and

text: Babson–Billera–Chan

text: is

text: A

text: C

text: Conjecture

text: d

text: d

text: An

text: analogue

text: of

text: Theorem

text: was

text: previously

text: known

text: for

text: the

text: much

text: wider

text: class

text: of

text: PL

text: cubical

text: spheres

text: Theorem

text: Also,

text: Conjecture

text: holds

text: for

text: d

text: by

text: combining

text: the

text: constructions

text: above

text: with

text: Steve

text: Klee’s

text: result

text: asserting

text: that

text: gc(Q)

text: for

text: any

text: cubical

text: polytope

text: Q

text: of

text: dimension

text: k

text: Sanyal

text: and

text: Ziegler

text: showed

text: how

text: to

text: construct,

text: from

text: any

text: simplicial

text: (d

text: P

text: on

text: n

text: vertices

text: and

text: a

text: total

text: order

text: v

text: v

text: v

text: on

text: its

text: vertices,

text: a

text: cubical

text: d-polytope

text: Q

text: Q(P,

text: on

text: n

text: R

text: vertices;

text: it

text: is

text: the

text: projection

text: of

text: a

text: deformed

text: n-cube

text: in

text: n

text: onto

text: the

text: last

text: d

text: coordinates.

text: Further,

text: they

text: showed

text: that

text: if

text: P

text: is

text: k-neighborly

text: then

text: the

text: k-skeleton

text: of

text: Q

text: is

text: isomorphic

text: to

text: the

text: k-skeleton

text: of

text: the

text: n-cube.

text: We

text: apply

text: their

text: construction

text: to

text: the

text: case

text: where

text: P

text: is

text: the

text: k-neighborly

text: k-stacked

text: (d

text: on

text: n

text: n

text: vertices

text: constructed

text: by

text: McMullen

text: and

text: Walkup

text: with

text: k

text: d

text: and

text: with

text: a

text: suitable

text: total

text: order

text: Analyzing

text: the

text: cubical

text: g-vectors

text: of

text: the

text: resulting

text: polytopes

text: Q

text: Q(P

text: as

text: n

text: tends

text: to

text: inﬁnity,

text: n

text: n

text: gives

text: Theorem

text: See

text: Theorem

text: and

text: Corollary

text: for

text: the

text: exact

text: values

text: and

text: asymptotic

text: behavior

text: of

text: the

text: cubical

text: g-vectors.

text: The

text: generalized

text: lower

text: bound

text: theorem

text: for

text: simplicial

text: polytopes

text: (GLBT),

text: conjectured

text: by

text: McMullen

text: and

text: Walkup

text: and

text: proved

text: by

text: Murai

text: and

text: Nevo

text: asserts

text: that

text: for

text: k

text: d

text: a

text: simplicial

text: d-

text: polytope

text: P

text: is

text: k-stacked

text: if

text: and

text: only

text: if

text: g

text: (P)

text: The

text: polytopes

text: Q

text: demonstrate

text: that

text: the

text: natural

text: cubical

text: n

text: analogue

text: of

text: the

text: GLBT

text: does

text: not

text: hold:

text: while

text: gc

text: (Q

text: Q

text: is

text: not

text: (k

text: according

text: to

text: the

text: n

text: n

text: deﬁnition

text: of

text: Babson,

text: Billera

text: and

text: Chan

text: The

text: paper

text: is

text: organized

text: as

text: follows.

text: Basic

text: deﬁnitions

text: and

text: notation

text: are

text: are

text: given

text: in

text: section

text: In

text: section

text: we

text: deﬁne

text: our

text: variant

text: of

text: the

text: McMullen–Walkup

text: polytopes.

text: A

text: sketch

text: of

text: the

text: Sanyal–Ziegler

text: construction

text: is

text: given

text: in

text: section

text: In

text: section

text: we

text: construct

text: the

text: polytopes

text: Q(k,

text: d,

text: n)

text: Q

text: mentioned

text: above

text: and

text: analyze

text: n

text: their

text: cubical

text: g-vector.

text: In

text: the

text: ﬁnal

text: section

text: we

text: show

text: that

text: a

text: cubical

text: GLBT

text: fails,

text: by

text: showing

text: that

text: Q

text: is

text: not

text: n

text: (k

text: Acknowledgments

text: The

text: authors

text: thank

text: Michael

text: Joswig

text: and

text: Isabella

text: Novik

text: for

text: helpful

text: comments.

text: This

text: work

text: was

text: partially

text: supported

text: by

text: the

text: National

text: Science

text: Foundation

text: under

text: Grant

text: No.

text: while

text: the

text: last

text: author

text: was

text: in

text: residence

text: at

text: the

text: Mathematical

text: Sciences

text: Research

text: Institute

text: in

text: Berkeley,

text: California,

text: during

text: the

text: Fall

text: semester.

text: Research

text: of

text: Kalmanovich

text: and

text: Nevo

text: partially

text: supported

text: by

text: Israel

text: Science

text: Foundation

text: grant

text: and

text: by

text: grant

text: of

text: the

text: ISF-NRF

text: Singapore

text: joint

text: research

text: program.

text: Preliminaries

text: The

text: purpose

text: of

text: this

text: section

text: is

text: mainly

text: to

text: set

text: the

text: notation

text: that

text: we

text: will

text: use

text: throughout

text: the

text: paper.

text: For

text: undeﬁned

text: terminology

text: we

text: refer

text: the

text: reader

text: to

text: A

text: d-dimensional

text: polytope

text: P

text: is

text: the

text: convex

text: hull

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: R. M. Adin, D. Kalmanovich and E. Nevo On the cubical g-vector • (cid:54)⊆ \ F is a face of P, with Vert(T) Vert(F), iﬀ Vert(F) Vert(T) is the set of vertices of a face F¯ of C. ∗ (cid:48) (cid:48) (cid:48) Now, let F be a face added when pulling v in P. Then F = v F , with F a face of P, and F a face 1 1 (cid:48) ∗ (cid:48) of P = conv(v , ..., v ). With the notation above, F¯ is a face of both C and C , and thus F¯ = v F¯ 1 2 m 1 1 ∗ is added when pulling v in C. Similarly, when pushing v in P, we add faces F (and v F) of P which 1 1 1 1 are non-faces of P, and so F¯ is a face of C , and a non-face of C, and thus is added when pushing v in 1 1 C. Reversing the argument shows that faces added when pushing/pulling v in C give corresponding added 1 faces in pushing/pulling in P. When constructing Lex (P) and Lex (C) sequentially, we always push/pull a a (cid:4) v in a suitable MW-polytope and the corresponding suitable cyclic polytope C, hence (10) follows. 1 We need some terminology, to be used in the following two propositions: let G be a subset of the set { } ⊆ v , . . . , v − of vertices of the cyclic polytope C. A block B G is a maximal (with respect to 1 n d+2k+1 inclusion) subset of G consisting of consecutive vertices, B is inner if min B > v and max B < v − = 1 n d+2k+1 x, and B is even / odd if its size is. We call a subset of vertices of C isolated if no two of them are − − consecutive. Recall that p denotes the apex of the diamond D of P = MW(2k, d 2, n 1). a ≤ ≤ − Proposition 17. For any 1 a n d + 1, the missing faces in D of size k + 1 or k + 2 are exactly the a sets of the following two types: { } ∪ (cid:48) (cid:48) ∈ (cid:48) (cid:48) (1) F = p F , with F of size k + 1, consisting of isolated vertices of C, x / F , and min F = a; or ∈ (cid:54) (2) F is of size k + 1, consisting of isolated vertices of C, x / F, min F = a. Proof. Any missing face F of D must contain a missing face of P, and is either of the form a { } ∪ (cid:48) (cid:48) (i) p F where F is an interior face of the ball Lex (P), or a (ii) a missing face of ∂P which is not a face of Lex (P), or a (cid:48) (iii) a missing face of Lex (P) containing a missing face F of ∂P as a strict subset. a Using Lemma 16 we can replace P by C in (i), (ii) and (iii), and F contains a missing face of C not containing x. By the Gale evenness criterion, a subset of i vertices of C forms a face of C iﬀ it contains at most − 2k i inner odd blocks. It follows that the missing faces of C are exactly the sets of k + 1 isolated vertices, { } containing at most one element of v , x . 1 (cid:48) ∈ (cid:48) (cid:48) { | ≥ } Let F be a missing face of C, x / F , with min F = v , and denote C = conv v j i , a cyclic i i j polytope contained in C. (cid:48) (cid:48) (cid:48) ∪ { } If i = a then F is a face in Lex (C ), thus F is also in Lex (C), yielding a missing face F = F v a a a a of D of type (i). a (cid:48) (cid:48) If i > a, then F is a missing face in Lex (C), hence F = F is a missing face of D of type (ii). a a (cid:48) \ (cid:48) (cid:48) If i < a, C blocks v from seeing F v , so F is missing in Lex (C), and so F = F is a missing face i+1 i i a of D of type (ii). a Now, we will show that missing faces of type (iii) do not exist, and thus the characterization in the { } \ (cid:48) statement will follow. Indeed, assume F to be of type (iii), and denote z = F F (indeed this diﬀerence | (cid:48)| | | ≤ (cid:48) ⊆ (cid:48) ∈ set must be a singleton as F = k + 1 < F k + 2 and F F). As F Lex (C), the discussion above a (cid:48) (cid:54) (cid:48) \ { } shows min(F ) = v . If z = min F, then min F = v , and z is adjacent to some vertex of F (else F v a a a would be missing in Lex (C)). But there is no missing face containing v in Lex (C ) so F is in Lex (C ), a a a a a a 13 ##LTLine##

text: R.

text: M.

text: Adin,

text: D.

text: Kalmanovich

text: and

text: E.

text: Nevo

text: On

text: the

text: cubical

text: g-vector

label: list

text: F

text: is

text: a

text: face

text: of

text: P,

text: with

text: Vert(T)

text: Vert(F),

text: iﬀ

text: Vert(F)

text: Vert(T)

text: is

text: the

text: set

text: of

text: vertices

text: of

text: a

text: face

text: F¯

text: of

text: C.

text: Now,

text: let

text: F

text: be

text: a

text: face

text: added

text: when

text: pulling

text: v

text: in

text: P.

text: Then

text: F

text: v

text: F

text: with

text: F

text: a

text: face

text: of

text: P,

text: and

text: F

text: a

text: face

text: of

text: P

text: conv(v

text: v

text: With

text: the

text: notation

text: above,

text: F¯

text: is

text: a

text: face

text: of

text: both

text: C

text: and

text: C

text: and

text: thus

text: F¯

text: v

text: F¯

text: m

text: is

text: added

text: when

text: pulling

text: v

text: in

text: C.

text: Similarly,

text: when

text: pushing

text: v

text: in

text: P,

text: we

text: add

text: faces

text: F

text: (and

text: v

text: F)

text: of

text: P

text: which

text: are

text: non-faces

text: of

text: P,

text: and

text: so

text: F¯

text: is

text: a

text: face

text: of

text: C

text: and

text: a

text: non-face

text: of

text: C,

text: and

text: thus

text: is

text: added

text: when

text: pushing

text: v

text: in

text: C.

text: Reversing

text: the

text: argument

text: shows

text: that

text: faces

text: added

text: when

text: pushing/pulling

text: v

text: in

text: C

text: give

text: corresponding

text: added

text: faces

text: in

text: pushing/pulling

text: in

text: P.

text: When

text: constructing

text: Lex

text: (P)

text: and

text: Lex

text: (C)

text: sequentially,

text: we

text: always

text: push/pull

text: a

text: a

text: v

text: in

text: a

text: suitable

text: MW-polytope

text: and

text: the

text: corresponding

text: suitable

text: cyclic

text: polytope

text: C,

text: hence

text: follows.

text: We

text: need

text: some

text: terminology,

text: to

text: be

text: used

text: in

text: the

text: following

text: two

text: propositions:

text: let

text: G

text: be

text: a

text: subset

text: of

text: the

text: set

text: v

text: v

text: of

text: vertices

text: of

text: the

text: cyclic

text: polytope

text: C.

text: A

text: block

text: B

text: G

text: is

text: a

text: maximal

text: (with

text: respect

text: to

text: n

text: inclusion)

text: subset

text: of

text: G

text: consisting

text: of

text: consecutive

text: vertices,

text: B

text: is

text: inner

text: if

text: min

text: B

text: v

text: and

text: max

text: B

text: v

text: n

text: x,

text: and

text: B

text: is

text: even

text: odd

text: if

text: its

text: size

text: is.

text: We

text: call

text: a

text: subset

text: of

text: vertices

text: of

text: C

text: isolated

text: if

text: no

text: two

text: of

text: them

text: are

text: consecutive.

text: Recall

text: that

text: p

text: denotes

text: the

text: apex

text: of

text: the

text: diamond

text: D

text: of

text: P

text: d

text: n

text: a

text: Proposition

text: For

text: any

text: a

text: n

text: d

text: the

text: missing

text: faces

text: in

text: D

text: of

text: size

text: k

text: or

text: k

text: are

text: exactly

text: the

text: a

text: sets

text: of

text: the

text: following

text: two

text: types:

text: F

text: p

text: F

text: with

text: F

text: of

text: size

text: k

text: consisting

text: of

text: isolated

text: vertices

text: of

text: C,

text: x

text: F

text: and

text: min

text: F

text: a;

text: or

text: F

text: is

text: of

text: size

text: k

text: consisting

text: of

text: isolated

text: vertices

text: of

text: C,

text: x

text: F,

text: min

text: F

text: a.

text: Proof.

text: Any

text: missing

text: face

text: F

text: of

text: D

text: must

text: contain

text: a

text: missing

text: face

text: of

text: P,

text: and

text: is

text: either

text: of

text: the

text: form

text: a

text: (i)

text: p

text: F

text: where

text: F

text: is

text: an

text: interior

text: face

text: of

text: the

text: ball

text: Lex

text: (P),

text: or

text: a

text: (ii)

text: a

text: missing

text: face

text: of

text: ∂P

text: which

text: is

text: not

text: a

text: face

text: of

text: Lex

text: (P),

text: or

text: a

text: (iii)

text: a

text: missing

text: face

text: of

text: Lex

text: (P)

text: containing

text: a

text: missing

text: face

text: F

text: of

text: ∂P

text: as

text: a

text: strict

text: subset.

text: a

text: Using

text: Lemma

text: we

text: can

text: replace

text: P

text: by

text: C

text: in

text: (i),

text: (ii)

text: and

text: (iii),

text: and

text: F

text: contains

text: a

text: missing

text: face

text: of

text: C

text: not

text: containing

text: x.

text: By

text: the

text: Gale

text: evenness

text: criterion,

text: a

text: subset

text: of

text: i

text: vertices

text: of

text: C

text: forms

text: a

text: face

text: of

text: C

text: iﬀ

text: it

text: contains

text: at

text: most

text: i

text: inner

text: odd

text: blocks.

text: It

text: follows

text: that

text: the

text: missing

text: faces

text: of

text: C

text: are

text: exactly

text: the

text: sets

text: of

text: k

text: isolated

text: vertices,

text: containing

text: at

text: most

text: one

text: element

text: of

text: v

text: x

text: Let

text: F

text: be

text: a

text: missing

text: face

text: of

text: C,

text: x

text: F

text: with

text: min

text: F

text: v

text: and

text: denote

text: C

text: conv

text: v

text: j

text: i

text: a

text: cyclic

text: i

text: i

text: j

text: polytope

text: contained

text: in

text: C.

text: If

text: i

text: a

text: then

text: F

text: is

text: a

text: face

text: in

text: Lex

text: (C

text: thus

text: F

text: is

text: also

text: in

text: Lex

text: (C),

text: yielding

text: a

text: missing

text: face

text: F

text: F

text: v

text: a

text: a

text: a

text: a

text: of

text: D

text: of

text: type

text: (i).

text: a

text: If

text: i

text: a,

text: then

text: F

text: is

text: a

text: missing

text: face

text: in

text: Lex

text: (C),

text: hence

text: F

text: F

text: is

text: a

text: missing

text: face

text: of

text: D

text: of

text: type

text: (ii).

text: a

text: a

text: If

text: i

text: a,

text: C

text: blocks

text: v

text: from

text: seeing

text: F

text: v

text: so

text: F

text: is

text: missing

text: in

text: Lex

text: (C),

text: and

text: so

text: F

text: F

text: is

text: a

text: missing

text: face

text: i

text: i

text: a

text: of

text: D

text: of

text: type

text: (ii).

text: a

text: Now,

text: we

text: will

text: show

text: that

text: missing

text: faces

text: of

text: type

text: (iii)

text: do

text: not

text: exist,

text: and

text: thus

text: the

text: characterization

text: in

text: the

text: statement

text: will

text: follow.

text: Indeed,

text: assume

text: F

text: to

text: be

text: of

text: type

text: (iii),

text: and

text: denote

text: z

text: F

text: F

text: (indeed

text: this

text: diﬀerence

text: set

text: must

text: be

text: a

text: singleton

text: as

text: F

text: k

text: F

text: k

text: and

text: F

text: F).

text: As

text: F

text: Lex

text: (C),

text: the

text: discussion

text: above

text: a

text: shows

text: min(F

text: v

text: If

text: z

text: min

text: F,

text: then

text: min

text: F

text: v

text: and

text: z

text: is

text: adjacent

text: to

text: some

text: vertex

text: of

text: F

text: (else

text: F

text: v

text: a

text: a

text: a

text: would

text: be

text: missing

text: in

text: Lex

text: (C)).

text: But

text: there

text: is

text: no

text: missing

text: face

text: containing

text: v

text: in

text: Lex

text: (C

text: so

text: F

text: is

text: in

text: Lex

text: (C

text: a

text: a

text: a

text: a

text: a

text: a

text: ##LTLine##


text: R. M. Adin, D. Kalmanovich and E. Nevo On the cubical g-vector ≤ ≤ −  ≤ ≤ (cid:98) − (cid:99) Proposition 9. For each 2 a n d + 1 a(cid:16)(cid:16)nd 0 i (cid:17)(cid:17) d21 : 0(cid:16)(cid:16), − −(cid:17)(cid:17) if i > k + 1; g (D ) = n d a+1 , if i = k + 1; i a k − ≤ n d , if i k. i ∩ Proof. Let v be the apex of the diamond D . The link condition lk (D ) = lk (D ) lk (D ) holds: 0 a v1v0 a v1 a v0 a indeed, lk (D ) = ∂P and the faces of lk (D ) not containing v form lk (Lex P), so v0 a v1 a 0 v1 a ∩ ∩ lk (D ) lk (D ) = lk (Lex P) ∂P, v1 a v0 a v1 a ≥ and as a 2, the right hand side equals lk (∂P). Further, v1 lk (∂P) = lk (lk (D )) = lk (D ), v1 v1 v0 a v1v0 a − thus the link condition holds. Denote by K the (d 2)-complex obtained from ∂D by contracting the edge a v v . Then, the h-polynomials are related by 1 0 · h(Da, t) = h(K, t) + t h(P /v1 , t), − see e.g. [Nev07, Eq.(10)]. Note that K is the boundary complex of a polytope, namely of the (a 1)- − − − lexicographic diamond over P = MW(2k, d 2, n 2); denote this diamond by Y − (n 1). Then, by 1 a 1 Lemma 5 − · − − − h(D , t) = h(Y − (n 1), t) + t h(MW(2k 1, d 3, n 2), t). a a 1 − We may again contract the edge between the ﬁrst vertex of P and the apex v in Y − (n 1) to obtain 1 0 a 1 − − Y − (n 2). We do these edge contractions a 1 t(cid:88)imes, and get a 2 − a 1 − · − − − − h(D , t) = h(Y (n a + 1), t) + t h(MW(2k 1, d 3, n 1 j), t), a 1 j=1 and therefore (cid:88) − a 1 − − − − − (8) g (D ) = g (Y (n a + 1)) + g − (MW(2k 1, d 3, n 1 j)). i a i 1 i 1 j=1 − − − As Y (n a + 1) is the 1-lexicographic diamond over MW(2k, d 2, n a), the claimed result follows 1 (cid:18)(cid:18) (cid:19)(cid:19) (cid:18)(cid:18) (cid:19)(cid:19) from Proposition 8, Lemm(cid:88)a 4, and the identity − − − − − a 1 n d a + 1 n d j + 1 (cid:18)(cid:18) + (cid:19)(cid:19) − (cid:20)(cid:18)(cid:18) (cid:19)(cid:19) (cid:18)(cid:18) (cid:19)(cid:19)(cid:21) (cid:18)(cid:18) (cid:19)(cid:19) (cid:88) i i 1 j=1 − − − − − − − − a 1 n d a + 1 n d j + 1 − n d j n d (cid:4) = + (cid:0)(cid:0) (cid:1)(cid:1) = . i i i i j=1 ≥ Combining (6) with Propositions 8 and 9, and noting that 0 = 0 for k 1, we conclude k 9 ##LTLine## ##LTLine##

text: R.

text: M.

text: Adin,

text: D.

text: Kalmanovich

text: and

text: E.

text: Nevo

text: On

text: the

text: cubical

text: g-vector

text: Proposition

text: For

text: each

text: a

text: n

text: d

text: 1

text: i

text: if

text: i

text: k

text: g

text: (D

text: n

text: d

text: if

text: i

text: k

text: i

text: a

text: k

text: n

text: d

text: if

text: i

text: k.

text: i

text: Proof.

text: Let

text: v

text: be

text: the

text: apex

text: of

text: the

text: diamond

text: D

text: The

text: link

text: condition

text: lk

text: (D

text: lk

text: (D

text: lk

text: (D

text: holds:

text: a

text: a

text: a

text: a

text: indeed,

text: lk

text: (D

text: ∂P

text: and

text: the

text: faces

text: of

text: lk

text: (D

text: not

text: containing

text: v

text: form

text: lk

text: (Lex

text: P),

text: so

text: a

text: a

text: a

text: lk

text: (D

text: lk

text: (D

text: lk

text: (Lex

text: P)

text: ∂P,

text: a

text: a

text: a

text: and

text: as

text: a

text: the

text: right

text: hand

text: side

text: equals

text: lk

text: (∂P).

text: Further,

text: lk

text: (∂P)

text: lk

text: (lk

text: (D

text: lk

text: (D

text: a

text: a

text: thus

text: the

text: link

text: condition

text: holds.

text: Denote

text: by

text: K

text: the

text: (d

text: obtained

text: from

text: ∂D

text: by

text: contracting

text: the

text: edge

text: a

text: v

text: v

text: Then,

text: the

text: h-polynomials

text: are

text: related

text: by

text: h(Da,

text: t)

text: h(K,

text: t)

text: t

text: h(P

text: t),

text: see

text: e.g.

text: Note

text: that

text: K

text: is

text: the

text: boundary

text: complex

text: of

text: a

text: polytope,

text: namely

text: of

text: the

text: (a

text: lexicographic

text: diamond

text: over

text: P

text: d

text: n

text: denote

text: this

text: diamond

text: by

text: Y

text: (n

text: Then,

text: by

text: a

text: Lemma

text: h(D

text: t)

text: h(Y

text: (n

text: t)

text: t

text: d

text: n

text: t).

text: a

text: a

text: We

text: may

text: again

text: contract

text: the

text: edge

text: between

text: the

text: ﬁrst

text: vertex

text: of

text: P

text: and

text: the

text: apex

text: v

text: in

text: Y

text: (n

text: to

text: obtain

text: a

text: Y

text: (n

text: We

text: do

text: these

text: edge

text: contractions

text: a

text: and

text: get

text: a

text: a

text: h(D

text: t)

text: h(Y

text: (n

text: a

text: t)

text: t

text: d

text: n

text: j),

text: t),

text: a

text: and

text: therefore

text: a

text: g

text: (D

text: g

text: (Y

text: (n

text: a

text: g

text: d

text: n

text: j)).

text: i

text: a

text: i

text: i

text: As

text: Y

text: (n

text: a

text: is

text: the

text: diamond

text: over

text: d

text: n

text: a),

text: the

text: claimed

text: result

text: follows

text: from

text: Proposition

text: and

text: the

text: identity

text: a

text: n

text: d

text: a

text: n

text: d

text: j

text: i

text: i

text: a

text: n

text: d

text: a

text: n

text: d

text: j

text: n

text: d

text: j

text: n

text: d

text: i

text: i

text: i

text: i

text: Combining

text: with

text: Propositions

text: and

text: and

text: noting

text: that

text: for

text: k

text: we

text: conclude

text: k

text: ##LTLine##

text: ##LTLine##


text: 5 − The ΦHI,JF r (Eq.18) term is an integration, and we Hydrogen atom Nr,mult Nang,max N need use the(cid:88)approximation as shown in Eq.(7). light 1 302 4740 tight 2 434 14450 ΦHF−r ≈ (cid:88) ∂w(r(cid:34))n(r)V (1)(r) (cid:35) really-tight 2 590 19502 I,J ∂R I J r TABLE I. Diﬀerent grid settings for the hydrogen atom. ∂n(r) ∂V (1)(r) + w(r) V (1)(r) + n(r) I (2.8) ∂R I ∂R J J r Here ﬁrst order den(cid:80)sity is In should be noted that, when using Gaussian basis set12,20,21, only the Pulay term contains xc part need to be considered, and the hˆ is replaced by vˆ . ∂∂nR(rJ) = (cid:80)µν ∂∂PR(cid:126)µJν χµ((cid:126)r)χν((cid:126)r) And ﬁnally, t(cid:88)he i(cid:88)ntegraKtSion form o(cid:20)f ΦPI,−J Sx(Ec q.26(cid:21)) is ∂χ ((cid:126)r) +(cid:80) µν Pµν ∂Rµ(cid:126) χν((cid:126)r) ΦP−S ≈ 2 (cid:88) (cid:88) W · ∂w(r) ∂χµ(r)χ (r) J I,J µ,ν ∂R ∂R ν + µν Pµνχµ((cid:126)r)∂∂χRν(cid:126)((cid:126)r) . (29) + (cid:20)2 µ,ν r W · w(r)J I (cid:21) J µ,ν µ,ν r When considering moving-grid eﬀect, the derivative of (cid:26) · ∂2χ (r) ∂χ (r) ∂χ (r) basis function is written as: µ χ (r) + µ ν . (34) ∂RI∂RJ ν ∂RI ∂R(cid:126)J −(cid:79) ∈ ∂χ ((cid:126)r) χ ((cid:126)r)δ if r / atom µ = (cid:79) µ− I(µ),J ∈ J (30) When considering moving-grid eﬀect, the second deriva- ∂R(cid:126)J χµ((cid:126)r)(1 δI(µ),J) if r atomJ tive of basis fu(cid:26)nction is written as, and the electron-io(cid:26)nic interaction is (cid:79) ∈ ∂2χ ((cid:126)r) 2χ ((cid:126)r)δ if r / atom ∂V∂I(R(cid:126)1)(r) = EEqq..((2302)) iiff rr ∈∈/ aattoommJJ (31) ∂R(cid:126)IµR(cid:126)J = −(cid:79)2χµ((cid:126)r)(1µ− δII(µ(µ),)J,J) if r ∈ atomJJ(35) J − − When r belong to atomJ, then it is moving grid, and Other Pulay terms ΦPI,J P (Eq.23) and ΦPI,J W (Eq.25) the derivative of electron-ionic interaction is contains the derivatives which similar to Pulay force, so the moving-grid eﬀect do not need to be considered as explained in Sec.. ∂VI(α1)(r) = −(1 − δ )δ | −ZI | III. RESULTS ∂R IJ αβ r R 3 Jβ I − − · − Z (R r) (R r) The grids setting in FHI-aims is charactered by light, + 3 (1 δ ) I I | − α | J β ) .(32) IJ r R 5 tight and really-tight by increasing the radial multi- I plier N and the maximum number of angular r,mult 2. Moving-grid eﬀect in Pulay term integration points N , e.g. for hydrogen atom, the ang,max number of total integration grids is shown in Tab.I. − Similarly, using Eq.(7),the ΦP H (Eq.24) term can be In Table II, we present force constants for hydrogen written as (cid:88) (cid:88) I,J(cid:20) (cid:21) dimer with a local approximation for exchange and corre- lation (LDA parametrization of Perdew and Zunger26 for the correlation energy density of the homogeneous elec- ΦP−H ≈ 2 (cid:88) (cid:88) P · ∂w(r) ∂χµ(r)hˆ χ (r) tron gas based on the data of Ceperley and Alder27). In I,J µ,ν ∂R ∂R KS ν J I all cases, the force constants calculations were performed µ,ν r (cid:20) · for the respective equilibrium geometry, i.e., the structure + 2 P w(r) − µ,ν obtained by relaxation (maximum force < 10 4 eV/˚A) µ,ν r using the exact same computational settings. In the fol- · ∂2χµ(r) hˆ χ (r) lowing, a minimal basis includes the radial functions of ∂R ∂R KS ν the occupied orbitals of free atoms with noble gas conﬁg- I J ∂χ (r) dhˆ (cid:21) uration and quantum numbers of the additional valence + µ KS χ (r) functions. And additional radial functions are added to ∂RI dR(cid:126)J ν make “tier 1” ,“tier 2”, and so on. See Ref.23 for more + ∂χµ(r)hˆ ∂χν(r) . (33) details. Here in Table II and Table III we use tier 2 basis ∂RI KS ∂R(cid:126)J set. ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: The

text: ΦHI,JF

text: r

text: term

text: is

text: an

text: integration,

text: and

text: we

text: Hydrogen

text: atom

text: Nr,mult

text: Nang,max

text: N

text: need

text: use

text: as

text: shown

text: in

text: light

text: tight

text: ΦHF−r

text: really-tight

text: I,J

text: ∂R

text: I

text: J

text: r

text: TABLE

text: I.

text: Diﬀerent

text: grid

text: settings

text: for

text: the

text: hydrogen

text: atom.

text: ∂n(r)

text: ∂V

text: w(r)

text: V

text: n(r)

text: I

text: ∂R

text: I

text: ∂R

text: J

text: J

text: r

text: Here

text: ﬁrst

text: order

text: is

text: In

text: should

text: be

text: noted

text: that,

text: when

text: using

text: Gaussian

text: basis

text: only

text: the

text: Pulay

text: term

text: contains

text: xc

text: part

text: need

text: to

text: be

text: considered,

text: and

text: the

text: hˆ

text: is

text: replaced

text: by

text: vˆ

text: ∂∂nR(rJ)

text: And

text: ﬁnally,

text: form

text: ΦPI,−J

text: Sx(Ec

text: is

text: ∂χ

text: µν

text: Pµν

text: ΦP−S

text: W

text: ∂w(r)

text: ∂χµ(r)χ

text: (r)

text: J

text: I,J

text: µ,ν

text: ∂R

text: ∂R

text: ν

text: µν

text: µ,ν

text: r

text: W

text: w(r)J

text: I

text: J

text: µ,ν

text: µ,ν

text: r

text: When

text: considering

text: moving-grid

text: eﬀect,

text: the

text: derivative

text: of

text: ∂2χ

text: (r)

text: ∂χ

text: (r)

text: ∂χ

text: (r)

text: basis

text: function

text: is

text: written

text: as:

text: µ

text: χ

text: (r)

text: µ

text: ν

text: ∂RI∂RJ

text: ν

text: ∂RI

text: ∂χ

text: χ

text: if

text: r

text: atom

text: µ

text: µ−

text: I(µ),J

text: J

text: When

text: considering

text: moving-grid

text: eﬀect,

text: the

text: second

text: deriva-

text: δI(µ),J)

text: if

text: r

text: atomJ

text: tive

text: of

text: basis

text: is

text: written

text: as,

text: and

text: the

text: interaction

text: is

text: ∂2χ

text: 2χ

text: if

text: r

text: atom

text: iiff

text: rr

text: aattoommJJ

text: δII(µ(µ),)J,J)

text: if

text: r

text: J

text: When

text: r

text: belong

text: to

text: atomJ,

text: then

text: it

text: is

text: moving

text: grid,

text: and

text: Other

text: Pulay

text: terms

text: ΦPI,J

text: P

text: and

text: ΦPI,J

text: W

text: the

text: derivative

text: of

text: electron-ionic

text: interaction

text: is

text: contains

text: the

text: derivatives

text: which

text: similar

text: to

text: Pulay

text: force,

text: so

text: the

text: moving-grid

text: eﬀect

text: do

text: not

text: need

text: to

text: be

text: considered

text: as

text: explained

text: in

text: Sec..

text: δ

text: )δ

text: −ZI

text: III.

text: RESULTS

text: ∂R

text: IJ

text: αβ

text: r

text: R

text: Jβ

text: I

text: Z

text: (R

text: r)

text: (R

text: r)

text: The

text: grids

text: setting

text: in

text: FHI-aims

text: is

text: charactered

text: by

text: light,

text: δ

text: I

text: I

text: α

text: J

text: β

text: IJ

text: r

text: R

text: tight

text: and

text: really-tight

text: by

text: increasing

text: the

text: radial

text: multi-

text: I

text: plier

text: N

text: and

text: the

text: maximum

text: number

text: of

text: angular

text: r,mult

text: Moving-grid

text: eﬀect

text: in

text: Pulay

text: term

text: integration

text: points

text: N

text: e.g.

text: for

text: hydrogen

text: atom,

text: the

text: ang,max

text: number

text: of

text: total

text: integration

text: grids

text: is

text: shown

text: in

text: Tab.I.

text: Similarly,

text: using

text: ΦP

text: H

text: term

text: can

text: be

text: In

text: Table

text: II,

text: we

text: present

text: force

text: constants

text: for

text: hydrogen

text: written

text: as

text: dimer

text: with

text: a

text: local

text: approximation

text: for

text: exchange

text: and

text: corre-

text: lation

text: (LDA

text: parametrization

text: of

text: Perdew

text: and

text: for

text: the

text: correlation

text: energy

text: density

text: of

text: the

text: homogeneous

text: elec-

text: ΦP−H

text: P

text: ∂w(r)

text: ∂χµ(r)hˆ

text: χ

text: (r)

text: tron

text: gas

text: based

text: on

text: the

text: data

text: of

text: Ceperley

text: and

text: In

text: I,J

text: µ,ν

text: ∂R

text: ∂R

text: KS

text: ν

text: J

text: I

text: all

text: cases,

text: the

text: force

text: constants

text: calculations

text: were

text: performed

text: µ,ν

text: r

text: for

text: the

text: respective

text: equilibrium

text: geometry,

text: i.e.,

text: the

text: structure

text: P

text: w(r)

text: µ,ν

text: obtained

text: by

text: relaxation

text: (maximum

text: force

text: eV/˚A)

text: µ,ν

text: r

text: using

text: the

text: exact

text: same

text: computational

text: settings.

text: In

text: the

text: fol-

text: hˆ

text: χ

text: (r)

text: lowing,

text: a

text: minimal

text: basis

text: includes

text: the

text: radial

text: functions

text: of

text: ∂R

text: ∂R

text: KS

text: ν

text: the

text: occupied

text: orbitals

text: of

text: free

text: atoms

text: with

text: noble

text: gas

text: conﬁg-

text: I

text: J

text: ∂χ

text: (r)

text: dhˆ

text: uration

text: and

text: quantum

text: numbers

text: of

text: the

text: additional

text: valence

text: µ

text: KS

text: χ

text: (r)

text: functions.

text: And

text: additional

text: radial

text: functions

text: are

text: added

text: to

text: ∂RI

text: ν

text: make

text: “tier

text: 1”

text: ,“tier

text: and

text: so

text: on.

text: See

text: for

text: more

text: ∂χµ(r)hˆ

text: ∂χν(r)

text: details.

text: Here

text: in

text: Table

text: II

text: and

text: Table

text: III

text: we

text: use

text: tier

text: basis

text: ∂RI

text: KS

text: set.

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: ##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: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 4 Proof of Theorem 1 We now split the proof of Theorem 1 into two main lemmas. The ﬁrst of these two lemmas follows. − Lemma 10. Let (cid:15) > c(n)N 1. Then the following concentration inequality holds: Pr(||P \ D | − E[|P \ D |]| ≥ (cid:15)) ≤ e−c(n)N1−n−2 1 ln(N)−2(1−n−1 1) n,N n n,N n 4.1 Proof of Lemma 10 \ Using Fubini’s t(cid:90)heorem, we express the volume of the set diﬀerence P D as n,N n | \ | (cid:90) (cid:90) Pn,N Dn = R \ 1{max(cid:104)Xi,x(cid:105)≤tn,N}dx n Dn ∞ | | (cid:90) (cid:90) − = ∂Dn S − rn 11{max(cid:104)Xi,x(cid:105)≤r−1tn,N}dr dσ(x) n∞ 1 1 | | − = ∂Dn (cid:18) (cid:90) rn 1 S − 1{max(cid:104)Xi,x(cid:105)≤(cid:90)r−1tn,N}dσ(x) dr 1 n 1 −2 − −2 = (cid:90)|∂Dn| 1+ln(N)n 1 N n(cid:90) 1 rn−1 S − 1{max(cid:104)Xi,x(cid:105)≤r−1tn,N}dσ(x) dr (8) 1 n 1 + (cid:90) 1+n2 −2(cid:90) − −2 rn−1 S − 1{max(cid:104)Xi,x(cid:105)≤r−1(cid:19)tn,N}dr dσ(x) 1+ln(N)n 1 N n 1 n 1 ∞ ‘ − + 1+ 2 rn 1 Sn−1 1{max(cid:104)Xi,x(cid:105)≤r−1tn,N}dr dσ(x) n = Y + Z + W, (cid:90) (cid:90) where −2 − −2 | | (cid:90) 1+ln(N)n 1 N n 1 − (cid:90) Y := ∂Dn rn 1 S − 1{max(cid:104)Xi,x(cid:105)≤r−1tn,N}dσ(x) dr 1 n 1 Z := |∂Dn| (cid:90) 1+n2 −2(cid:90) − −2 rn−1 S − 1{max(cid:104)Xi,x(cid:105)≤r−1tn,N}dσ(x) dr 1+ln(N)n 1 N n 1 n 1 ∞ | | − W := ∂Dn 1+ 2 rn 1 Sn−1 1{max(cid:104)Xi,x(cid:105)≤r−1tn,N}dσ(x) dr. n − First we show that the polytope P lies inside the ball (1 + 2 )D with probability of 1 − n,N n n ⊂ e c(n)N, which implies that W = 0. Observe from the deﬁnition of W that if P (1+ 2 )D , ∈ S − (cid:104) (cid:105) ≥ − n,N n n then for every y n 1 there exists an X such that X , x r 1t . Thus it’s enough to show that with probability of 1 − e−c(n)N, ithe points X aire a √C -netn,oNf the unit sphere. This i n claim is proven in Lemma 12 at the end of this section. In order to prove Theorem 1, it’s ⊂ enough to restrict our attention to the event in which P (1 + 2 )D . n,N n n Now we show that with probability 1 − e−c(n)N1−n−2 1 ln(N)−2(1−n−1 1), the random variable − Z is of the order c(n)N 1. Observe that t < 1, which implies that for any r > 1 we have n,N 8 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Proof

text: of

text: Theorem

text: We

text: now

text: split

text: the

text: proof

text: of

text: Theorem

text: into

text: two

text: main

text: lemmas.

text: The

text: ﬁrst

text: of

text: these

text: two

text: lemmas

text: follows.

text: Lemma

text: Let

text: c(n)N

text: Then

text: the

text: following

text: concentration

text: inequality

text: holds:

text: Pr(||P

text: D

text: E[|P

text: D

text: n,N

text: n

text: n,N

text: n

text: Proof

text: of

text: Lemma

text: Using

text: Fubini’s

text: we

text: express

text: the

text: volume

text: of

text: the

text: set

text: diﬀerence

text: P

text: D

text: as

text: n,N

text: n

text: Pn,N

text: Dn

text: R

text: n

text: Dn

text: ∂Dn

text: S

text: rn

text: dσ(x)

text: n∞

text: ∂Dn

text: rn

text: S

text: dr

text: n

text: −2

text: −2

text: N

text: S

text: dr

text: n

text: −2

text: S

text: dσ(x)

text: N

text: n

text: n

text: rn

text: dσ(x)

text: n

text: Y

text: Z

text: W,

text: where

text: −2

text: −2

text: N

text: n

text: Y

text: ∂Dn

text: rn

text: S

text: dr

text: n

text: Z

text: |∂Dn|

text: −2

text: S

text: dr

text: N

text: n

text: n

text: W

text: ∂Dn

text: rn

text: dr.

text: n

text: First

text: we

text: show

text: that

text: the

text: polytope

text: P

text: lies

text: inside

text: the

text: ball

text: )D

text: with

text: probability

text: of

text: n,N

text: n

text: n

text: e

text: c(n)N,

text: which

text: implies

text: that

text: W

text: Observe

text: from

text: the

text: deﬁnition

text: of

text: W

text: that

text: if

text: P

text: )D

text: S

text: n,N

text: n

text: n

text: then

text: for

text: every

text: y

text: n

text: there

text: exists

text: an

text: X

text: such

text: that

text: X

text: x

text: r

text: Thus

text: it’s

text: enough

text: to

text: show

text: that

text: with

text: probability

text: of

text: e−c(n)N,

text: ithe

text: points

text: X

text: aire

text: a

text: √C

text: -netn,oNf

text: the

text: unit

text: sphere.

text: This

text: i

text: n

text: claim

text: is

text: proven

text: in

text: Lemma

text: at

text: the

text: end

text: of

text: this

text: section.

text: In

text: order

text: to

text: prove

text: Theorem

text: it’s

text: enough

text: to

text: restrict

text: our

text: attention

text: to

text: the

text: event

text: in

text: which

text: P

text: )D

text: n,N

text: n

text: n

text: Now

text: we

text: show

text: that

text: with

text: probability

text: the

text: random

text: variable

text: Z

text: is

text: of

text: the

text: order

text: c(n)N

text: Observe

text: that

text: t

text: which

text: implies

text: that

text: for

text: any

text: r

text: we

text: have

text: n,N

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: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: where c = 1/(2M). Similarly, the probability of observing the transition v from r to s is j i ← | p (s r ) = p (s r )p (r ) v i j v i j v j = c a (µ ω )φ. v ij i j Therefore the stationary probability of observing a transition between s and i r in any direction (from s to r or from r to s ) is j i j j i → ← p(s , r ) = p (s r ) + p (s r ) i j v i j v i j = 2c a (µ ω )φ. v ij i j = ca (µ ω )φ. ij i j with c = 1/M, as we wanted to show. Finally, we will link p(s ), the probability of a word that is used in the main i text to derive the meaning-frequency law, with p (s ). Notice that p(s ) = | v i i p (s S), the latter being the probability of visiting vertex s knowing that it v i i belongs to the partition S, the partition of words. Since the graph is bipartite, p (S), probability that the random walk is visiting a vertex of partition S, is v 1/2. The joint probability of visiting vertex s and that the vertex belongs to i S is | p (s , S) = p (S s )p (s ) v i v i v i = p (s ). v i Therefore, | p(s ) = p (s S) i v i p (s , S) = v i p (S) v = 2p (s ). v i Then p(s ) is the stationary probability of visiting s in a biased random i i walk knowing that the vertex is in S. B Mutual information maximization Suppose that I(S, R) is the mutual information between words (S) and meanings (R), that can be deﬁned as 16 ##LTLine##

text: where

text: c

text: Similarly,

text: the

text: probability

text: of

text: observing

text: the

text: transition

text: v

text: from

text: r

text: to

text: s

text: is

text: j

text: i

text: p

text: (s

text: r

text: p

text: (s

text: r

text: )p

text: (r

text: v

text: i

text: j

text: v

text: i

text: j

text: v

text: j

text: c

text: a

text: (µ

text: ω

text: )φ.

text: v

text: ij

text: i

text: j

text: Therefore

text: the

text: stationary

text: probability

text: of

text: observing

text: a

text: transition

text: between

text: s

text: and

text: i

text: r

text: in

text: any

text: direction

text: (from

text: s

text: to

text: r

text: or

text: from

text: r

text: to

text: s

text: is

text: j

text: i

text: j

text: j

text: i

text: p(s

text: r

text: p

text: (s

text: r

text: p

text: (s

text: r

text: i

text: j

text: v

text: i

text: j

text: v

text: i

text: j

text: a

text: (µ

text: ω

text: )φ.

text: v

text: ij

text: i

text: j

text: ca

text: (µ

text: ω

text: )φ.

text: ij

text: i

text: j

text: with

text: c

text: as

text: we

text: wanted

text: to

text: show.

text: Finally,

text: we

text: will

text: link

text: p(s

text: the

text: probability

text: of

text: a

text: word

text: that

text: is

text: used

text: in

text: the

text: main

text: i

text: text

text: to

text: derive

text: the

text: meaning-frequency

text: law,

text: with

text: p

text: (s

text: Notice

text: that

text: p(s

text: v

text: i

text: i

text: p

text: (s

text: S),

text: the

text: latter

text: being

text: the

text: probability

text: of

text: visiting

text: vertex

text: s

text: knowing

text: that

text: it

text: v

text: i

text: i

text: belongs

text: to

text: the

text: partition

text: S,

text: the

text: partition

text: of

text: words.

text: Since

text: the

text: graph

text: is

text: bipartite,

text: p

text: (S),

text: probability

text: that

text: the

text: random

text: walk

text: is

text: visiting

text: a

text: vertex

text: of

text: partition

text: S,

text: is

text: v

text: The

text: joint

text: probability

text: of

text: visiting

text: vertex

text: s

text: and

text: that

text: the

text: vertex

text: belongs

text: to

text: i

text: S

text: is

text: p

text: (s

text: S)

text: p

text: (S

text: s

text: )p

text: (s

text: v

text: i

text: v

text: i

text: v

text: i

text: p

text: (s

text: v

text: i

text: Therefore,

text: p(s

text: p

text: (s

text: S)

text: i

text: v

text: i

text: p

text: (s

text: S)

text: v

text: i

text: p

text: (S)

text: v

text: (s

text: v

text: i

text: Then

text: p(s

text: is

text: the

text: stationary

text: probability

text: of

text: visiting

text: s

text: in

text: a

text: biased

text: random

text: i

text: i

text: walk

text: knowing

text: that

text: the

text: vertex

text: is

text: in

text: S.

text: B

text: Mutual

text: information

text: maximization

text: Suppose

text: that

text: I(S,

text: R)

text: is

text: the

text: mutual

text: information

text: between

text: words

text: (S)

text: and

text: meanings

text: (R),

text: that

text: can

text: be

text: deﬁned

text: as

text: ##LTLine##