text: 4 ⊗ ψ ψ ψ ψ ψ ψ 1 2 3 4 5 6 ψ ψ ψ ψ ψ ψ ψ 1 1 2 3 4 5 6 ⊕ ⊕ ⊕ ⊕ ⊕ ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ 2 2 1 5 6 3 4 3 4 2 5 2 ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ 3 3 3 4 1 2 5 2 5 6 3 4 4 ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ 4 4 3 4 2 5 6 1 2 5 3 4 3 ⊕ ⊕ ⊕ ⊕ ⊕ ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ 5 5 2 5 3 4 3 4 1 2 6 5 ψ ψ ψ ψ ψ ψ ψ 6 6 2 4 3 5 1 TABLE I. The fusion rules for SO(5) anyons. 2 For more than three objects diﬀerent decompositions of the fusion can be related by distinct sequences of F -moves. Their consistency for arbitrary number of factors is guaranteed by the Pentagon equation satisﬁed by the F -moves. There also must be a mapping that braids two objects, ⊗ → ⊗ R : ψ ψ ψ ψ . a b b a Note that while fusion is commutative, states of the system may pick up a phase Rab under c braiding. Graphically this is represented by a b b a = Rab c c c (2.2) For a consistent anyon theory braids have to commute with fusion and satisfy the Yang-Baxter relation. Both properties follow from the Hexagon equation for F - and R-moves. In this paper we consider a system of particles which satisfy the fusion rules for SO(5) shown 2 in Table I. This fusion algebra is the truncation of the category of irreducible representations of 2iπ the quantum group Uq(so(5)) where q = e 5 [30]. For the SO(5)2 fusion rules there exist four known sets of inequivalent unitary F -moves, which can be found by solving the pentagon equations directly or by utilizing the representation theory of U (so(5)). Associated with each set of F -moves q ##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##

text: ψ

text: ψ

text: ψ

text: ψ

text: ψ

text: ψ

text: ψ

text: ψ

text: ψ

text: ψ

text: ψ

text: ψ

text: ψ

text: ψ

text: ψ

text: ψ

text: ψ

text: ψ

text: ψ

text: ψ

text: ψ

text: ψ

text: ψ

text: ψ

text: ψ

text: ψ

text: ψ

text: ψ

text: ψ

text: ψ

text: ψ

text: ψ

text: ψ

text: ψ

text: ψ

text: ψ

text: ψ

text: ψ

text: ψ

text: ψ

text: ψ

text: ψ

text: ψ

text: ψ

text: ψ

text: ψ

text: ψ

text: ψ

text: ψ

text: ψ

text: ψ

text: ψ

text: ψ

text: ψ

text: ψ

text: ψ

text: ψ

text: ψ

text: ψ

text: ψ

text: ψ

text: ψ

text: ψ

text: ψ

text: ψ

text: ψ

text: ψ

text: ψ

text: ψ

text: ψ

text: TABLE

text: I.

text: The

text: fusion

text: rules

text: for

text: anyons.

text: For

text: more

text: than

text: three

text: objects

text: diﬀerent

text: decompositions

text: of

text: the

text: fusion

text: can

text: be

text: related

text: by

text: distinct

text: sequences

text: of

text: F

text: -moves.

text: Their

text: consistency

text: for

text: arbitrary

text: number

text: of

text: factors

text: is

text: guaranteed

text: by

text: the

text: Pentagon

text: equation

text: satisﬁed

text: by

text: the

text: F

text: -moves.

text: There

text: also

text: must

text: be

text: a

text: mapping

text: that

text: braids

text: two

text: objects,

text: R

text: ψ

text: ψ

text: ψ

text: ψ

text: a

text: b

text: b

text: a

text: Note

text: that

text: while

text: fusion

text: is

text: commutative,

text: states

text: of

text: the

text: system

text: may

text: pick

text: up

text: a

text: phase

text: Rab

text: under

text: c

text: braiding.

text: Graphically

text: this

text: is

text: represented

text: by

text: a

text: b

text: b

text: a

text: Rab

text: c

text: c

text: c

text: For

text: a

text: consistent

text: anyon

text: theory

text: braids

text: have

text: to

text: commute

text: with

text: fusion

text: and

text: satisfy

text: the

text: Yang-Baxter

text: relation.

text: Both

text: properties

text: follow

text: from

text: the

text: Hexagon

text: equation

text: for

text: F

text: and

text: R-moves.

text: In

text: this

text: paper

text: we

text: consider

text: a

text: system

text: of

text: particles

text: which

text: satisfy

text: the

text: fusion

text: rules

text: for

text: shown

text: in

text: Table

text: I.

text: This

text: fusion

text: algebra

text: is

text: the

text: truncation

text: of

text: the

text: category

text: of

text: irreducible

text: representations

text: of

text: the

text: quantum

text: group

text: where

text: q

text: e

text: For

text: the

text: fusion

text: rules

text: there

text: exist

text: four

text: known

text: sets

text: of

text: inequivalent

text: unitary

text: F

text: -moves,

text: which

text: can

text: be

text: found

text: by

text: solving

text: the

text: pentagon

text: equations

text: directly

text: or

text: by

text: utilizing

text: the

text: representation

text: theory

text: of

text: U

text: Associated

text: with

text: each

text: set

text: of

text: F

text: -moves

text: q

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text: 33 Appendix A: Transfer matrices for the BMW integrable point To construct the family (3.13) of transfer matrices t((cid:96))(u) we deﬁne the functions appearing in the generalized Boltzmann weights (3.12) to be w1,3(u) = 1 , 3 w2,3(u) = i cosh(u) , w2,3(u) = sinh(u) , 3 4 iπ 3iπ w3,3(u) = sinh(u + ) sinh(u + ) , 1 10 10 − iπ − 3iπ w3,3(u) = sinh(u ) sinh(u ) , 2 10 10 9iπ 3iπ w3,3(u) = sinh(u + ) sinh(u + ) , 5 10 10 − 2iπ iπ w4,3(u) = sinh(u + ) sinh(u + ) , 2 5 5 − 2iπ − iπ w4,3(u) = sinh(u + ) sinh(u ) , 3 5 5 − 2iπ − iπ w4,3(u) = sinh(u ) sinh(u ) , 6 5 5 w35,3(u) = e81i0π , w45,3(u) = e31i0π , − w6,3(u) = i . 4 The additional weights are constructed from (3.10) as descendents [50]. Among the resulting transfer matrices t((cid:96))(u), the ones for (cid:96) = 1, 5, 6 are found to be independent ⊗ ⊗ of the spectral parameter u. For t(1)(u) and t(6)(u) this follows from ψ ψ and ψ ψ both 1 3 6 3 being isomorphic to simple objects. The absence of a parameter in t(5)(u) is connected to our choice of representation of the BMW algebra. The diﬀerent representations and their corresponding R- matrices lead to diﬀerent sets of transfer matrices and we ﬁnd that it is always the case that either t(2)(u) or t(5)(u) is parameter independent. L ≤ From the analysis of small systems, 10, we ﬁnd that the topological Y -operators (2.5) are obtained as limits of the transfer matrices, i.e. ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: paragraph

text: Appendix

label: section

text: A:

text: Transfer

text: matrices

text: for

text: the

text: BMW

text: integrable

text: point

text: To

text: construct

text: the

text: family

text: of

text: transfer

text: matrices

text: we

text: deﬁne

text: the

text: functions

text: appearing

text: in

text: the

text: generalized

text: Boltzmann

text: weights

text: to

text: be

text: i

text: cosh(u)

text: sinh(u)

text: iπ

text: sinh(u

text: sinh(u

text: iπ

text: sinh(u

text: sinh(u

text: sinh(u

text: sinh(u

text: iπ

text: sinh(u

text: sinh(u

text: iπ

text: sinh(u

text: sinh(u

text: iπ

text: sinh(u

text: sinh(u

text: i

text: The

text: additional

text: weights

text: are

text: constructed

text: from

text: as

text: descendents

text: Among

text: the

text: resulting

text: transfer

text: matrices

text: the

text: ones

text: for

text: are

text: found

text: to

text: be

text: independent

text: of

text: the

text: spectral

text: parameter

text: u.

text: For

text: and

text: this

text: follows

text: from

text: ψ

text: ψ

text: and

text: ψ

text: ψ

text: both

text: being

text: isomorphic

text: to

text: simple

text: objects.

text: The

text: absence

text: of

text: a

text: parameter

text: in

text: is

text: connected

text: to

text: our

text: choice

text: of

text: representation

text: of

text: the

text: BMW

text: algebra.

text: The

text: diﬀerent

text: representations

text: and

text: their

text: corresponding

text: R-

text: matrices

text: lead

text: to

text: diﬀerent

text: sets

text: of

text: transfer

text: matrices

text: and

text: we

text: ﬁnd

text: that

text: it

text: is

text: always

text: the

text: case

text: that

text: either

text: or

text: is

text: parameter

text: independent.

text: L

text: From

text: the

text: analysis

text: of

text: small

text: systems,

text: we

text: ﬁnd

text: that

text: the

text: topological

text: Y

text: -operators

text: are

text: obtained

text: as

text: limits

text: of

text: the

text: transfer

text: matrices,

text: i.e.

text: ##LTLine##

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text: 35 ≥ This is true in particular for rational conformal ﬁeld theories with c 1: these theories having a ﬁnite number of admissible highest weight representations with respect to its maximally extended W chiral symmetry algebra (d = 2, d , . . . , d ). Here d denote the conformal scaling dimensions 1 2 (cid:96) k (cid:12) (cid:11) (cid:12) ≡ of chiral primary ﬁelds W (dk) which, together with the energy-momentum tensor T W (d1), (cid:12) (cid:11) ≡ (cid:12) generate the chiral symmetry algebra. A highest weight state h w(1), w(2), w(3), . . . w((cid:96)) has then (cid:12) (cid:11) (cid:12) (cid:96) quantum numbers w(k) with respect to the zero modes W (dk), i.e. W (dk) h, w(2), w(3), . . . w((cid:96)) = 0 0 w(k) h, w(2), w(3), . . . w((cid:96)) . The highest weight representations are closed under fusion and hence Z give rise to a ﬁnite-dimensional representation of the modular group PSL(2, ). As all ﬁelds in the symmetry algebra must be mutually local, the scaling dimensions are restricted to be integers ∈ N or half-integers, 2 d . Therefore, chiral symmetry algebras constitute meromorphic conformal k ﬁeld theories. A class of chiral symmetry algebras whose representation theory is particularly well understood W are -algebras associated with Lie-algebras g, where the scaling dimensions d are related to the k exponents of the Lie-algebra g, i.e. where one has one W (dk) ﬁeld associated to each independent W Casimir operator. For a Lie-algebra of rank (cid:96), the -algebra is generated by (cid:96) such ﬁelds, one of those always being the Virasoro ﬁeld associated to the quadratic Casimir operator. For this W class of -algebras, free ﬁeld realizations are known and hence the algebras can, in principle, W be explicitly constructed [63]. For all semi-simple Lie-algebras the corresponding -algebras are known together with their so-called minimal series of rational theories they admit. In the following let g denote a simple Lie-algebra of rank (cid:96), and h a Cartan subalgebra of it. ∗ We denote by d the dimension of the Lie-algebra and by g its dual Coxeter number. Let Φ g + (cid:80) denote the set of positive roots α, and ht α the height of the root α. The height of a positive root is (cid:80) determined from its unique decomposition α = r n α into a linear combination of the simple i=1 i i roots α with non-negative integers n : ht α = r n . Finally, the maximal root may be denoted i i i=1 i | | by ρ. Under aﬃnization of the Lie-algebra to gˆ, we introduce the level x = 2k/ ρ 2, where k is the central extension of gˆ. Choosing the Killing form as our invariant symmetric bilinear form on g, | | ∗ W one ﬁnds that ρ 2 = 1/g and the central charge for at level x is given by [60] ˆg (cid:88) (cid:88) ∗ − d g − 12 ∗ c (x) = (cid:96) g ∗ ∗ (x + g ) (ht α)2 + 12 ht α . (B1) g x + g g ∈ ∈ α Φ+ α Φ+ We see that the central charge is entirely expressed in terms of basic properties of the Lie-algebra and the sums of the (squares of the) heights of the positive roots. The former are standard and can ##LTLine## ##LTLine##

text: This

text: is

text: true

text: in

text: particular

text: for

text: rational

text: conformal

text: ﬁeld

text: theories

text: with

text: c

text: these

text: theories

text: having

text: a

text: ﬁnite

text: number

text: of

text: admissible

text: highest

text: weight

text: representations

text: with

text: respect

text: to

text: its

text: maximally

text: extended

text: W

text: chiral

text: symmetry

text: algebra

text: (d

text: d

text: d

text: Here

text: d

text: denote

text: the

text: conformal

text: scaling

text: dimensions

text: k

text: of

text: chiral

text: primary

text: ﬁelds

text: W

text: (dk)

text: which,

text: together

text: with

text: the

text: energy-momentum

text: tensor

text: T

text: W

text: generate

text: the

text: chiral

text: symmetry

text: algebra.

text: A

text: highest

text: weight

text: state

text: h

text: has

text: then

text: quantum

text: numbers

text: w(k)

text: with

text: respect

text: to

text: the

text: zero

text: modes

text: W

text: (dk),

text: i.e.

text: W

text: (dk)

text: h,

text: w(k)

text: h,

text: The

text: highest

text: weight

text: representations

text: are

text: closed

text: under

text: fusion

text: and

text: hence

text: Z

text: give

text: rise

text: to

text: a

text: ﬁnite-dimensional

text: representation

text: of

text: the

text: modular

text: group

text: As

text: all

text: ﬁelds

text: in

text: the

text: symmetry

text: algebra

text: must

text: be

text: mutually

text: local,

text: the

text: scaling

text: dimensions

text: are

text: restricted

text: to

text: be

text: integers

text: N

text: or

text: half-integers,

text: d

text: Therefore,

text: chiral

text: symmetry

text: algebras

text: constitute

text: meromorphic

text: conformal

text: k

text: ﬁeld

text: theories.

text: A

text: class

text: of

text: chiral

text: symmetry

text: algebras

text: whose

text: representation

text: theory

text: is

text: particularly

text: well

text: understood

text: W

text: are

text: -algebras

text: associated

text: with

text: Lie-algebras

text: g,

text: where

text: the

text: scaling

text: dimensions

text: d

text: are

text: related

text: to

text: the

text: k

text: exponents

text: of

text: the

text: Lie-algebra

text: g,

text: i.e.

text: where

text: one

text: has

text: one

text: W

text: (dk)

text: ﬁeld

text: associated

text: to

text: each

text: independent

text: W

text: Casimir

text: operator.

text: For

text: a

text: Lie-algebra

text: of

text: rank

text: the

text: -algebra

text: is

text: generated

text: by

text: such

text: ﬁelds,

text: one

text: of

text: those

text: always

text: being

text: the

text: Virasoro

text: ﬁeld

text: associated

text: to

text: the

text: quadratic

text: Casimir

text: operator.

text: For

text: this

text: W

text: class

text: of

text: -algebras,

text: free

text: ﬁeld

text: realizations

text: are

text: known

text: and

text: hence

text: the

text: algebras

text: can,

text: in

text: principle,

text: W

text: be

text: explicitly

text: constructed

text: For

text: all

text: semi-simple

text: Lie-algebras

text: the

text: corresponding

text: -algebras

text: are

text: known

text: together

text: with

text: their

text: so-called

text: minimal

text: series

text: of

text: rational

text: theories

text: they

text: admit.

text: In

text: the

text: following

text: let

text: g

text: denote

text: a

text: simple

text: Lie-algebra

text: of

text: rank

text: and

text: h

text: a

text: Cartan

text: subalgebra

text: of

text: it.

text: We

text: denote

text: by

text: d

text: the

text: dimension

text: of

text: the

text: Lie-algebra

text: and

text: by

text: g

text: its

text: dual

text: Coxeter

text: number.

text: Let

text: Φ

text: g

text: denote

text: the

text: set

text: of

text: positive

text: roots

text: α,

text: and

text: ht

text: α

text: the

text: height

text: of

text: the

text: root

text: α.

text: The

text: height

text: of

text: a

text: positive

text: root

text: is

text: determined

text: from

text: its

text: unique

text: decomposition

text: α

text: r

text: n

text: α

text: into

text: a

text: linear

text: combination

text: of

text: the

text: simple

text: i

text: i

text: roots

text: α

text: with

text: non-negative

text: integers

text: n

text: ht

text: α

text: r

text: n

text: Finally,

text: the

text: maximal

text: root

text: may

text: be

text: denoted

text: i

text: i

text: i

text: by

text: ρ.

text: Under

text: aﬃnization

text: of

text: the

text: Lie-algebra

text: to

text: gˆ,

text: we

text: introduce

text: the

text: level

text: x

text: ρ

text: where

text: k

text: is

text: the

text: central

text: extension

text: of

text: gˆ.

text: Choosing

text: the

text: Killing

text: form

text: as

text: our

text: invariant

text: symmetric

text: bilinear

text: form

text: on

text: g,

text: W

text: one

text: ﬁnds

text: that

text: ρ

text: and

text: the

text: central

text: charge

text: for

text: at

text: level

text: x

text: is

text: given

text: by

text: ˆg

text: d

text: g

text: c

text: (x)

text: g

text: (x

text: g

text: (ht

text: ht

text: α

text: g

text: x

text: g

text: g

text: α

text: Φ+

text: α

text: Φ+

text: We

text: see

text: that

text: the

text: central

text: charge

text: is

text: entirely

text: expressed

text: in

text: terms

text: of

text: basic

text: properties

text: of

text: the

text: Lie-algebra

text: and

text: the

text: sums

text: of

text: the

text: (squares

text: of

text: the)

text: heights

text: of

text: the

text: positive

text: roots.

text: The

text: former

text: are

text: standard

text: and

text: can

text: ##LTLine##

text: ##LTLine##


text: arXiv:1408.1285 Renormalization-group inﬂationary scalar electrodynamics and SU(5) scenarios confronted with Planck2013 and BICEP2 results ∗ † ‡ § E. Elizalde1 , S.D. Odintsov1,2,3 , E.O. Pozdeeva4 , S.Yu. Vernov4 1Instituto de Ciencias del Espacio (ICE/CSIC) and Institut d’Estudis Espacials de Catalunya (IEEC) Campus UAB, Facultat de Ci`encies, Torre C5-Parell-2a planta, 08193 Bellaterra (Barcelona), Spain 2Instituci´o Catalana de Recerca i Estudis Avancats (ICREA), Barcelona, Spain 3King Abdulaziz University, Jeddah, 22254, Saudi Arabia 4Skobeltsyn Institute of Nuclear Physics, Lomonosov Moscow State University, Leninskie Gory, GSP-1, 119991, Moscow, Russia The possibility to construct inﬂationary models for the renormalization-group improved potentials corresponding to scalar electrodynamics and to SU(2) and SU(5) models is inves- tigated. In all cases, the tree-level potential, which corresponds to the cosmological constant in the Einstein frame, is seen to be non-suitable for inﬂation. Rather than adding the Hilbert–Einstein term to the action, quantum corrections to the potential, coming from to the RG-equation, are included. The inﬂationary scenario is analyzed with unstable de Sitter solutions which correspond to positive values of the coupling function, only. We show that, for the ﬁnite SU(2) model and SU(2) gauge model, there are no de Sitter solutions suitable for inﬂation, unless exit from it occurs according to some weird, non-standard scenarios. Inﬂation is realized both for scalar electrodynamics and for SU(5) RG-improved potentials, and the corresponding values of the coupling function are seen to be positive. It is shown that, for quite reasonable values of the parameters, the inﬂationary models obtained both from scalar electrodynamics and from the SU(5) RG-improved potentials, are in good agree- ment with the most recent observational data coming from the Planck2013 and BICEP2 collaborations. PACS numbers: 04.50.Kd, 11.10.Hi, 98.80.-k, 98.80.Cq I. INTRODUCTION Precise astronomical data coming from recent observational missions [1–3] (see also [4]) support ∗ E-mail address: elizalde@ieec.uab.es † E-mail address: odintsov@ieec.uab.es ‡ E-mail address: pozdeeva@www-hep.sinp.msu.ru § E-mail address: svernov@theory.sinp.msu.ru ##LTLine##

text: Renormalization-group

label: title

text: inﬂationary

text: scalar

text: electrodynamics

text: and

text: scenarios

text: confronted

text: with

text: and

text: results

label: author

text: E.

text: S.D.

text: E.O.

text: S.Yu.

text: de

text: Ciencias

text: del

text: Espacio

text: (ICE/CSIC)

text: and

text: Institut

text: d’Estudis

text: Espacials

text: de

text: Catalunya

text: (IEEC)

text: Campus

text: UAB,

text: Facultat

text: de

text: Ci`encies,

text: Torre

text: planta,

text: Bellaterra

text: (Barcelona),

text: Spain

text: Catalana

text: de

text: Recerca

text: i

text: Estudis

text: Avancats

text: (ICREA),

text: Barcelona,

text: Spain

text: Abdulaziz

text: University,

text: Jeddah,

text: Saudi

text: Arabia

text: Institute

text: of

text: Nuclear

text: Physics,

text: Lomonosov

text: Moscow

text: State

text: University,

text: Leninskie

text: Gory,

text: Moscow,

text: Russia

text: The

label: abstract

text: possibility

text: to

text: construct

text: inﬂationary

text: models

text: for

text: the

text: renormalization-group

text: improved

text: potentials

text: corresponding

text: to

text: scalar

text: electrodynamics

text: and

text: to

text: and

text: models

text: is

text: inves-

text: tigated.

text: In

text: all

text: cases,

text: the

text: tree-level

text: potential,

text: which

text: corresponds

text: to

text: the

text: cosmological

text: constant

text: in

text: the

text: Einstein

text: frame,

text: is

text: seen

text: to

text: be

text: non-suitable

text: for

text: inﬂation.

text: Rather

text: than

text: adding

text: the

text: Hilbert–Einstein

text: term

text: to

text: the

text: action,

text: quantum

text: corrections

text: to

text: the

text: potential,

text: coming

text: from

text: to

text: the

text: RG-equation,

text: are

text: included.

text: The

text: inﬂationary

text: scenario

text: is

text: analyzed

text: with

text: unstable

text: de

text: Sitter

text: solutions

text: which

text: correspond

text: to

text: positive

text: values

text: of

text: the

text: coupling

text: function,

text: only.

text: We

text: show

text: that,

text: for

text: the

text: ﬁnite

text: model

text: and

text: gauge

text: model,

text: there

text: are

text: no

text: de

text: Sitter

text: solutions

text: suitable

text: for

text: inﬂation,

text: unless

text: exit

text: from

text: it

text: occurs

text: according

text: to

text: some

text: weird,

text: non-standard

text: scenarios.

text: Inﬂation

text: is

text: realized

text: both

text: for

text: scalar

text: electrodynamics

text: and

text: for

text: RG-improved

text: potentials,

text: and

text: the

text: corresponding

text: values

text: of

text: the

text: coupling

text: function

text: are

text: seen

text: to

text: be

text: positive.

text: It

text: is

text: shown

text: that,

text: for

text: quite

text: reasonable

text: values

text: of

text: the

text: parameters,

text: the

text: inﬂationary

text: models

text: obtained

text: both

text: from

text: scalar

text: electrodynamics

text: and

text: from

text: the

text: RG-improved

text: potentials,

text: are

text: in

text: good

text: agree-

text: ment

text: with

text: the

text: most

text: recent

text: observational

text: data

text: coming

text: from

text: the

text: and

text: collaborations.

text: PACS

text: numbers:

text: I.

text: INTRODUCTION

text: Precise

text: astronomical

text: data

text: coming

text: from

text: recent

text: observational

text: missions

text: (see

text: also

text: support

text: E-mail

text: address:

text: elizalde@ieec.uab.es

text: E-mail

text: address:

text: odintsov@ieec.uab.es

text: E-mail

text: address:

text: pozdeeva@www-hep.sinp.msu.ru

text: E-mail

text: address:

text: svernov@theory.sinp.msu.ru

text: ##LTLine##


text: 12 Another possibility is to keep the gauge arbitrary; then we cannot demand that γ¯ vanishes. We get in this case − W = aκ g2f4(ϑ)φ4 bξ(ϑ)f2(ϑ)φ2R (28) 1 and γ¯ = C g2, C being some constant which depends on the gauge parameter and on the 1 1 − features of the theory, f(ϑ) = exp( C g2ϑ), and ξ(ϑ) is given by (24). 1 From V = aκ g2f4(ϑ)φ4, U = bξ(ϑ)f2(ϑ)φ2, (29) 1 (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) where − φ C1g2 1 − 1 φ Cg2 f(ϑ) = , ξ(ϑ) = + ξ , 0 µ 6 6 µ we get (cid:32) (cid:33) (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) − − φ 4C1g2 1 − 1 φ Cg2 φ 2C1g2 V = ak g2 φ4, U = b + ξ φ2, (30) 1 0 µ 6 6 µ µ and (cid:16) (cid:17) (cid:20) (cid:21) (cid:16) (cid:17) U(cid:48) 2(1 − C1g2) (6ξ0 − 1)Cg2 µφ22 Cg2/2 V (cid:48) 4(1 − C1g2) = + , = . (31) U φ − Cg2/2 V φ φ2 1 + (6ξ 1) φ 0 µ2 Now, we use the de Sitter condition (9) and conclude that this equation has as only solution C = 0. Summing up, in both cases there is no de Sitter solution for a nonconstant ξ. In the case of a constant ξ, we get a model with a power-law potential V and power-law coupling function U which satisﬁes the condition (19). Thus, if the signs of the constants are such that ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Another

text: possibility

text: is

text: to

text: keep

text: the

text: gauge

text: arbitrary;

text: then

text: we

text: cannot

text: demand

text: that

text: γ¯

text: vanishes.

text: We

text: get

text: in

text: this

text: case

label: equation

text: W

text: aκ

text: and

text: γ¯

text: C

text: C

text: being

text: some

text: constant

text: which

text: depends

text: on

text: the

text: gauge

text: parameter

text: and

text: on

text: the

text: features

text: of

text: the

text: theory,

text: f(ϑ)

text: exp(

text: C

text: and

text: ξ(ϑ)

text: is

text: given

text: by

text: From

text: V

text: aκ

text: U

text: where

text: φ

text: φ

text: f(ϑ)

text: ξ(ϑ)

text: ξ

text: µ

text: µ

text: we

text: get

text: φ

text: φ

text: φ

text: V

text: ak

text: U

text: b

text: ξ

text: µ

text: µ

text: µ

text: and

text: µφ22

text: V

text: U

text: φ

text: V

text: φ

text: φ2

text: φ

text: µ2

text: Now,

text: we

text: use

text: the

text: de

text: Sitter

text: condition

text: and

text: conclude

text: that

text: this

text: equation

text: has

text: as

text: only

text: solution

text: C

text: Summing

text: up,

text: in

text: both

text: cases

text: there

text: is

text: no

text: de

text: Sitter

text: solution

text: for

text: a

text: nonconstant

text: ξ.

text: In

text: the

text: case

text: of

text: a

text: constant

text: ξ,

text: we

text: get

text: a

text: model

text: with

text: a

text: power-law

text: potential

text: V

text: and

text: power-law

text: coupling

text: function

text: U

text: which

text: satisﬁes

text: the

text: condition

text: Thus,

text: if

text: the

text: signs

text: of

text: the

text: constants

text: are

text: such

text: that

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text: 14 (cid:54) (cid:54) Note that Θ˜ = 0, hence k = 1/2. Using (7), we calculate the Hubble parameter H for the de f 3 Sitter solution H2 = g2k1(2k3 − 1)φ2Θ˜ 2(6−a˜24k2)−1 . (36) f 24k f f 3 ∼ In [63] the authors show that asymptotically free models exist only when f2(ϑ) g4(ϑ), what − (cid:104) (cid:105) corresponds to a˜2 = 4k 6. For this choice of the parameter a˜2, we obtain 2 V = g2k1φ4f Θ˜ −5, U = φ2f Θ˜ −2 1 + (6ξ − 1) Θ˜ −k3 = φ2f−k3 . f 24 f f 12 f f 6(2k 1)Θ˜ 2 3 f − − − − − Note that k = 2 5k /(3(4k 6)) = 2 5k /(3a˜2). Therefore, 3 1 2 1 − − g2k (1 2k ) − 5k (2k + 3a˜2) − − − H2 = 1 3 φ2Θ˜ 3 = 1 1 g2φ2 [(2k 1) (6ξ 1)] 3/k3 . (37) f 24k f f 24(5k + 6a˜2) f 3 3 1 − − From here, we get that H2 > 0 provided either 3a˜2/2 < k < 6a˜2/5 that is equivalent to f 1 ∞ −∞ 0 < k < 1/2, or 0 < k < that correspond to < k < 0. 3 1 3 Let us consider the stability of the de Sitter solutions in the case V > 0 and U > 0. In this f f (cid:18) (cid:19) (cid:18) (cid:19) case, the sign of K coincides with the sign of f (cid:48) (cid:48) (cid:48) (cid:48) − U | − V | a˜4g4(2k 1) 3 2 = . U φ=φf V φ=φf 512π4φ2Θ˜ 2 f f De Sitter solutions exist only for k < 1/2. The additional condition U > 0 gives k < 0. We 3 f 3 can see that K < 0 at any k < 0. Thus, for this model with U > 0 we have stable de Sitter f 3 f solutions only. VIII. THE SU(5) RG-IMPROVED POTENTIAL Now we study the RG-improved potential for the SU(5) GUT [68]. In ﬂat space this theory has been used for the discussion of inﬂationary cosmology [9, 21]. We assume that the breaking → × × SU(5) SU(3) SU(2) U(1) has taken place. The RG-improved eﬀective potential has the following form: (cid:32) (cid:33) (cid:20) (cid:18) (cid:19) (cid:21) 3375 − g2 15 1 − 1 − V = g2 φ4f4, U = + ξ Θ˘ 9/8 φ2f2, (38) 512 16/9 5 4 6 6 5 f 5 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

font: default

text: Note

text: that

text: Θ˜

text: hence

text: k

text: Using

text: we

text: calculate

text: the

text: Hubble

text: parameter

text: H

text: for

text: the

text: de

text: f

text: Sitter

text: solution

text: 1)φ2Θ˜

text: f

text: f

text: f

text: In

text: the

text: authors

text: show

text: that

text: asymptotically

text: free

text: models

text: exist

text: only

text: when

text: what

text: corresponds

text: to

text: For

text: this

text: choice

text: of

text: the

text: parameter

text: we

text: obtain

text: V

text: Θ˜

text: U

text: Θ˜

text: −2

text: Θ˜

text: f

text: f

text: f

text: f

text: f

text: 1)Θ˜

text: f

text: Note

text: that

text: k

text: Therefore,

text: φ2Θ˜

text: f

text: f

text: f

text: f

text: From

text: here,

text: we

text: get

text: that

text: provided

text: either

text: k

text: that

text: is

text: equivalent

text: to

text: f

text: k

text: or

text: k

text: that

text: correspond

text: to

text: k

text: Let

text: us

text: consider

text: the

text: stability

text: of

text: the

text: de

text: Sitter

text: solutions

text: in

text: the

text: case

text: V

text: and

text: U

text: In

text: this

text: f

text: f

text: case,

text: the

text: sign

text: of

text: K

text: coincides

text: with

text: the

text: sign

text: of

text: f

text: U

text: V

text: U

text: φ=φf

text: V

text: φ=φf

text: f

text: f

text: De

text: Sitter

text: solutions

text: exist

text: only

text: for

text: k

text: The

text: additional

text: condition

text: U

text: gives

text: k

text: We

text: f

text: can

text: see

text: that

text: K

text: at

text: any

text: k

text: Thus,

text: for

text: this

text: model

text: with

text: U

text: we

text: have

text: stable

text: de

text: Sitter

text: f

text: f

text: solutions

text: only.

text: VIII.

text: THE

text: RG-IMPROVED

text: POTENTIAL

text: Now

text: we

text: study

text: the

text: RG-improved

text: potential

text: for

text: the

text: GUT

text: In

text: ﬂat

text: space

text: this

text: theory

text: has

text: been

text: used

text: for

text: the

text: discussion

text: of

text: inﬂationary

text: cosmology

text: We

text: assume

text: that

text: the

text: breaking

text: has

text: taken

text: place.

text: The

text: RG-improved

text: eﬀective

text: potential

text: has

text: the

text: following

text: form:

text: V

text: U

text: ξ

text: Θ˘

text: f

text: ##LTLine##

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text: 20 ≡ where we use the dimensionless quantity σ φ/µ. Note that the slow-roll parameters (cid:15) and η do not depend on the dimensionless combination κµ. We choose the parameters so that U > 0, f (cid:48) what means λ < 36e2ξ. Note that the de Sitter solutions correspond to the condition V (φ ) = 0. E f Solutions are unstable; so, in this point the potential V has a maximum. In Fig. 2 we see that E the potential is very ﬂat near the maximum and decreases more rapidly than σ, when it tends to zero. σ FIG. 2: The potential V (σ) multiplied by κ4 in the scalar electrodynamics model. In both pictures E λ = 18e2ξ. This means that σ = 1. The parameter values are e = 10, ξ = 10 (red dashed line) and f e = 10, ξ = 5 (blue solid line). Note that the potential V with such choice of λ depends on the combination E ξ/e2 only. ± ± In our calculations, we put λ = 18e2ξ what gives φ = µ (σ = 1). At the de Sitter point f f φ the value of the RG-improved potential coincides with the value of the tree-level one. We f consider positive values for σ and for the parameters in the action. The corresponding inﬂationary parameters are listed in Table I. One can see that, for the values of the parameters e and ξ presented in Table I, the corresponding values of n and r are in good agreement with the observational data [2, 4]. The potential V (σ) s E with these values of parameters are presented in Fig. 2. C. The SU(5) model In the case of a SU(5) RG-improved potential, ##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##

text: where

text: we

text: use

text: the

text: dimensionless

text: quantity

text: σ

text: φ/µ.

text: Note

text: that

text: the

text: slow-roll

text: parameters

text: and

text: η

text: do

text: not

text: depend

text: on

text: the

text: dimensionless

text: combination

text: κµ.

text: We

text: choose

text: the

text: parameters

text: so

text: that

text: U

text: f

text: what

text: means

text: λ

text: Note

text: that

text: the

text: de

text: Sitter

text: solutions

text: correspond

text: to

text: the

text: condition

text: V

text: (φ

text: E

text: f

text: Solutions

text: are

text: unstable;

text: so,

text: in

text: this

text: point

text: the

text: potential

text: V

text: has

text: a

text: maximum.

text: In

text: Fig.

text: we

text: see

text: that

text: E

text: the

text: potential

text: is

text: very

text: ﬂat

text: near

text: the

text: maximum

text: and

text: decreases

text: more

text: rapidly

text: than

text: σ,

text: when

text: it

text: tends

text: to

text: zero.

text: σ

text: FIG.

text: The

text: potential

text: V

text: (σ)

text: multiplied

text: by

text: κ4

text: in

text: the

text: scalar

text: electrodynamics

text: model.

text: In

text: both

text: pictures

text: E

text: λ

text: This

text: means

text: that

text: σ

text: The

text: parameter

text: values

text: are

text: e

text: ξ

text: (red

text: dashed

text: line)

text: and

text: f

text: e

text: ξ

text: (blue

text: solid

text: line).

text: Note

text: that

text: the

text: potential

text: V

text: with

text: such

text: choice

text: of

text: λ

text: depends

text: on

text: the

text: combination

text: E

text: only.

text: In

text: our

text: calculations,

text: we

text: put

text: λ

text: what

text: gives

text: φ

text: µ

text: (σ

text: At

text: the

text: de

text: Sitter

text: point

text: f

text: f

text: φ

text: the

text: value

text: of

text: the

text: RG-improved

text: potential

text: coincides

text: with

text: the

text: value

text: of

text: the

text: tree-level

text: one.

text: We

text: f

text: consider

text: positive

text: values

text: for

text: σ

text: and

text: for

text: the

text: parameters

text: in

text: the

text: action.

text: The

text: corresponding

text: inﬂationary

text: parameters

text: are

text: listed

text: in

text: Table

text: I.

text: One

text: can

text: see

text: that,

text: for

text: the

text: values

text: of

text: the

text: parameters

text: e

text: and

text: ξ

text: presented

text: in

text: Table

text: I,

text: the

text: corresponding

text: values

text: of

text: n

text: and

text: r

text: are

text: in

text: good

text: agreement

text: with

text: the

text: observational

text: data

text: The

text: potential

text: V

text: (σ)

text: s

text: E

text: with

text: these

text: values

text: of

text: parameters

text: are

text: presented

text: in

text: Fig.

text: C.

text: The

text: model

text: In

text: the

text: case

text: of

text: a

text: RG-improved

text: potential,

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text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 23 In this paper we have considered the possibility to construct inﬂationary models for the renormalization-group improved potentials corresponding to scalar electrodynamics and to the − SU(2) and SU(5) models. In all cases, the tree-level potential is λφ4 ξφ2R, what corresponds to the cosmological constant in the Einstein frame, and is in no case suitable for inﬂation. The standard way to get an inﬂationary model is to add the Hilbert–Einstein term to the action [17, 18]. Actually we did not add this term here, but included instead the quantum corrections to the po- tential coming from to the RG-equation. We then analyzed the corresponding inﬂationary scenario with unstable de Sitter solutions, only. This means that the corresponding potential in the Einstein frame should have a maximum. We have found that, for some reasonable values of the parameters, this is indeed the case, both for scalar electrodynamics and for the SU(5) model, and that the corresponding values of the coupling function are indeed positive, U > 0. For the ﬁnite SU(2) f model de Sitter solutions exist, if it corresponds to the model with a minimally coupled scalar ﬁeld and a cosmological constant; thus, this case is not suitable for inﬂation. In the SU(2) gauge model there exist stable de Sitter solutions for U > 0, only. Note also that a stable de Sitter f solution may appear to be a true inﬂationary solution, but only under the condition that the exit from inﬂation occurs according to some other particular scenarios; but we did not consider such inﬂationary scenarios in this paper. In the inﬂationary models, both for scalar electrodynamics and the SU(5) RG-improved po- tentials, we have got that these models are in good agreement with the most recent observational data [2, 4] provided some reasonable values are taken for the parameters. Our study indicates that inﬂation could well be caused by quantum eﬀects of the scalar sector of some convenient GUT theory. We believe this is quite a remarkable result. In this respect, it would be of interest to investigate the possibility of inﬂation in GUTs with other gauge groups, as the exceptional E8 group, or GUTs which proceed from the string framework. From another side, adding a RG-improved eﬀective potential to the classical GR action may lead to a qualitative change of the inﬂationary dynamics which occur in such models. This issue will be discussed elsewhere. Acknowledgements. E.E. and S.D.O. are supported in part by MINECO (Spain), project FIS2010-15640, and by the CPAN Consolider Ingenio Project. E.O.P. and S.Yu.V. are partially supported by RFBR grant 14-01-00707 and by the Russian Ministry of Education and Science

text: In

text: this

text: paper

text: we

text: have

text: considered

text: the

text: possibility

text: to

text: construct

text: inﬂationary

text: models

text: for

text: the

text: renormalization-group

text: improved

text: potentials

text: corresponding

text: to

text: scalar

text: electrodynamics

text: and

text: to

text: the

text: and

text: models.

text: In

text: all

text: cases,

text: the

text: tree-level

text: potential

text: is

text: λφ4

text: what

text: corresponds

text: to

text: the

text: cosmological

text: constant

text: in

text: the

text: Einstein

text: frame,

text: and

text: is

text: in

text: no

text: case

text: suitable

text: for

text: inﬂation.

text: The

text: standard

text: way

text: to

text: get

text: an

text: inﬂationary

text: model

text: is

text: to

text: add

text: the

text: Hilbert–Einstein

text: term

text: to

text: the

text: action

text: Actually

text: we

text: did

text: not

text: add

text: this

text: term

text: here,

text: but

text: included

text: instead

text: the

text: quantum

text: corrections

text: to

text: the

text: po-

text: tential

text: coming

text: from

text: to

text: the

text: RG-equation.

text: We

text: then

text: analyzed

text: the

text: corresponding

text: inﬂationary

text: scenario

text: with

text: unstable

text: de

text: Sitter

text: solutions,

text: only.

text: This

text: means

text: that

text: the

text: corresponding

text: potential

text: in

text: the

text: Einstein

text: frame

text: should

text: have

text: a

text: maximum.

text: We

text: have

text: found

text: that,

text: for

text: some

text: reasonable

text: values

text: of

text: the

text: parameters,

text: this

text: is

text: indeed

text: the

text: case,

text: both

text: for

text: scalar

text: electrodynamics

text: and

text: for

text: the

text: model,

text: and

text: that

text: the

text: corresponding

text: values

text: of

text: the

text: coupling

text: function

text: are

text: indeed

text: positive,

text: U

text: For

text: the

text: ﬁnite

text: f

text: model

text: de

text: Sitter

text: solutions

text: exist,

text: if

text: it

text: corresponds

text: to

text: the

text: model

text: with

text: a

text: minimally

text: coupled

text: scalar

text: ﬁeld

text: and

text: a

text: cosmological

text: constant;

text: thus,

text: this

text: case

text: is

text: not

text: suitable

text: for

text: inﬂation.

text: In

text: the

text: gauge

text: model

text: there

text: exist

text: stable

text: de

text: Sitter

text: solutions

text: for

text: U

text: only.

text: Note

text: also

text: that

text: a

text: stable

text: de

text: Sitter

text: f

text: solution

text: may

text: appear

text: to

text: be

text: a

text: true

text: inﬂationary

text: solution,

text: but

text: only

text: under

text: the

text: condition

text: that

text: the

text: exit

text: from

text: inﬂation

text: occurs

text: according

text: to

text: some

text: other

text: particular

text: scenarios;

text: but

text: we

text: did

text: not

text: consider

text: such

text: inﬂationary

text: scenarios

text: in

text: this

text: paper.

text: In

text: the

text: inﬂationary

text: models,

text: both

text: for

text: scalar

text: electrodynamics

text: and

text: the

text: RG-improved

text: po-

text: tentials,

text: we

text: have

text: got

text: that

text: these

text: models

text: are

text: in

text: good

text: agreement

text: with

text: the

text: most

text: recent

text: observational

text: data

text: provided

text: some

text: reasonable

text: values

text: are

text: taken

text: for

text: the

text: parameters.

text: Our

text: study

text: indicates

text: that

text: inﬂation

text: could

text: well

text: be

text: caused

text: by

text: quantum

text: eﬀects

text: of

text: the

text: scalar

text: sector

text: of

text: some

text: convenient

text: GUT

text: theory.

text: We

text: believe

text: this

text: is

text: quite

text: a

text: remarkable

text: result.

text: In

text: this

text: respect,

text: it

text: would

text: be

text: of

text: interest

text: to

text: investigate

text: the

text: possibility

text: of

text: inﬂation

text: in

text: GUTs

text: with

text: other

text: gauge

text: groups,

text: as

text: the

text: exceptional

text: group,

text: or

text: GUTs

text: which

text: proceed

text: from

text: the

text: string

text: framework.

text: From

text: another

text: side,

text: adding

text: a

text: RG-improved

text: eﬀective

text: potential

text: to

text: the

text: classical

text: GR

text: action

text: may

text: lead

text: to

text: a

text: qualitative

text: change

text: of

text: the

text: inﬂationary

text: dynamics

text: which

text: occur

text: in

text: such

text: models.

text: This

text: issue

text: will

text: be

text: discussed

text: elsewhere.

text: Acknowledgements.

text: E.E.

text: and

text: S.D.O.

text: are

text: supported

text: in

text: part

text: by

text: MINECO

text: (Spain),

text: project

text: and

text: by

text: the

text: CPAN

text: Consolider

text: Ingenio

text: Project.

text: E.O.P.

text: and

text: S.Yu.V.

text: are

text: partially

text: supported

text: by

text: RFBR

text: grant

text: and

text: by

text: the

text: Russian

text: Ministry

text: of

text: Education

text: and

text: Science


text: 8 The renormalization-group improved eﬀective potential for an arbitrary renormalizable massless gauge theory in curved space-time was discussed in detail in [41]. In this section we will just remind the reader of the basic steps for the construction of the renormalization-group improved eﬀective potential. The tree-level potential reads as follows [41] − − W (0)(φ) = aλφ4 bξφ2R = V U R, (16) 0 0 where a and b are positive constants and ξ is the conformal coupling. The potential W (0) includes both the potential V and the function U multiplied by the scalar curvature. 0 0 As is known, see [39, 41], the renormalization-group equation for the eﬀective potential in curved space-time has the form (cid:18) (cid:19) (cid:101) (cid:101) ∂ ∂ ∂ ∂ − ∂ µ + β + δ + β γφ W = 0, (17) g ξ ∂µ ∂g ∂α ∂ξ ∂φ where α is the gauge parameter and g˜ is the set of all coupling constants of the theory (Higgs, gauge and Yukawa ones). The standard ﬂat-space renormalization-group equation [61, 62] is mod- iﬁed in curved space-time, for instance, it has an additional term related with the contribution from the nonminimal coupling constant ξ and the corresponding β function [? ]. ξ It is natural to split W into two parts, namely ≡ − ≡ − W V UR af (p, φ, µ)φ4 bf (p, φ, µ)φ2R, (18) 1 2 (cid:101) { } where f and f are some unknown functions, and p = g, α, ξ . Actually, in [41] the authors 1 2 imposed the additional restriction that, not only the function W satisﬁes (17), but also that the functions V and U satisfy it, separately. It is easy to see [52] that, for V (φ) = CU2(φ), (19) 0 0 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: The

text: renormalization-group

text: improved

text: eﬀective

text: potential

text: for

text: an

text: arbitrary

text: renormalizable

text: massless

text: gauge

text: theory

text: in

text: curved

text: space-time

text: was

text: discussed

text: in

text: detail

text: in

text: In

text: this

text: section

text: we

text: will

text: just

text: remind

text: the

text: reader

text: of

text: the

text: basic

text: steps

text: for

text: the

text: construction

text: of

text: the

text: renormalization-group

text: improved

text: eﬀective

text: potential.

text: The

text: tree-level

text: potential

text: reads

text: as

text: follows

text: W

text: aλφ4

text: V

text: U

text: R,

text: where

text: a

text: and

text: b

text: are

text: positive

text: constants

text: and

text: ξ

text: is

text: the

text: conformal

text: coupling.

text: The

text: potential

text: W

text: includes

text: both

text: the

text: potential

text: V

text: and

text: the

text: function

text: U

text: multiplied

text: by

text: the

text: scalar

text: curvature.

text: As

text: is

text: known,

text: see

text: the

text: renormalization-group

text: equation

text: for

text: the

text: eﬀective

text: potential

text: in

text: curved

text: space-time

text: has

text: the

text: form

text: µ

text: β

text: δ

text: β

text: γφ

text: W

text: g

text: ξ

text: ∂µ

text: ∂g

text: ∂α

text: ∂ξ

text: ∂φ

text: where

text: α

text: is

text: the

text: gauge

text: parameter

text: and

text: g˜

text: is

text: the

text: set

text: of

text: all

text: coupling

text: constants

text: of

text: the

text: theory

text: (Higgs,

text: gauge

text: and

text: Yukawa

text: ones).

text: The

text: standard

text: ﬂat-space

text: renormalization-group

text: equation

text: is

text: mod-

text: iﬁed

text: in

text: curved

text: space-time,

text: for

text: instance,

text: it

text: has

text: an

text: additional

text: term

text: related

text: with

text: the

text: contribution

text: from

text: the

text: nonminimal

text: coupling

text: constant

text: ξ

text: and

text: the

text: corresponding

text: β

text: function

text: ξ

text: It

text: is

text: natural

text: to

text: split

text: W

text: into

text: two

text: parts,

text: namely

text: W

text: V

text: UR

text: af

text: (p,

text: φ,

text: µ)φ4

text: bf

text: (p,

text: φ,

text: where

text: f

text: and

text: f

text: are

text: some

text: unknown

text: functions,

text: and

text: p

text: g,

text: α,

text: ξ

text: Actually,

text: in

text: the

text: authors

text: imposed

text: the

text: additional

text: restriction

text: that,

text: not

text: only

text: the

text: function

text: W

text: satisﬁes

text: but

text: also

text: that

text: the

text: functions

text: V

text: and

text: U

text: satisfy

text: it,

text: separately.

text: It

text: is

text: easy

text: to

text: see

text: that,

text: for

text: V

text: (φ)

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 4 J. HU AND R. MA The Raviart–Thomas element method of Problem (2.2) seeks (σ , u ) ∈ RT(T ) × RT RT U (T ) such that RT (σ , τ) + (u , div τ) = 0 for any τ ∈ RT(T ), RT RT (2.6) (cid:82) (div σ , v) = (−f, v) for any v ∈ U (T ). RT RT Given K ∈ T and f ∈ L2(K), deﬁne f = | 1 | fdx. Given f ∈ L2(Ω), deﬁne the K K K piecewise constant projection Π f by 0 (Π f)| = f . 0 K K Because of the deﬁnition of U (T ), f in the second equation of (2.6) can be replaced RT by Π f. We deﬁne the auxiliary method: Find u¯ ∈ V such that 0 CR CR (2.7) (∇ u¯ , ∇ v) = (Π f, v) for all v ∈ V . NC CR NC 0 CR Note that this method diﬀers from (2.3) only by the presence of the projection in the right hand side. Marini [22] proved its equivalence to the Raviart–Thomas element method (2.6): f K (2.8) σ | = ∇u¯ | − (x − Mid(K)) x ∈ K for any K ∈ T , RT K CR K 2 where Mid(K) denotes the center of K. Subtracting (2.7) from (2.3) with v = u − u¯ yields that CR CR (∇ (u − u¯ ), ∇ (u − u¯ )) = (f − Π f, u − u¯ ) NC CR CR NC CR CR 0 CR CR = (f − Π f, u − u¯ − Π (u − u¯ )). 0 CR CR 0 CR CR Hence, the Poincar´e inequality from [17] yields h2 (2.9) (cid:107)∇ (u − u¯ )(cid:107) ≤ |f| , NC CR CR 0,Ω j2 1,Ω 1,1 where j = 3.8317 denotes the ﬁrst positive root of the Bessel function of the ﬁrst 1,1 kind. 3. Superconvergence analysis of the Crouzeix–Raviart element In this section, we ﬁrst present the superconvergence result of the Raviart–Thomas element by Brandts [4]. Then, based on this result and the equivalence (2.8), we derive the superconvergence result of the Crouzeix–Raviart element. We introduce a result on Sobolev spaces in the following lemma, which describes the behavior of functions near the boundary. Deﬁne Ω as the subset of points in Ω having h distance less that h from the boundary: Ω = {x ∈ Ω : ∃y ∈ ∂Ω, dist(x, y) ≤ h}. h ##LTLine## ##LTLine## ##LTLine##

text: J.

text: HU

text: AND

text: R.

text: MA

text: The

text: Raviart–Thomas

text: element

text: method

text: of

text: Problem

text: seeks

text: (σ

text: u

text: RT(T

text: RT

text: RT

text: U

text: (T

text: such

text: that

text: RT

text: (σ

text: τ)

text: (u

text: div

text: τ)

text: for

text: any

text: τ

text: RT(T

text: RT

text: RT

text: (div

text: σ

text: v)

text: (−f,

text: v)

text: for

text: any

text: v

text: U

text: (T

text: RT

text: RT

text: Given

text: K

text: T

text: and

text: f

text: deﬁne

text: f

text: fdx.

text: Given

text: f

text: deﬁne

text: the

text: K

text: K

text: K

text: piecewise

text: constant

text: projection

text: Π

text: f

text: by

text: (Π

text: f)|

text: f

text: K

text: K

text: Because

text: of

text: the

text: deﬁnition

text: of

text: U

text: (T

text: f

text: in

text: the

text: second

text: equation

text: of

text: can

text: be

text: replaced

text: RT

text: by

text: Π

text: f.

text: We

text: deﬁne

text: the

text: auxiliary

text: method:

text: Find

text: u¯

text: V

text: such

text: that

text: CR

text: CR

text: u¯

text: v)

text: (Π

text: f,

text: v)

text: for

text: all

text: v

text: V

text: NC

text: CR

text: NC

text: CR

text: Note

text: that

text: this

text: method

text: diﬀers

text: from

text: only

text: by

text: the

text: presence

text: of

text: the

text: projection

text: in

text: the

text: right

text: hand

text: side.

text: Marini

text: proved

text: its

text: equivalence

text: to

text: the

text: Raviart–Thomas

text: element

text: method

text: f

text: K

text: σ

text: ∇u¯

text: (x

text: Mid(K))

text: x

text: K

text: for

text: any

text: K

text: T

text: RT

text: K

text: CR

text: K

text: where

text: Mid(K)

text: denotes

text: the

text: center

text: of

text: K.

text: Subtracting

text: from

text: with

text: v

text: u

text: u¯

text: yields

text: that

text: CR

text: CR

text: (u

text: u¯

text: (u

text: u¯

text: (f

text: Π

text: f,

text: u

text: u¯

text: NC

text: CR

text: CR

text: NC

text: CR

text: CR

text: CR

text: CR

text: (f

text: Π

text: f,

text: u

text: u¯

text: Π

text: (u

text: u¯

text: CR

text: CR

text: CR

text: CR

text: Hence,

text: the

text: Poincar´e

text: inequality

text: from

text: yields

text: (u

text: u¯

text: |f|

text: NC

text: CR

text: CR

text: where

text: j

text: denotes

text: the

text: ﬁrst

text: positive

text: root

text: of

text: the

text: Bessel

text: function

text: of

text: the

text: ﬁrst

text: kind.

text: Superconvergence

text: analysis

text: of

text: the

text: Crouzeix–Raviart

text: element

text: In

text: this

text: section,

text: we

text: ﬁrst

text: present

text: the

text: superconvergence

text: result

text: of

text: the

text: Raviart–Thomas

text: element

text: by

text: Brandts

text: Then,

text: based

text: on

text: this

text: result

text: and

text: the

text: equivalence

text: we

text: derive

text: the

text: superconvergence

text: result

text: of

text: the

text: Crouzeix–Raviart

text: element.

text: We

text: introduce

text: a

text: result

text: on

text: Sobolev

text: spaces

text: in

text: the

text: following

text: lemma,

text: which

text: describes

text: the

text: behavior

text: of

text: functions

text: near

text: the

text: boundary.

text: Deﬁne

text: Ω

text: as

text: the

text: subset

text: of

text: points

text: in

text: Ω

text: having

text: h

text: distance

text: less

text: that

text: h

text: from

text: the

text: boundary:

text: Ω

text: {x

text: Ω

text: ∃y

text: ∂Ω,

text: dist(x,

text: y)

text: h}.

text: h

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 6 J. HU AND R. MA and the center N of the parallelogram intersects the boundary of N in another c point P˜. Deﬁne K q(P ) = 2K q(N ) − K q(P˜). h h c h Brandts [4] proved that the vector K Π q is a higher order approximation of q than h RT Π q itself. RT Theorem 3.3. Suppose q ∈ (H2(Ω))2, then there holds that (cid:46) (cid:107)q − K Π q(cid:107) h2|q| . h RT 0,Ω 2,Ω Combining Theorem 3.2 and Theorem 3.3 concludes that the post-processing oper- ator K also improves the order of approximation of σ . h RT Corollary 3.4. Let σ ∈ (H2(Ω))2 and σ be the solutions of (2.2) and (2.6), respec- RT tively. There holds that (cid:46) 3 1 (cid:107)σ − KhσRT(cid:107)0,Ω h2 ((cid:107)σ(cid:107)3,Ω + h2 |σ|2,Ω). 2 Theorem 3.5. Let u ∈ H3(Ω) and u be the solutions of (2.1) and (2.3), respectively. ∞ CR Further, suppose that f ∈ W 1, (Ω), then we have (cid:46) (3.1) (cid:107)∇u − Kh∇NCuCR(cid:107)0,Ω h32 ((cid:107)u(cid:107)5,Ω + h12 |u|3,Ω + h12 |f|1,∞,Ω). 2 Proof. Using the equivalence equality (2.8), for e = K ∩ K and the center P of e, 1 2 there holds that f f |K (∇ u¯ − σ )(P )| = | K1 (P − Mid(K )) + K2 (P − Mid(K ))| h NC CR RT 1 2 4 4 Since K and K form a parallelogram, we have P − Mid(K ) = Mid(K ) − P . This 1 2 1 2 yields that 1 |K (∇ u¯ − σ )(P )| = |(f − f )(P − Mid(K ))| h NC CR RT 4 K1 K2 1 (cid:46) h2|f| ∞ . 1, ,Ω Suppose that φ , 1 ≤ i ≤ 3 denote the nodal basis functions on K of (W )2. Hence, i (cid:88) CR by the deﬁnition of K and scaling arguments, there holds that h 3 (cid:46) (cid:46) (cid:107)K (∇ u¯ − σ )(cid:107)2 h4|f|2 ∞ (cid:107)φ (cid:107)2 h6|f|2 ∞ . h NC CR RT 0,K 1, ,Ω i 0,K 1, ,Ω i=1 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: J.

text: HU

text: AND

text: R.

text: MA

text: and

label: list

text: the

text: center

text: N

text: of

text: the

text: parallelogram

text: intersects

text: the

text: boundary

text: of

text: N

text: in

text: another

text: c

text: point

text: P˜.

text: Deﬁne

text: K

text: q(P

text: q(N

text: K

text: q(P˜).

text: h

text: h

text: c

text: h

text: Brandts

text: proved

text: that

text: the

text: vector

text: K

text: Π

text: q

text: is

text: a

text: higher

text: order

text: approximation

text: of

text: q

text: than

text: h

text: RT

text: Π

text: q

text: itself.

text: RT

text: Theorem

text: Suppose

text: q

text: then

text: there

text: holds

text: that

text: K

text: Π

text: h

text: RT

text: Combining

text: Theorem

text: and

text: Theorem

text: concludes

text: that

text: the

text: post-processing

text: oper-

text: ator

text: K

text: also

text: improves

text: the

text: order

text: of

text: approximation

text: of

text: σ

text: h

text: RT

text: Corollary

text: Let

text: σ

text: and

text: σ

text: be

text: the

text: solutions

text: of

text: and

text: respec-

text: RT

text: tively.

text: There

text: holds

text: that

text: Theorem

text: Let

text: u

text: and

text: u

text: be

text: the

text: solutions

text: of

text: and

text: respectively.

text: CR

text: Further,

text: suppose

text: that

text: f

text: W

text: (Ω),

text: then

text: we

text: have

text: Proof.

text: Using

text: the

text: equivalence

text: equality

text: for

text: e

text: K

text: K

text: and

text: the

text: center

text: P

text: of

text: e,

text: there

text: holds

text: that

text: f

text: f

text: |K

text: u¯

text: σ

text: )(P

text: (P

text: Mid(K

text: (P

text: Mid(K

text: h

text: NC

text: CR

text: RT

text: Since

text: K

text: and

text: K

text: form

text: a

text: parallelogram,

text: we

text: have

text: P

text: Mid(K

text: Mid(K

text: P

text: This

text: yields

text: that

text: |K

text: u¯

text: σ

text: )(P

text: |(f

text: f

text: )(P

text: Mid(K

text: h

text: NC

text: CR

text: RT

text: ,Ω

text: Suppose

text: that

text: φ

text: i

text: denote

text: the

text: nodal

text: basis

text: functions

text: on

text: K

text: of

text: (W

text: Hence,

text: i

text: CR

text: by

text: the

text: deﬁnition

text: of

text: K

text: and

text: scaling

text: arguments,

text: there

text: holds

text: that

text: h

text: u¯

text: σ

text: h

text: NC

text: CR

text: RT

text: ,Ω

text: i

text: ,Ω

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: (cid:80) (cid:80) | | ≤ Proof. Since n α = r and D (log n)3, it follows by convexity that ∈ \ α2 | | x=1 x − ≤ − (cid:32) (cid:33) x [n] D x is at most D /(log n) 4 (log n(cid:88)) 1 = o(1). Consequently, (cid:88)   A ≥ N2 − ≥ N2 − e( ) 1 + α2 1 + o(1) α2 , 2 x 2 x ∈ ∈ x [n] x D A and so the probability that a family as in the statement of the lemma induces an independent set is at most (cid:32) (cid:32) (cid:33)(cid:33) (cid:88) − A ≤ −pN2 − (1 p)e( ) e(cid:32)xp 1 + o(cid:18)(1) (cid:19) (cid:33)α2 2 (cid:89) x ∈ x D ≤ r(1+o(1))r nr+1 α2x N/2 . (3) n(1+o(1))(r+1) rr ∈ x D A Next, we boun(cid:12)d(cid:91)the n(cid:12)umb(cid:88)er of ways i(cid:88)n which we can choose as in Lemma 3.1. (cid:12) ≤(cid:12) | | ≤ Using the fact th(cid:12)at r (cid:12)log n and D (log n)3, we ﬁrst note that ≥ A ≥ |A | − |A ∩ A | N (cid:18) (cid:19) x (cid:88) x x y ∈ ∈ ∈ x D x D x,y D x<y − ≥ (cid:32) |A | − | | (cid:33)n 2 (cid:32) (cid:33) (cid:88) x D 2 − (cid:88) r 2 ∈ x D | | ≥ − D 2r ≥ α N α + o(1) N. x x n ∈ ∈ x D x D It follows that (cid:88) ≤ α 1 + o(1) < 1 + 1/10 (4) x ∈ x D (cid:32) (cid:33) (cid:88) and |A\ ∪ A | − ( ∈ ) < N 1 + 1/5 α . x D x x ∈ x D (Here, the choice of the constants 1/10 and 1/5 was arbitrary; any two suﬃciently A small constants would have suﬃced.) Hence, the number of ways to choose is at most 11 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Proof.

text: Since

text: n

text: α

text: r

text: and

text: D

text: (log

text: it

text: follows

text: by

text: convexity

text: that

text: α2

text: x

text: x

text: [n]

text: D

text: x

text: is

text: at

text: most

text: D

text: /(log

text: n)

text: (log

text: Consequently,

text: A

text: e(

text: α2

text: α2

text: x

text: x

text: x

text: [n]

text: x

text: D

text: A

text: and

text: so

text: the

text: probability

text: that

text: a

text: family

text: as

text: in

text: the

text: statement

text: of

text: the

text: lemma

text: induces

text: an

text: independent

text: set

text: is

text: at

text: most

text: A

text: p)e(

text: x

text: x

text: D

text: rr

text: x

text: D

text: A

text: Next,

text: we

text: of

text: ways

text: which

text: we

text: can

text: choose

text: as

text: in

text: Lemma

text: Using

text: the

text: fact

text: r

text: n

text: and

text: D

text: (log

text: we

text: ﬁrst

text: note

text: that

text: A

text: |A

text: |A

text: A

text: N

text: x

text: x

text: x

text: y

text: x

text: D

text: x

text: D

text: x,y

text: D

text: x<y

text: |A

text: x

text: D

text: r

text: x

text: D

text: D

text: α

text: N

text: α

text: N.

text: x

text: x

text: n

text: x

text: D

text: x

text: D

text: It

text: follows

text: that

text: α

text: x

text: x

text: D

text: and

text: |A\

text: A

text: N

text: α

text: x

text: D

text: x

text: x

text: x

text: D

text: (Here,

text: the

text: choice

text: of

text: the

text: constants

text: and

text: was

text: arbitrary;

text: any

text: two

text: suﬃciently

text: A

text: small

text: constants

text: would

text: have

text: suﬃced.)

text: Hence,

text: the

text: number

text: of

text: ways

text: to

text: choose

text: is

text: at

text: most

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Phenomena of this kind have also been observed in additive combinatorics. Roth’s theorem [25], a central result in additive combinatorics, states that for every δ > 0 { } and all suﬃciently large n, every subset of [n] = 1, 2, . . . , n of density δ contains a three-term arithmetic progression. Kohayakawa, Ro¨dl and L(cid:32) uczak [18] proved a random analogue, showing that such a statement holds not only for [n] but also, − with high probability, for random subsets of [n] of density at least Cn 1/2, where C > 0 is an absolute constant. Another classical result in additive combinatorics, due to Diananda and Yap [10], Z is that the largest sum-free subset of is the set of odd numbers. Balogh, Morris 2n Z and Samotij [3] proved that the same is true of random subsets of of density 2n greater than (log n/3n)1/2 with high probability, and also that this no longer the case when the density is less than (log n/3n)1/2. Thus, there is a sharp threshold at (log n/3n)1/2 for the stability of this extremal result; an extension of this sharp threshold result to all even-order Abelian groups has recently been proved by Bushaw, Neto, Morris and Smith [7]. Perhaps the most striking application of such transference principles in additive combinatorics is the Green-Tao theorem [15] on primes in arithmetic progressions. These results constitute a tiny sample of the large number of beautiful results which have been proved in this setting. Very general transference theorems have been proved by Conlon and Gowers [8] and Schacht [26]. We refer the interested reader to the surveys of L(cid:32) uczak [20] and Ro¨dl and Schacht [24] for a more detailed account of such results. Returning to the question at hand, our aim, as we remarked before, is to investigate the independence number of random subgraphs of K(n, r). (The independence number of random induced subgraphs of K(n, r), i.e., the size of the largest intersecting subfamily of a random r-uniform hypergraph, has been investigated by Balogh, Bohman and Mubayi [2].) As we mentioned earlier, an analogue of the Erdo˝s-Ko-Rado theorem continues to hold for very sparse random subgraphs of the K(n, r) and so it will be more convenient to retain (rather than delete) the edges of K(n, r) independently with some probability p; to this end, let K (n, r) denote the random subgraph of K(n, r) obtained by retaining each edge p of K(n, r) independently with probability p. The(cid:0) ma(cid:1)in question of interest is the following. − Problem 1.1. For what p > 0 is α(K (n, r)) = n−1 with high probability? p r 1 3

text: Phenomena

text: of

text: this

text: kind

text: have

text: also

text: been

text: observed

text: in

text: additive

text: combinatorics.

text: Roth’s

text: theorem

text: a

text: central

text: result

text: in

text: additive

text: combinatorics,

text: states

text: that

text: for

text: every

text: δ

text: and

text: all

text: suﬃciently

text: large

text: n,

text: every

text: subset

text: of

text: [n]

text: n

text: of

text: density

text: δ

text: contains

text: a

text: three-term

text: arithmetic

text: progression.

text: Kohayakawa,

text: Ro¨dl

text: and

text: uczak

text: proved

text: a

text: random

text: analogue,

text: showing

text: that

text: such

text: a

text: statement

text: holds

text: not

text: only

text: for

text: [n]

text: but

text: also,

text: with

text: high

text: probability,

text: for

text: random

text: subsets

text: of

text: [n]

text: of

text: density

text: at

text: least

text: Cn

text: where

text: C

text: is

text: an

text: absolute

text: constant.

text: Another

text: classical

text: result

text: in

text: additive

text: combinatorics,

text: due

text: to

text: Diananda

text: and

text: Yap

text: Z

text: is

text: that

text: the

text: largest

text: sum-free

text: subset

text: of

text: is

text: the

text: set

text: of

text: odd

text: numbers.

text: Balogh,

text: Morris

text: Z

text: and

text: Samotij

text: proved

text: that

text: the

text: same

text: is

text: true

text: of

text: random

text: subsets

text: of

text: of

text: density

text: greater

text: than

text: (log

text: with

text: high

text: probability,

text: and

text: also

text: that

text: this

text: no

text: longer

text: the

text: case

text: when

text: the

text: density

text: is

text: less

text: than

text: (log

text: Thus,

text: there

text: is

text: a

text: sharp

text: threshold

text: at

text: (log

text: for

text: the

text: stability

text: of

text: this

text: extremal

text: result;

text: an

text: extension

text: of

text: this

text: sharp

text: threshold

text: result

text: to

text: all

text: even-order

text: Abelian

text: groups

text: has

text: recently

text: been

text: proved

text: by

text: Bushaw,

text: Neto,

text: Morris

text: and

text: Smith

text: Perhaps

text: the

text: most

text: striking

text: application

text: of

text: such

text: transference

text: principles

text: in

text: additive

text: combinatorics

text: is

text: the

text: Green-Tao

text: theorem

text: on

text: primes

text: in

text: arithmetic

text: progressions.

text: These

text: results

text: constitute

text: a

text: tiny

text: sample

text: of

text: the

text: large

text: number

text: of

text: beautiful

text: results

text: which

text: have

text: been

text: proved

text: in

text: this

text: setting.

text: Very

text: general

text: transference

text: theorems

text: have

text: been

text: proved

text: by

text: Conlon

text: and

text: Gowers

text: and

text: Schacht

text: We

text: refer

text: the

text: interested

text: reader

text: to

text: the

text: surveys

text: of

text: uczak

text: and

text: Ro¨dl

text: and

text: Schacht

text: for

text: a

text: more

text: detailed

text: account

text: of

text: such

text: results.

text: Returning

text: to

text: the

text: question

text: at

text: hand,

text: our

text: aim,

text: as

text: we

text: remarked

text: before,

text: is

text: to

text: investigate

text: the

text: independence

text: number

text: of

text: random

text: subgraphs

text: of

text: K(n,

text: r).

text: (The

text: independence

text: number

text: of

text: random

text: induced

text: subgraphs

text: of

text: K(n,

text: r),

text: i.e.,

text: the

text: size

text: of

text: the

text: largest

text: intersecting

text: subfamily

text: of

text: a

text: random

text: r-uniform

text: hypergraph,

text: has

text: been

text: investigated

text: by

text: Balogh,

text: Bohman

text: and

text: Mubayi

text: As

text: we

text: mentioned

text: earlier,

text: an

text: analogue

text: of

text: the

text: Erdo˝s-Ko-Rado

text: theorem

text: continues

text: to

text: hold

text: for

text: very

text: sparse

text: random

text: subgraphs

text: of

text: the

text: K(n,

text: r)

text: and

text: so

text: it

text: will

text: be

text: more

text: convenient

text: to

text: retain

text: (rather

text: than

text: delete)

text: the

text: edges

text: of

text: K(n,

text: r)

text: independently

text: with

text: some

text: probability

text: p;

text: to

text: this

text: end,

text: let

text: K

text: (n,

text: r)

text: denote

text: the

text: random

text: subgraph

text: of

text: K(n,

text: r)

text: obtained

text: by

text: retaining

text: each

text: edge

text: p

text: of

text: K(n,

text: r)

text: independently

text: with

text: probability

text: p.

text: question

text: of

text: interest

text: is

text: the

text: following.

text: Problem

text: For

text: what

text: p

text: is

text: α(K

text: (n,

text: r))

text: with

text: high

text: probability?

text: p

text: r


text: ∈ R | | ≤ Proposition 2.4. For every x such that x 1/2, − (cid:3) ex x2 < 1 + x < ex. (cid:0) (cid:1) (cid:0) (cid:1) Although our last proposition is also very simple, we prove it here for the sake − − − of completeness. Recall that N = n−1 and M = n −r 1 . r 1 r 1 − Proposition 2.5. If r = r(n) = o(n1/2), then N M = o(N). Furthermore, if − r = o(n1/3), then N M = o(N/r). (cid:18) (cid:19) (cid:18) (cid:19) Proof. Both claims follow from the observation that − − − − n (cid:18) 1 −(cid:19) n (cid:18) r 1 (cid:19) N M = (cid:88)− − r 1 r 1 − − − r n i − n i 1 (cid:18) (cid:19) = (cid:88) − − r 1 r 1 i=1 − − r (cid:18) n (cid:19)i 1 = − r 2 i=1 − − ≤ n 2 r(r 1) (cid:3) r − = − N. r 2 n 1 3. Upper bound for the critical threshold We now turn to our proof of Theorem 1.2. In this section, we shall bound the critical threshold from above, i.e., we shall prove that a random analogue of the Erdo˝s-Ko-Rado theorem holds if p > (1 + ε)p (n, r) where p (n, r) is given by (1). c c Proof of the upper bound in Theorem 1.2. Let 0 < ε < 1/2 and set p = p(n) = (1 + ε)p (n, r). We shall prove that with high probability, the independence number c of K (n, r) is N, and that furthermore, the only independent sets of size N in p K (n, r) are stars. Since we are working with monotone properties, it suﬃces to p prove this result for ε small enough and so we lose nothing by assuming 0 < ε < 1/2. ≥ A ⊂ For each i 1, let X be the number of families [n](r) inducing an i |A| A − independent set in K (n, r) such that = N and d( ) = N i. Also, let p 6 ##LTLine##

text: R

text: Proposition

text: For

text: every

text: x

text: such

text: that

text: x

text: ex

text: x

text: ex.

text: Although

text: our

text: last

text: proposition

text: is

text: also

text: very

text: simple,

text: we

text: prove

text: it

text: here

text: for

text: the

text: sake

text: of

text: completeness.

text: Recall

text: that

text: N

text: and

text: M

text: n

text: −r

text: r

text: r

text: Proposition

text: If

text: r

text: r(n)

text: then

text: N

text: M

text: o(N).

text: Furthermore,

text: if

text: r

text: then

text: N

text: M

text: o(N/r).

text: Proof.

text: Both

text: claims

text: follow

text: from

text: the

text: observation

text: that

text: n

text: n

text: r

text: N

text: M

text: r

text: r

text: r

text: n

text: i

text: n

text: i

text: r

text: r

text: r

text: n

text: r

text: n

text: r(r

text: r

text: N.

text: r

text: n

text: Upper

text: bound

text: for

text: the

text: critical

text: threshold

text: We

text: now

text: turn

text: to

text: our

text: proof

text: of

text: Theorem

text: In

text: this

text: section,

text: we

text: shall

text: bound

text: the

text: critical

text: threshold

text: from

text: above,

text: i.e.,

text: we

text: shall

text: prove

text: that

text: a

text: random

text: analogue

text: of

text: the

text: Erdo˝s-Ko-Rado

text: theorem

text: holds

text: if

text: p

text: ε)p

text: (n,

text: r)

text: where

text: p

text: (n,

text: r)

text: is

text: given

text: by

text: c

text: c

text: Proof

text: of

text: the

text: upper

text: bound

text: in

text: Theorem

text: Let

text: ε

text: and

text: set

text: p

text: p(n)

text: ε)p

text: (n,

text: r).

text: We

text: shall

text: prove

text: that

text: with

text: high

text: probability,

text: the

text: independence

text: number

text: c

text: of

text: K

text: (n,

text: r)

text: is

text: N,

text: and

text: that

text: furthermore,

text: the

text: only

text: independent

text: sets

text: of

text: size

text: N

text: in

text: p

text: K

text: (n,

text: r)

text: are

text: stars.

text: Since

text: we

text: are

text: working

text: with

text: monotone

text: properties,

text: it

text: suﬃces

text: to

text: p

text: prove

text: this

text: result

text: for

text: ε

text: small

text: enough

text: and

text: so

text: we

text: lose

text: nothing

text: by

text: assuming

text: ε

text: A

text: For

text: each

text: i

text: let

text: X

text: be

text: the

text: number

text: of

text: families

text: [n](r)

text: inducing

text: an

text: i

text: |A|

text: A

text: independent

text: set

text: in

text: K

text: (n,

text: r)

text: such

text: that

text: N

text: and

text: d(

text: N

text: i.

text: Also,

text: let

text: p

text: ##LTLine##


text: S For any star , we know that the number of edges in K(n, r) between a set ∈ \S A ⊂ S |A| − − A [n](r) and a family is at least (N M). We know from − Proposition 2.5 that N M = o(N/r) when r = o(n1/3); consequently, it follows A ⊂ A ≥ − A ≥ that if [n](r) has cardinality N and d( ) (1 ε/2)N/(r + 1), then e( ) A − A (1 + o(1))d( )(N d( )). To simplify calculations, let us set i = αN = αrR/n where ≤ ε/2 < α (r + ε/2)/(r + 1). (cid:18) (cid:19)(cid:18) (cid:19) In this range, we see that E ≤ (cid:16) N(cid:17) R(cid:16) (cid:17)− A(cid:0) (cid:1) [X ] n (1 p)e( ) i i i ≤ (cid:18)e αN en αN (cid:19) (cid:16) (cid:17) − n exp (1 + ε + o(1))α(1 α)p (n, r)N2 c α rα ≤ nr(1+ε+o(1))(1−α)r αN ≤ r (ε2/4−ε3/4+o(1))N n − n , rn(1+ε+o(1))(1 α)(r+1) n and it follows tha(cid:88)t (cid:16) (cid:17) (2r+ε)N 2(r+1) P ≤ r (ε2/4−ε3/4+o(1))N (X > 0) nN = o(1), i n i=εN/2 ≤ and so with high probability, for each εN/2 < i (r + ε/2)N/(r + 1), the random variable X is zero. i Case 3: Small maximum d(cid:88)egree. We shall complete the proof of the lower bound by showing that E [X ] = o(1). i i>(2r+ε)N 2(r+1) It turns out that in this range of i, somewhat surprisingly, it is signiﬁcantly easier to deal with the case where r tends to inﬁnity with n as opposed to the case where r is small. ≥ Suppose ﬁrst that r log n. Then − (1 ε/2) 1 < r + 1 r + 4 9 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: S

text: For

text: any

text: star

text: we

text: know

text: that

text: the

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