text: 3 density of the C region, additional contributions are in- cluded for each pair of PAOs φ and φ located close to i j a boundary when at least one of them is centered at the neighboring reservoir region. In total, two new contribu- tions must be added to the electronic density evaluated using Eqs. 1-2 for each pair of PAOs at each boundary. For the L/C boundary, t(cid:88)hese are (Fig. 1b): nLL = (cid:88) DLLφLφL, i,j i j i,j (3) nLC = DLCφLφC, i,j i j i,j which can be further distinguished based on whether both (D¯LL) or just one (D¯LC) of the two PAOs involved is centered at the L region. In the calculations presented in this work, the “spill-in” terms are independent of the applied bias voltage. This is justiﬁed as we checked that the non-periodic KS potential at the boundary of the C region for each value of the applied bias matches smoothly with the periodic KS potential of the neigh- boring reservoir, i.e. that the “screening approximation” is veriﬁed – see Ref. 26 for additional details. We stress that including these “spill-in” terms is essential to ensure a stable and well-behaved convergence behavior of the SCF cycle, which turns out to be especially important for heterogeneous systems such as the Ag(100)/Si(100) interface investigated in this work. B. Exchange-correlation potential Further complications in describing the Ag(100)/Si(100) interface arise from the fact that one of its sides is semi- conducting. In fact, a major problem aﬀecting the de- scription of metal-semiconductor interfaces is the severe underestimation of the semiconductor band gap in DFT calculations using (semi-)local xc-functionals based on the local density approximation (LDA) or on the gen- FIG. 1. Scheme showing the two-center “spill-in” terms used eralized gradient approximation (GGA) [45]. For model for the evaluation of the Hamiltonian (a) and the electronic calculations based on few-layer thick fully periodic sys- density (b) of the C region for a pair of s type PAOs φ j tems, such an underestimation has been shown to result and φ located close to the L/C boundary. In (a,b) the blue i in unrealistically low Schottky barriers at the interface shaded regions indicate the integrals performed in the C re- [15, 46]. In order to remedy this drawback, we have gion, whereas the red and green regions indicate the “spill-in” evaluated the electronic structure of the LDA-optimized terms for the Hamiltonian and for the electronic density, re- interface geometries using the Tran-Blaha meta-GGA spectively. xc-functional (TB09) [27]. The TB09 xc-functional has been shown to provide band gaps in excellent agreement with the experiments for a wide range of semiconductors “spill-in”. For the Hamiltonian, corrective terms are ap- including silicon, at a computational cost comparable plied to both the two-center and three-center integrals. to that of conventional (semi-)local functionals. In the Speciﬁcally, if two PAOs φ and φ centered in the C i j TB09 xc-functional, the exchange potential υTB09(r) region lie close to a boundary, e.g. the L/C one, the cor- x depends explicitly on electron kinetic e(cid:115)nergy τ(r), rected Hamiltonian in Eq. 2 will include also the matrix (cid:48) (cid:104) | | (cid:105) element H = φ V LL(r) φ associated with the tail i,j i eff j of the PAOs protruding into the L region, V LL(r) being − eff 3c 2 4τ(r) the periodic KS potential of the semi-inﬁnite L reservoir υTB09(r) = cυBR(r) + , (4) x x π 6ρ(r) (Fig. 1a). Similar arguments hold also for the Hamil- tonian three-center non-local terms. For the electronic ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine##

text: density

text: of

text: the

text: C

text: region,

text: additional

text: contributions

text: are

text: in-

text: cluded

text: for

text: each

text: pair

text: of

text: PAOs

text: φ

text: and

text: φ

text: located

text: close

text: to

text: i

text: j

text: a

text: boundary

text: when

text: at

text: least

text: one

text: of

text: them

text: is

text: centered

text: at

text: the

text: neighboring

text: reservoir

text: region.

text: In

text: total,

text: two

text: new

text: contribu-

text: tions

text: must

text: be

text: added

text: to

text: the

text: electronic

text: density

text: evaluated

text: using

text: Eqs.

text: for

text: each

text: pair

text: of

text: PAOs

text: at

text: each

text: boundary.

text: For

text: the

text: L/C

text: boundary,

text: are

text: (Fig.

text: nLL

text: DLLφLφL,

text: i,j

text: i

text: j

text: i,j

text: nLC

text: DLCφLφC,

text: i,j

text: i

text: j

text: i,j

text: which

text: can

text: be

text: further

text: distinguished

text: based

text: on

text: whether

text: both

text: (D¯LL)

text: or

text: just

text: one

text: (D¯LC)

text: of

text: the

text: two

text: PAOs

text: involved

text: is

text: centered

text: at

text: the

text: L

text: region.

text: In

text: the

text: calculations

text: presented

text: in

text: this

text: work,

text: the

text: “spill-in”

text: terms

text: are

text: independent

text: of

text: the

text: applied

text: bias

text: voltage.

text: This

text: is

text: justiﬁed

text: as

text: we

text: checked

text: that

text: the

text: non-periodic

text: KS

text: potential

text: at

text: the

text: boundary

text: of

text: the

text: C

text: region

text: for

text: each

text: value

text: of

text: the

text: applied

text: bias

text: matches

text: smoothly

text: with

text: the

text: periodic

text: KS

text: potential

text: of

text: the

text: neigh-

text: boring

text: reservoir,

text: i.e.

text: that

text: the

text: “screening

text: approximation”

text: is

text: veriﬁed

text: see

text: Ref.

text: for

text: additional

text: details.

text: We

text: stress

text: that

text: including

text: these

text: “spill-in”

text: terms

text: is

text: essential

text: to

text: ensure

text: a

text: stable

text: and

text: well-behaved

text: convergence

text: behavior

text: of

text: the

text: SCF

text: cycle,

text: which

text: turns

text: out

text: to

text: be

text: especially

text: important

text: for

text: heterogeneous

text: systems

text: such

text: as

text: the

text: interface

text: investigated

text: in

text: this

text: work.

text: B.

text: Exchange-correlation

text: potential

text: Further

text: complications

text: in

text: describing

text: the

text: interface

text: arise

text: from

text: the

text: fact

text: that

text: one

text: of

text: its

text: sides

text: is

text: semi-

text: conducting.

text: In

text: fact,

text: a

text: major

text: problem

text: aﬀecting

text: the

text: de-

text: scription

text: of

text: metal-semiconductor

text: interfaces

text: is

text: the

text: severe

text: underestimation

text: of

text: the

text: semiconductor

text: band

text: gap

text: in

text: DFT

text: calculations

text: using

text: (semi-)local

text: xc-functionals

text: based

text: on

text: the

text: local

text: density

text: approximation

text: (LDA)

text: or

text: on

text: the

text: gen-

text: FIG.

text: Scheme

text: showing

text: the

text: two-center

text: “spill-in”

text: terms

text: used

text: eralized

text: gradient

text: approximation

text: (GGA)

text: For

text: model

text: for

text: the

text: evaluation

text: of

text: the

text: Hamiltonian

text: (a)

text: and

text: the

text: electronic

text: calculations

text: based

text: on

text: few-layer

text: thick

text: fully

text: periodic

text: sys-

text: density

text: (b)

text: of

text: the

text: C

text: region

text: for

text: a

text: pair

text: of

text: s

text: type

text: PAOs

text: φ

text: j

text: tems,

text: such

text: an

text: underestimation

text: has

text: been

text: shown

text: to

text: result

text: and

text: φ

text: located

text: close

text: to

text: the

text: L/C

text: boundary.

text: In

text: (a,b)

text: the

text: blue

text: i

text: in

text: unrealistically

text: low

text: Schottky

text: barriers

text: at

text: the

text: interface

text: shaded

text: regions

text: indicate

text: the

text: integrals

text: performed

text: in

text: the

text: C

text: re-

text: In

text: order

text: to

text: remedy

text: this

text: drawback,

text: we

text: have

text: gion,

text: whereas

text: the

text: red

text: and

text: green

text: regions

text: indicate

text: the

text: “spill-in”

text: evaluated

text: the

text: electronic

text: structure

text: of

text: the

text: LDA-optimized

text: terms

text: for

text: the

text: Hamiltonian

text: and

text: for

text: the

text: electronic

text: density,

text: re-

text: interface

text: geometries

text: using

text: the

text: Tran-Blaha

text: meta-GGA

text: spectively.

text: xc-functional

text: The

text: xc-functional

text: has

text: been

text: shown

text: to

text: provide

text: band

text: gaps

text: in

text: excellent

text: agreement

text: with

text: the

text: experiments

text: for

text: a

text: wide

text: range

text: of

text: semiconductors

text: “spill-in”.

text: For

text: the

text: Hamiltonian,

text: corrective

text: terms

text: are

text: ap-

text: including

text: silicon,

text: at

text: a

text: computational

text: cost

text: comparable

text: plied

text: to

text: both

text: the

text: two-center

text: and

text: three-center

text: integrals.

text: to

text: that

text: of

text: conventional

text: (semi-)local

text: functionals.

text: In

text: the

text: Speciﬁcally,

text: if

text: two

text: PAOs

text: φ

text: and

text: φ

text: centered

text: in

text: the

text: C

text: i

text: j

text: xc-functional,

text: the

text: exchange

text: potential

text: region

text: lie

text: close

text: to

text: a

text: boundary,

text: e.g.

text: the

text: L/C

text: one,

text: the

text: cor-

text: x

text: depends

text: explicitly

text: on

text: electron

text: kinetic

text: τ(r),

text: rected

text: Hamiltonian

text: in

text: Eq.

text: will

text: include

text: also

text: the

text: matrix

text: element

text: H

text: φ

text: V

text: LL(r)

text: φ

text: associated

text: with

text: the

text: tail

text: i,j

text: i

text: eff

text: j

text: of

text: the

text: PAOs

text: protruding

text: into

text: the

text: L

text: region,

text: V

text: LL(r)

text: being

text: eff

text: the

text: periodic

text: KS

text: potential

text: of

text: the

text: semi-inﬁnite

text: L

text: reservoir

text: cυBR(r)

text: x

text: x

text: π

text: (Fig.

text: Similar

text: arguments

text: hold

text: also

text: for

text: the

text: Hamil-

text: tonian

text: three-center

text: non-local

text: terms.

text: For

text: the

text: electronic

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: J Stat Phys manuscript No. (will be inserted by the editor) · Bernard Derrida Zhan Shi Large deviations for the branching Brownian motion in presence of selection or coalescence Received: date / Accepted: date Abstract The large deviation function has been known for a long time in the literature for the displacement of the rightmost particle in a branching random walk (BRW), or in a branching Brownian motion (BBM). More recently a number of generalizations of the BBM and of the BRW have been considered where selection or coalescence mech- anisms tend to limit the exponential growth of the number of particles. Here we try to estimate the large deviation function of the position of the rightmost particle for several such generalizations: the L-BBM, the N-BBM, and the CBRW (coalescing branching random walk) which is closely related to the noisy FKPP equation. Our approach allows us to obtain only upper bounds on these large deviation functions. One noticeable feature of our results is their non analytic dependence on the parameters (such as the coalescence rate in the CBRW). PACS 02.50.-r, 05.40.-a May 28, 2020 1 Introduction Branching Brownian motions (BBM) and branching random walks (BRW) are among the simplest stochastic models of a growing population in space and time. They describe particles which perform Brownian motions or random walks and branch independently B. Derrida Collège de France, 11 place Marcelin Berthelot, 75231 Paris Cedex 05 - France and Laboratoire de Physique Statistique, École Normale Supérieure, Université Pierre et Marie Curie, Université Denis Diderot, CNRS 24 rue Lhomond, 75231 Paris Cedex 05 - France Z. Shi LPMA, Université Pierre et Marie Curie 4 place Jussieu, 75252 Paris Cedex 05 - France · E-mail: derrida@lps.ens.fr E-mail: zhan.shi@upmc.fr ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: J

label: paragraph

text: Stat

text: Phys

text: manuscript

text: No.

text: (will

text: be

text: inserted

text: by

text: the

text: editor)

label: author

text: Bernard

text: Derrida

text: Zhan

text: Shi

text: Large

label: title

text: deviations

text: for

text: the

text: branching

text: Brownian

text: motion

text: in

text: presence

text: of

text: selection

text: or

text: coalescence

text: Received:

label: date

text: date

text: Accepted:

text: date

text: Abstract

text: The

label: abstract

text: large

text: deviation

text: function

text: has

text: been

text: known

text: for

text: a

text: long

text: time

text: in

text: the

text: literature

text: for

text: the

text: displacement

text: of

text: the

text: rightmost

text: particle

text: in

text: a

text: branching

text: random

text: walk

text: (BRW),

text: or

text: in

text: a

text: branching

text: Brownian

text: motion

text: (BBM).

text: More

text: recently

text: a

text: number

text: of

text: generalizations

text: of

text: the

text: BBM

text: and

text: of

text: the

text: BRW

text: have

text: been

text: considered

text: where

text: selection

text: or

text: coalescence

text: mech-

text: anisms

text: tend

text: to

text: limit

text: the

text: exponential

text: growth

text: of

text: the

text: number

text: of

text: particles.

text: Here

text: we

text: try

text: to

text: estimate

text: the

text: large

text: deviation

text: function

text: of

text: the

text: position

text: of

text: the

text: rightmost

text: particle

text: for

text: several

text: such

text: generalizations:

text: the

text: L-BBM,

text: the

text: N-BBM,

text: and

text: the

text: CBRW

text: (coalescing

text: branching

text: random

text: walk)

text: which

text: is

text: closely

text: related

text: to

text: the

text: noisy

text: FKPP

text: equation.

text: Our

text: approach

text: allows

text: us

text: to

text: obtain

text: only

text: upper

text: bounds

text: on

text: these

text: large

text: deviation

text: functions.

text: One

text: noticeable

text: feature

text: of

text: our

text: results

text: is

text: their

text: non

text: analytic

text: dependence

text: on

text: the

text: parameters

text: (such

text: as

text: the

text: coalescence

text: rate

text: in

text: the

text: CBRW).

text: PACS

text: May

text: Introduction

text: Branching

text: Brownian

text: motions

text: (BBM)

text: and

text: branching

text: random

text: walks

text: (BRW)

text: are

text: among

text: the

text: simplest

text: stochastic

text: models

text: of

text: a

text: growing

text: population

text: in

text: space

text: and

text: time.

text: They

text: describe

text: particles

text: which

text: perform

text: Brownian

text: motions

text: or

text: random

text: walks

text: and

text: branch

text: independently

text: B.

text: Derrida

text: Collège

text: de

text: France,

text: place

text: Marcelin

text: Berthelot,

text: Paris

text: Cedex

text: France

text: and

text: Laboratoire

text: de

text: Physique

text: Statistique,

text: École

text: Normale

text: Supérieure,

text: Université

text: Pierre

text: et

text: Marie

text: Curie,

text: Université

text: Denis

text: Diderot,

text: CNRS

text: rue

text: Lhomond,

text: Paris

text: Cedex

text: France

text: Z.

text: Shi

text: LPMA,

text: Université

text: Pierre

text: et

text: Marie

text: Curie

text: place

text: Jussieu,

text: Paris

text: Cedex

text: France

text: E-mail:

text: derrida@lps.ens.fr

text: E-mail:

text: zhan.shi@upmc.fr

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 11 2.3 The CBRW (branching random walk with coalescence) For a branching random walk on a lattice, the probability that a red particle reaches the position vt with v > v at time t is of the form c − et(r f(v)) . For example if the random walk is characterized by the probability ρ(y) that the walker jumps a distance y from the site it occupies, f(v) is given in a parametric form as − (cid:48) (cid:48) f(v) = g(λ) + λg (λ) ; v = g (λ) (2.24) with g(λ) given by (1.24). Given that the red particle m(cid:104)ov(cid:16)es on(cid:16)av(cid:17)erage at velo(cid:16)city v du(cid:17)ring tim(cid:17)(cid:105)e t, the probability (cid:28) (cid:28) P(x, s) that it moves a distance x during a time interval 1 s t is as before (2.5) by ∼ − x x − (cid:48) P(x, s) exp s f + f(v) + v f (v) . s s On the other hand the number of particles produced by the subtree at position x at time (cid:46) − s is es(r f(x/s)). Therefore the probabil(cid:104)ity that the red pa(cid:105)rticle is killed by a subtree of age s is (cid:46) − Q(x, s) min 1, µes(r f(x/s)) . (2.25) As for the L-BBM, one needs to distinguish two cases: (cid:16) (cid:16) (cid:17)(cid:17) – If 1 dominates in (2.25) this means that x/s satisﬁes the relation − x s r f + ln µ = 0 (2.26) s then one has to maximize P(x, s) given by (2.5) over s and x given the constraint (2.26). This leads to the fact that x = sy where y is solution of − − f(y) r − f(v) r − (cid:48) y = (cid:48) v (2.27) f (y) f (v) and after some algebra to p ∼ µff(cid:48)(cid:48)((yv))−1. This leads to (1.21). − – The other case, when (2.25) is dominated by µes(r f(x/s)), is much easier. The opti- ∼ mum over s gives s = 0 and therefore p µ. 3 Proof: Existence of the large deviation function In this section we prove the existence of the large deviation functions (1.5,1.9,1.18). 3.1 The L-BBM We ﬁrst establish two elementary properties of the L-BBM if one starts at time 0 with ≥ N surviving particles. In view of the statement, we can assume N 2. ≥ N For any s 0, let (s) be the number of surviving particles of the L-BBM at time s N (so that (0) = N). ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: section

text: The

text: CBRW

text: (branching

text: random

text: walk

text: with

text: coalescence)

text: For

text: a

text: branching

text: random

text: walk

text: on

text: a

text: lattice,

text: the

text: probability

text: that

text: a

text: red

text: particle

text: reaches

text: the

text: position

text: vt

text: with

text: v

text: v

text: at

text: time

text: t

text: is

text: of

text: the

text: form

text: c

text: et(r

text: f(v))

text: For

text: example

text: if

text: the

text: random

text: walk

text: is

text: characterized

text: by

text: the

text: probability

text: ρ(y)

text: that

text: the

text: walker

text: jumps

text: a

text: distance

text: y

text: from

text: the

text: site

text: it

text: occupies,

text: f(v)

text: is

text: given

text: in

text: a

text: parametric

text: form

text: as

label: equation

text: f(v)

text: g(λ)

text: λg

text: (λ)

text: v

text: g

text: (λ)

text: with

text: g(λ)

text: given

text: by

text: Given

text: that

text: the

text: red

text: particle

text: at

text: v

text: t,

text: the

text: probability

text: P(x,

text: s)

text: that

text: it

text: moves

text: a

text: distance

text: x

text: during

text: a

text: time

text: interval

text: s

text: t

text: is

text: as

text: before

text: by

text: x

text: x

text: P(x,

text: s)

text: exp

text: s

text: f

text: f(v)

text: v

text: f

text: (v)

text: s

text: s

text: On

text: the

text: other

text: hand

text: the

text: number

text: of

text: particles

text: produced

text: by

text: the

text: subtree

text: at

text: position

text: x

text: at

text: time

text: s

text: is

text: es(r

text: f(x/s)).

text: Therefore

text: the

text: that

text: the

text: red

text: is

text: killed

text: by

text: a

text: subtree

text: of

text: age

text: s

text: is

text: Q(x,

text: s)

text: min

text: µes(r

text: f(x/s))

text: As

text: for

text: the

text: L-BBM,

text: one

text: needs

text: to

text: distinguish

text: two

text: cases:

text: If

text: dominates

text: in

text: this

text: means

text: that

text: x/s

text: satisﬁes

text: the

text: relation

text: x

text: s

text: r

text: f

text: ln

text: µ

text: s

text: then

text: one

text: has

text: to

text: maximize

text: P(x,

text: s)

text: given

text: by

text: over

text: s

text: and

text: x

text: given

text: the

text: constraint

text: This

text: leads

text: to

text: the

text: fact

text: that

text: x

text: sy

text: where

text: y

text: is

text: solution

text: of

text: f(y)

text: r

text: f(v)

text: r

text: y

text: v

text: f

text: (y)

text: f

text: (v)

text: and

text: after

text: some

text: algebra

text: to

text: p

text: This

text: leads

text: to

text: The

text: other

text: case,

text: when

text: is

text: dominated

text: by

text: µes(r

text: f(x/s)),

text: is

text: much

text: easier.

text: The

text: opti-

text: mum

text: over

text: s

text: gives

text: s

text: and

text: therefore

text: p

text: µ.

text: Proof:

text: Existence

text: of

text: the

text: large

text: deviation

text: function

text: In

text: this

text: section

text: we

text: prove

text: the

text: existence

text: of

text: the

text: large

text: deviation

text: functions

text: The

text: L-BBM

text: We

text: ﬁrst

text: establish

text: two

text: elementary

text: properties

text: of

text: the

text: L-BBM

text: if

text: one

text: starts

text: at

text: time

text: with

text: N

text: surviving

text: particles.

text: In

text: view

text: of

text: the

text: statement,

text: we

text: can

text: assume

text: N

text: N

text: For

text: any

text: s

text: let

text: (s)

text: be

text: the

text: number

text: of

text: surviving

text: particles

text: of

text: the

text: L-BBM

text: at

text: time

text: s

text: N

text: (so

text: that

text: N).

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 13 (cid:20) (cid:21) ≥ (cid:34) (cid:35) (cid:20) (cid:21) (cid:90) Q P max x (s) > bL ≤ ≤ i 1 i M √ − M ≥ ≥ − (b+1)L/ 2τσ2 e √u2du P max y (s) > bL 1 . 1≤i≤M i −∞ π (cid:90) Using the fact that for x > 2 − x e √u2du − − − − < 1 e 2x2 < exp[ e 2x2] −∞ π and that for y > 0 − 1 e y < y one gets that Q > 1 − N 2(b+a1)2 −λ . (3.4) (cid:98) To complete the proof of (3.3), we now show that there is a small probability that the N − number of particles of the L-BBM on the right of position (b 1)L at time τ exceeds (cid:98) (cid:101) M. To do so, we ﬁrst notice that (cid:101) N N P[ > M] < P[ > M] N − where is the number o(cid:101)f particles on the right of (b 1)L at time τ generated by N independent BBM’s (with no selection) starting all at time 0 at position L. One can N (cid:90) calculate the expectation (cid:101) ∞ − E[N] = Neτ e √u2du < 2N1−(b−a1)2 (√b−1)L π (cid:90) 2σ2τ where we have used that for x > 0 ∞ − e √u2du − < e x2. π x Therefore by the Markov ineq(cid:98)uality one gets P[N > M] < 2N1−λ−(b−a1)2 . (cid:98) Now we know that, at time τ, there is a probability Q close to 1 that there is at least N one particle on the right of bL and a probability also close to 1 that < M. Therefore, because when there is at least one particle on the left of bL and no more than M particles − on the right of (b 1)L, one knows that the total number of surviving particles of the L-BBM does not exceed M. Consequently, (cid:98) N − N − P[ (τ) > M] < 1 Q + P[ > M] < 3N µ if we choose − − (b 1)2 − 2(b + 1)2 µ = 1 + λ + = λ . (3.5) a a ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Q

text: P

text: max

text: x

text: (s)

text: bL

text: i

text: i

text: M

text: M

text: 2τσ2

text: e

text: P

text: max

text: y

text: (s)

text: bL

text: i

text: π

text: Using

text: the

text: fact

text: that

text: for

text: x

text: x

text: e

text: e

text: exp[

text: e

text: π

text: and

text: that

text: for

text: y

text: e

text: y

text: y

text: one

text: gets

text: that

text: Q

text: N

text: −λ

text: To

text: complete

text: the

text: proof

text: of

text: we

text: now

text: show

text: that

text: there

text: is

text: a

text: small

text: probability

text: that

text: the

text: N

text: number

text: of

text: particles

text: of

text: the

text: L-BBM

text: on

text: the

text: right

text: of

text: position

text: (b

text: at

text: time

text: τ

text: exceeds

text: M.

text: To

text: do

text: so,

text: we

text: ﬁrst

text: notice

text: that

text: N

text: N

text: P[

text: M]

text: P[

text: M]

text: N

text: where

text: is

text: the

text: number

text: particles

text: on

text: the

text: right

text: of

text: (b

text: at

text: time

text: τ

text: generated

text: by

text: N

text: independent

text: BBM’s

text: (with

text: no

text: selection)

text: starting

text: all

text: at

text: time

text: at

text: position

text: L.

text: One

text: can

text: N

text: calculate

text: the

text: expectation

text: E[N]

text: Neτ

text: e

text: π

text: 2σ2τ

text: where

text: we

text: have

text: used

text: that

text: for

text: x

text: e

text: e

text: π

text: x

text: Therefore

text: by

text: the

text: Markov

text: one

text: gets

text: P[N

text: M]

text: Now

text: we

text: know

text: that,

text: at

text: time

text: τ,

text: there

text: is

text: a

text: probability

text: Q

text: close

text: to

text: that

text: there

text: is

text: at

text: least

text: N

text: one

text: particle

text: on

text: the

text: right

text: of

text: bL

text: and

text: a

text: probability

text: also

text: close

text: to

text: that

text: M.

text: Therefore,

text: because

text: when

text: there

text: is

text: at

text: least

text: one

text: particle

text: on

text: the

text: left

text: of

text: bL

text: and

text: no

text: more

text: than

text: M

text: particles

text: on

text: the

text: right

text: of

text: (b

text: one

text: knows

text: that

text: the

text: total

text: number

text: of

text: surviving

text: particles

text: of

text: the

text: L-BBM

text: does

text: not

text: exceed

text: M.

text: Consequently,

text: N

text: N

text: P[

text: (τ)

text: M]

text: Q

text: P[

text: M]

text: µ

text: if

text: we

text: choose

text: (b

text: µ

text: λ

text: λ

text: a

text: a

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: 15 3.2 The N-BBM (cid:80) (cid:80) For the N-BBM, we start with two simple but useful monotonicity properties, which are borrowed from [17]. We include the elementary proof for the sake of self-containedness. ≥ We say that (u ) ≤ ≤ dominates (v ) ≤ ≤ if M 1{ ≥ } N 1{ ≥ } for all a ∈ (−∞, ∞) i(s1o iin pMarticular, M ≥iN1).i N i=1 ui a i=1 vi a ≥ · · · ≥ Lemma 3 (First monotonicity property for the N-BBM) Let x x and ≥ · · · ≥ ≥ ≤ ≤ 1 N y y be such that x y for all 1 i N. There exists a coupling for 1 N i i two N-BBM systems on a same probability space, starting at positions (x ) ≤ ≤ and i 1 i N (y ) ≤ ≤ respectively, such that the ﬁrst system dominates the second at all time. i 1 i N Proof. Consider two N-BBM systems, the ﬁrst starting at positions (x ) ≤ ≤ , and the i 1 i N second at (y ) ≤ ≤ . We attach the same Brownian motion to particles starting at x and ≤ i 1≤i N i y (for 1 i N) respectively in the two systems, and also attach the same Poisson i process which determines the branching times along the paths. As such, the ﬁrst branch- ing time is identical in the two systems, and before this time, the x-system obviously dominates the y-system. It is also easy to check that right after the ﬁrst branching time, the x-system still dominates the y-system. Then by attaching as before the same Brow- nian motions and the same Poissonian clocks to the x- and the y-particles, the x-system will continue to dominate the y-system. And so on. The procedure leads to the desired (cid:3) coupling. (cid:48) ≥ ≥ Lemma 4 (Second monotonicity property for the N-BBM) Let N N. Let x · · · ≥ ≥ · · · ≥ ≥ ≤ ≤ 1 x (cid:48) and y y be such that x y for all 1 i N. There exists N 1 (cid:48) N i i a coupling for an N -BBM and an N-BBM on a same probability space, with initial (cid:48) positions (x ) ≤ ≤ (cid:48) and (y ) ≤ ≤ respectively, such that the N -BBM dominates the i 1 i N i 1 i N N-BBM all time. (cid:48) (cid:48) Proof. If N = N, this amounts to the previous lemma. So let us assume N > N. Then, as in the proof of the previous lemma, if initially the N rightmost particles of the (cid:48) system with N particles dominates the other system, this remains true subsequently. The (cid:48) − (cid:3) remaining N N particles can only reinforce this domination. Let us now turn to the proof of the existence of the large deviation function for the ∈ (cid:110) (cid:111) N-BBM. Let v R. Consider the following event for the N-BBM: ∃ ∞ E := particle whose position at time t lies in [vt, ) . t To prove the existence of the large deviation function, we need to show that the limit lim →∞ 1 ln P(E ) exists. We prove this by an argument of superadditivity. By removing t t t all particles at time t except the rightmost one, the second monotonicity property stated in Lemma 4 tells us that ≥ ∀ ∀ (cid:48) P(E (cid:48)) P(E ) P(E (cid:48)), t > 0, t > 0 . t+t t t (cid:55)→ ∞ So the function t ln P(E ) is superadditive on (0, ). In particular, t 1 1 ∈ −∞ lim ln P(E ) = sup ln P(E ) ( , 0] , t→∞ t t t>0 t t exists. ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: The

text: N-BBM

text: For

text: the

text: N-BBM,

text: we

text: start

text: with

text: two

text: simple

text: but

text: useful

text: monotonicity

text: properties,

text: which

text: are

text: borrowed

text: from

text: We

text: include

text: the

text: elementary

text: proof

text: for

text: the

text: sake

text: of

text: self-containedness.

text: We

text: say

text: that

text: (u

text: dominates

text: (v

text: if

text: M

text: N

text: for

text: all

text: a

text: iin

text: pMarticular,

text: M

text: N

text: ui

text: a

text: vi

text: a

text: Lemma

text: (First

text: monotonicity

text: property

text: for

text: the

text: N-BBM)

text: Let

text: x

text: x

text: and

text: N

text: y

text: y

text: be

text: such

text: that

text: x

text: y

text: for

text: all

text: i

text: N.

text: There

text: exists

text: a

text: coupling

text: for

text: N

text: i

text: i

text: two

text: N-BBM

text: systems

text: on

text: a

text: same

text: probability

text: space,

text: starting

text: at

text: positions

text: (x

text: and

text: i

text: i

text: N

text: (y

text: respectively,

text: such

text: that

text: the

text: ﬁrst

text: system

text: dominates

text: the

text: second

text: at

text: all

text: time.

text: i

text: i

text: N

text: Proof.

text: Consider

text: two

text: N-BBM

text: systems,

text: the

text: ﬁrst

text: starting

text: at

text: positions

text: (x

text: and

text: the

text: i

text: i

text: N

text: second

text: at

text: (y

text: We

text: attach

text: the

text: same

text: Brownian

text: motion

text: to

text: particles

text: starting

text: at

text: x

text: and

text: i

text: N

text: i

text: y

text: (for

text: i

text: N)

text: respectively

text: in

text: the

text: two

text: systems,

text: and

text: also

text: attach

text: the

text: same

text: Poisson

text: i

text: process

text: which

text: determines

text: the

text: branching

text: times

text: along

text: the

text: paths.

text: As

text: such,

text: the

text: ﬁrst

text: branch-

text: ing

text: time

text: is

text: identical

text: in

text: the

text: two

text: systems,

text: and

text: before

text: this

text: time,

text: the

text: x-system

text: obviously

text: dominates

text: the

text: y-system.

text: It

text: is

text: also

text: easy

text: to

text: check

text: that

text: right

text: after

text: the

text: ﬁrst

text: branching

text: time,

text: the

text: x-system

text: still

text: dominates

text: the

text: y-system.

text: Then

text: by

text: attaching

text: as

text: before

text: the

text: same

text: Brow-

text: nian

text: motions

text: and

text: the

text: same

text: Poissonian

text: clocks

text: to

text: the

text: x-

text: and

text: the

text: y-particles,

text: the

text: x-system

text: will

text: continue

text: to

text: dominate

text: the

text: y-system.

text: And

text: so

text: on.

text: The

text: procedure

text: leads

text: to

text: the

text: desired

text: coupling.

text: Lemma

text: (Second

text: monotonicity

text: property

text: for

text: the

text: N-BBM)

text: Let

text: N

text: N.

text: Let

text: x

text: x

text: and

text: y

text: y

text: be

text: such

text: that

text: x

text: y

text: for

text: all

text: i

text: N.

text: There

text: exists

text: N

text: N

text: i

text: i

text: a

text: coupling

text: for

text: an

text: N

text: -BBM

text: and

text: an

text: N-BBM

text: on

text: a

text: same

text: probability

text: space,

text: with

text: initial

text: positions

text: (x

text: and

text: (y

text: respectively,

text: such

text: that

text: the

text: N

text: -BBM

text: dominates

text: the

text: i

text: i

text: N

text: i

text: i

text: N

text: N-BBM

text: all

text: time.

text: Proof.

text: If

text: N

text: N,

text: this

text: amounts

text: to

text: the

text: previous

text: lemma.

text: So

text: let

text: us

text: assume

text: N

text: N.

text: Then,

text: as

text: in

text: the

text: proof

text: of

text: the

text: previous

text: lemma,

text: if

text: initially

text: the

text: N

text: rightmost

text: particles

text: of

text: the

text: system

text: with

text: N

text: particles

text: dominates

text: the

text: other

text: system,

text: this

text: remains

text: true

text: subsequently.

text: The

text: remaining

text: N

text: N

text: particles

text: can

text: only

text: reinforce

text: this

text: domination.

text: Let

text: us

text: now

text: turn

text: to

text: the

text: proof

text: of

text: the

text: existence

text: of

text: the

text: large

text: deviation

text: function

text: for

text: the

text: N-BBM.

text: Let

text: v

text: R.

text: Consider

text: the

text: following

text: event

text: for

text: the

text: N-BBM:

text: E

text: particle

text: whose

text: position

text: at

text: time

text: t

text: lies

text: in

text: [vt,

text: t

text: To

text: prove

text: the

text: existence

text: of

text: the

text: large

text: deviation

text: function,

text: we

text: need

text: to

text: show

text: that

text: the

text: limit

text: lim

text: ln

text: P(E

text: exists.

text: We

text: prove

text: this

text: by

text: an

text: argument

text: of

text: superadditivity.

text: By

text: removing

text: t

text: t

text: t

text: all

text: particles

text: at

text: time

text: t

text: except

text: the

text: rightmost

text: one,

text: the

text: second

text: monotonicity

text: property

text: stated

text: in

text: Lemma

text: tells

text: us

text: that

text: P(E

text: P(E

text: P(E

text: t

text: t

text: t+t

text: t

text: t

text: So

text: the

text: function

text: t

text: ln

text: P(E

text: is

text: superadditive

text: on

text: In

text: particular,

text: t

text: lim

text: ln

text: P(E

text: sup

text: ln

text: P(E

text: t→∞

text: t

text: t

text: t

text: t

text: exists.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 23 − With the notation X = vu+σ(W uW ) (where W denotes again a standard Brownian u u t t− − − motion with variance 1, we have X X = vs + σ(W W sW ). The − − s+u u N s+u− u t t random variable W W sW has the Gaussian (0, s(1 s)) law. As for the s+u u t t t L-BBM, some elementary computations yield that, in case v > (4σ2)1/2, the subtrees ≤ move forward faster than the usual speed (2σ2)1/2, whereas if (2σ2)1/2 < v (4σ2)1/2, these subtrees make no particular effort, and wait only for the occasions when the red → ∞ → ∞ particle makes some ﬂuctuatio(cid:20)ns toward the left.5 Letting t and then N(cid:21) , we obtain: (cid:101) (cid:38) − v2 − − − E(#ENBBM) exp ( 1 + e c4(lnN)2 + M β(v)+oN(1))t , t 2σ2 where β(v) is deﬁned in (1.11), and o (1) stands for a term not depending on t such that N − − limN→∞ oN(1) = 0. Note that e c4(lnN)2 is negligible compared to M β(v)+oN(1). This yields the desired lower bound (4.11). 4.4 First moment computations for the CBRW The proof for the CBRW is along the lines of the proof for the L-BBM and for the N-BBM. Let (cid:91) (cid:101) N (t) { ∞ } ECBRW := particle i lies in [vt, ), leans to the left, does not coalesce . t i=1 −∞ (cid:48) The meaning of "leans to the left" is as before: the path of the particle lies in ( , t v + (cid:48) ∈ t2/3] for al(cid:101)l t [0, t]. By "does not coalesce", we mean that at no time(cid:101)during [0, t] does the particle coalesce with any other particle. Let #ECBRW denote the number of i satisfying the conditions in ECBRW. The main t t estimates for the CBRW are: (cid:104) (cid:105) (cid:101) (cid:101) (cid:38) − − E(#ECBRW) exp (f(v) r + µγ(v)+oµ(1))t , (4.13) t (cid:46) − − E[(#ECBRW)2] exp [ (f(v) r)t] , (4.14) t where γ(v) and f(v) are deﬁned in (1.20) and (1.15) respectively, and o (1) stands for a µ term not depending on t such that lim → o (1) = 0. Equations (4.13) and (4.14) together µ 0 µ with the Cauchy–Schwarz inequality will yield the upper bound stated in (1.19) for the large deviation function for the CBRW. The proof of (4.14), which is identical for all the three models, is postponed to Sub- section . The rest of this subsection is devoted to the proof of (4.13). ∈ Wr{iting ≤X = (Xu, (cid:88)u∀ ∈[0, t])}again for th(cid:16)e trajec(cid:89)tory of the red p(cid:12)(cid:12)article i, (cid:17)and A := (cid:101)X uv + t2/3, u [0, t] as in (4.6), we have (cid:12) t u E(#ECBRW) = ert P(kσ; t) E 1 1 X = kσ , t ≤ ≤ At ≤ DtCBRW(τj) t k:vt kσ vt+t2/3 j:τj t where, P(kσ; t) is the probability that a random walk is at position kσ at time t, and for all ∈ u [0, t], DCBRW(u) stands for the event that none of the particles in the subtree of BBM t ≥ branched at time u on the path of X coalesces with the red particle. Here, (τ , j 1) is j as before the atoms of a rate-2 Poisson process. 5 As we shall point out in Section , this picture is probably inaccurate, and is only due to the fact that our upper bound for Q(x, s) is not optimal. We conjecture that regardless of the value of v, the subtrees never make any particular effort in the N-BBM, which would be in complete contrast with the L-BBM. ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: With

text: the

text: notation

text: X

text: vu+σ(W

text: uW

text: (where

text: W

text: denotes

text: again

text: a

text: standard

text: Brownian

text: u

text: u

text: t

text: t−

text: motion

text: with

text: variance

text: we

text: have

text: X

text: X

text: vs

text: σ(W

text: W

text: sW

text: The

text: s+u

text: u

text: N

text: s+u−

text: u

text: t

text: t

text: random

text: variable

text: W

text: W

text: sW

text: has

text: the

text: Gaussian

text: s))

text: law.

text: As

text: for

text: the

text: s+u

text: u

text: t

text: t

text: t

text: L-BBM,

text: some

text: elementary

text: computations

text: yield

text: that,

text: in

text: case

text: v

text: the

text: subtrees

text: move

text: forward

text: faster

text: than

text: the

text: usual

text: speed

text: whereas

text: if

text: v

text: these

text: subtrees

text: make

text: no

text: particular

text: effort,

text: and

text: wait

text: only

text: for

text: the

text: occasions

text: when

text: the

text: red

text: particle

text: makes

text: some

text: toward

text: the

text: Letting

text: t

text: and

text: then

text: we

text: obtain:

text: E(#ENBBM)

text: exp

text: e

text: M

text: t

text: where

text: β(v)

text: is

text: deﬁned

text: in

text: and

text: o

text: stands

text: for

text: a

text: term

text: not

text: depending

text: on

text: t

text: such

text: that

text: N

text: limN→∞

text: Note

text: that

text: e

text: is

text: negligible

text: compared

text: to

text: M

text: This

text: yields

text: the

text: desired

text: lower

text: bound

text: First

text: moment

text: computations

text: for

text: the

text: CBRW

text: The

text: proof

text: for

text: the

text: CBRW

text: is

text: along

text: the

text: lines

text: of

text: the

text: proof

text: for

text: the

text: L-BBM

text: and

text: for

text: the

text: N-BBM.

text: Let

text: N

text: (t)

text: ECBRW

text: particle

text: i

text: lies

text: in

text: [vt,

text: leans

text: to

text: the

text: left,

text: does

text: not

text: coalesce

text: t

text: The

text: meaning

text: of

text: "leans

text: to

text: the

text: left"

text: is

text: as

text: before:

text: the

text: path

text: of

text: the

text: particle

text: lies

text: in

text: t

text: v

text: for

text: t

text: t].

text: By

text: "does

text: not

text: coalesce",

text: we

text: mean

text: that

text: at

text: no

text: t]

text: does

text: the

text: particle

text: coalesce

text: with

text: any

text: other

text: particle.

text: Let

text: #ECBRW

text: denote

text: the

text: number

text: of

text: i

text: satisfying

text: the

text: conditions

text: in

text: ECBRW.

text: The

text: main

text: t

text: t

text: estimates

text: for

text: the

text: CBRW

text: are:

text: E(#ECBRW)

text: exp

text: (f(v)

text: r

text: t

text: exp

text: (f(v)

text: r)t]

text: t

text: where

text: γ(v)

text: and

text: f(v)

text: are

text: deﬁned

text: in

text: and

text: respectively,

text: and

text: o

text: stands

text: for

text: a

text: µ

text: term

text: not

text: depending

text: on

text: t

text: such

text: that

text: lim

text: o

text: Equations

text: and

text: together

text: µ

text: µ

text: with

text: the

text: Cauchy–Schwarz

text: inequality

text: will

text: yield

text: the

text: upper

text: bound

text: stated

text: in

text: for

text: the

text: large

text: deviation

text: function

text: for

text: the

text: CBRW.

text: The

text: proof

text: of

text: which

text: is

text: identical

text: for

text: all

text: the

text: three

text: models,

text: is

text: postponed

text: to

text: Sub-

text: section

text: The

text: rest

text: of

text: this

text: subsection

text: is

text: devoted

text: to

text: the

text: proof

text: of

text: Wr{iting

text: ≤X

text: (Xu,

text: t])}again

text: for

text: of

text: the

text: red

text: i,

text: A

text: uv

text: u

text: t]

text: as

text: in

text: we

text: have

text: t

text: u

text: E(#ECBRW)

text: ert

text: P(kσ;

text: t)

text: E

text: X

text: kσ

text: t

text: At

text: DtCBRW(τj)

text: t

text: k:vt

text: kσ

text: j:τj

text: t

text: where,

text: P(kσ;

text: t)

text: is

text: the

text: probability

text: that

text: a

text: random

text: walk

text: is

text: at

text: position

text: kσ

text: at

text: time

text: t,

text: and

text: for

text: all

text: u

text: t],

text: DCBRW(u)

text: stands

text: for

text: the

text: event

text: that

text: of

text: the

text: particles

text: in

text: the

text: subtree

text: of

text: BBM

text: t

text: branched

text: at

text: time

text: u

text: on

text: the

text: path

text: of

text: X

text: coalesces

text: with

text: the

text: red

text: particle.

text: Here,

text: (τ

text: j

text: is

text: j

text: as

text: before

text: the

text: atoms

text: of

text: a

text: Poisson

text: process.

text: As

text: we

text: shall

text: point

text: out

text: in

text: Section

text: this

text: picture

text: is

text: probably

text: inaccurate,

text: and

text: is

text: only

text: due

text: to

text: the

text: fact

text: that

text: our

text: upper

text: bound

text: for

text: Q(x,

text: s)

text: is

text: not

text: optimal.

text: We

text: conjecture

text: that

text: regardless

text: of

text: the

text: value

text: of

text: v,

text: the

text: subtrees

text: never

text: make

text: any

text: particular

text: effort

text: in

text: the

text: N-BBM,

text: which

text: would

text: be

text: in

text: complete

text: contrast

text: with

text: the

text: L-BBM.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 6 2 The physical picture In this section we explain a heuristic way of understanding the claims in the introduc- tion. The main idea is rather similar for the three problems (see Figure 1). 2.1 The L-BBM Consider ﬁrst all the possible trees of a BBM which, starting with a single particle at the origin, contain at least one particle which reaches, at time t, a position on the right of vt at time t. √ Here we focus on velocities v > v (for the BBM one knows that v = 2σ). The c c probability that the tree has a(cid:20)t ti(cid:18)me t at lea(cid:19)st(cid:21)one parti(cid:20)cle(cid:18)on the r(cid:19)ig(cid:21)ht of vt is (1.2,1.3) for v > v c ∼ − v2 − v2 P exp t 1 = exp t 1 . (2.1) 2σ2 v2 c For each such tree event, we will call red particles all the particles which end up on the right of vt. Given its position at time t, the trajectory of a red particle is, up to a shift (linear in time), a Brownian bridge (in fact it is more like a Brownian excursion [14,1] but this has no incidence on the discussion below). When one goes from the BBM to the L-BBM, a red particle will survive if between time 0 and time t no other particle of the BBM overtakes it by a distance L. Any tree of the BBM for which a red particle survives contributes to the event that the the rightmost particle of L-BBM is on the right of vt. So the probability that a tree of the BBM reaches position vt and that at least one red particle is never overtaken by any other particle of the BBM by a distance L is a lower bound for the probablity that a L-BBM reaches position vt. This is why in the following, by estimating the survival probability of a red particle of a BBM, we will get an upper bound on the large deviation function (1.5) of the L-BBM. As a red particle is moving on average faster than v the only possibility for it to be c (cid:28) killed is that for a relatively short time interval s, i.e. a time s t, either this red particle moves slower than v, or one of the other particles of the tree moves sufﬁciently fast to overtake it by a distance L or both. So the picture is the following. A red particle moves at velocity v. Along its trajectory, branching events occur which give rise to subtrees. This red particle is then killed if, Fig. 1 A tree of the BBM which contributes to the large deviation function. The thick trajectory is the trajectory of a red particle. This red particle is a particle which ends up on the right of position vt. This tree will contribute to the large deviation function of the L-BBM, N-BBM or CBRW if the red particle is not killed by one of the subtrees which branch from its trajectory. ##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##

text: The

text: physical

text: picture

text: In

text: this

text: section

text: we

text: explain

text: a

text: heuristic

text: way

text: of

text: understanding

text: the

text: claims

text: in

text: the

text: introduc-

text: tion.

text: The

text: main

text: idea

text: is

text: rather

text: similar

text: for

text: the

text: three

text: problems

text: (see

text: Figure

text: The

text: L-BBM

text: Consider

text: ﬁrst

text: all

text: the

text: possible

text: trees

text: of

text: a

text: BBM

text: which,

text: starting

text: with

text: a

text: single

text: particle

text: at

text: the

text: origin,

text: contain

text: at

text: least

text: one

text: particle

text: which

text: reaches,

text: at

text: time

text: t,

text: a

text: position

text: on

text: the

text: right

text: of

text: vt

text: at

text: time

text: t.

text: Here

text: we

text: focus

text: on

text: velocities

text: v

text: v

text: (for

text: the

text: BBM

text: one

text: knows

text: that

text: v

text: The

text: c

text: c

text: probability

text: that

text: the

text: tree

text: has

text: t

text: at

text: the

text: of

text: vt

text: is

text: for

text: v

text: v

text: c

text: P

text: exp

text: t

text: exp

text: t

text: c

text: For

text: each

text: such

text: tree

text: event,

text: we

text: will

text: call

text: red

text: particles

text: all

text: the

text: particles

text: which

text: end

text: up

text: on

text: the

text: right

text: of

text: vt.

text: Given

text: its

text: position

text: at

text: time

text: t,

text: the

text: trajectory

text: of

text: a

text: red

text: particle

text: is,

text: up

text: to

text: a

text: shift

text: (linear

text: in

text: time),

text: a

text: Brownian

text: bridge

text: (in

text: fact

text: it

text: is

text: more

text: like

text: a

text: Brownian

text: excursion

text: but

text: this

text: has

text: no

text: incidence

text: on

text: the

text: discussion

text: below).

text: When

text: one

text: goes

text: from

text: the

text: BBM

text: to

text: the

text: L-BBM,

text: a

text: red

text: particle

text: will

text: survive

text: if

text: between

text: time

text: and

text: time

text: t

text: no

text: other

text: particle

text: of

text: the

text: BBM

text: overtakes

text: it

text: by

text: a

text: distance

text: L.

text: Any

text: tree

text: of

text: the

text: BBM

text: for

text: which

text: a

text: red

text: particle

text: survives

text: contributes

text: to

text: the

text: event

text: that

text: the

text: the

text: rightmost

text: particle

text: of

text: L-BBM

text: is

text: on

text: the

text: right

text: of

text: vt.

text: So

text: the

text: probability

text: that

text: a

text: tree

text: of

text: the

text: BBM

text: reaches

text: position

text: vt

text: and

text: that

text: at

text: least

text: one

text: red

text: particle

text: is

text: never

text: overtaken

text: by

text: any

text: other

text: particle

text: of

text: the

text: BBM

text: by

text: a

text: distance

text: L

text: is

text: a

text: lower

text: bound

text: for

text: the

text: probablity

text: that

text: a

text: L-BBM

text: reaches

text: position

text: vt.

text: This

text: is

text: why

text: in

text: the

text: following,

text: by

text: estimating

text: the

text: survival

text: probability

text: of

text: a

text: red

text: particle

text: of

text: a

text: BBM,

text: we

text: will

text: get

text: an

text: upper

text: bound

text: on

text: the

text: large

text: deviation

text: function

text: of

text: the

text: L-BBM.

text: As

text: a

text: red

text: particle

text: is

text: moving

text: on

text: average

text: faster

text: than

text: v

text: the

text: only

text: possibility

text: for

text: it

text: to

text: be

text: c

text: killed

text: is

text: that

text: for

text: a

text: relatively

text: short

text: time

text: interval

text: s,

text: i.e.

text: a

text: time

text: s

text: t,

text: either

text: this

text: red

text: particle

text: moves

text: slower

text: than

text: v,

text: or

text: one

text: of

text: the

text: other

text: particles

text: of

text: the

text: tree

text: moves

text: sufﬁciently

text: fast

text: to

text: overtake

text: it

text: by

text: a

text: distance

text: L

text: or

text: both.

text: So

text: the

text: picture

text: is

text: the

text: following.

text: A

text: red

text: particle

text: moves

text: at

text: velocity

text: v.

text: Along

text: its

text: trajectory,

text: branching

text: events

text: occur

text: which

text: give

text: rise

text: to

text: subtrees.

text: This

text: red

text: particle

text: is

text: then

text: killed

text: if,

text: Fig.

text: A

text: tree

text: of

text: the

text: BBM

text: which

text: contributes

text: to

text: the

text: large

text: deviation

text: function.

text: The

text: thick

text: trajectory

text: is

text: the

text: trajectory

text: of

text: a

text: red

text: particle.

text: This

text: red

text: particle

text: is

text: a

text: particle

text: which

text: ends

text: up

text: on

text: the

text: right

text: of

text: position

text: vt.

text: This

text: tree

text: will

text: contribute

text: to

text: the

text: large

text: deviation

text: function

text: of

text: the

text: L-BBM,

text: N-BBM

text: or

text: CBRW

text: if

text: the

text: red

text: particle

text: is

text: not

text: killed

text: by

text: one

text: of

text: the

text: subtrees

text: which

text: branch

text: from

text: its

text: trajectory.

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: 9 As v is solution of (2.3), one can check that the solution y of (2.12) reduces to the c → solution of (2.9) when y + L/s v , meaning that the rightmost particle of the (cid:112) c subtree moves at the velocity v . c − In the particular case where√f(v) = v2/(2σ2) the solution of (2.12) is y = (v − v2 2v2)/2 (where v = 2σ) and this leads to the second line of (1.7). c c 2.2 The N-BBM In the N-BBM, the picture is rather similar and one has to estimate the probability p that a subtree will kill a red particle. To do so one needs the red particle to slow down so that the subtree produces N particles ahead of the red particle to eliminate it. The probability that the red particle moves a distance x during time s is still given by (2.5). We now need to estimate the probability Q(x, s) that the subtree produces, at time s, N particles on the right of postion x. We do not have an expression for Q(x, s) (see the discusion in the conclusion for a conjecture). One can however obtain an easy upper bound (using the Markov inequality) (cid:104)N (cid:105) (x, s) Q(x, s) < N (cid:20) (cid:21) N where (x, s) is the number of particles of a subtree (of age s) on the right of position x. One has (cid:104)N (cid:105) ∼ − x2 (x, s) exp s 2σ2s (cid:104) (cid:105) so that Q(x, s) (cid:46) min 1, es−lnN−2σx22s (2.13) (cid:104) (cid:105) which, as for the L-BBM, we can write for more generality (cid:46) − − Q(x, s) min 1, es lnN sf(xs) (2.14) to treat the case of an arbitrary N-BBM. Now we need to ﬁnd a bound for p given by (2.7) and the discussion is very similar to what we did for the L-BBM: – If 1 dominates in (2.14), then x = ys where s and y are related by − − s ln N sf(y) = 0 . (2.15) The optimisation of (2.7) under the constraint (2.15) leads to ∼ − − (cid:48) p es[ f(y)+f(v)+(y v)f (v)] where y is solution of − (cid:48) − (cid:48) 1 f(y) + yf (y) 1 f(v) + vf (v) (cid:48) = (cid:48) . (2.16) f (y) f (v) (cid:54) [A solution y = v exists for v > v for the same reason as in (1.21).] One gets after c some algebra ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: As

label: list

text: v

text: is

text: solution

text: of

text: one

text: can

text: check

text: that

text: the

text: solution

text: y

text: of

text: reduces

text: to

text: the

text: c

text: solution

text: of

text: when

text: y

text: L/s

text: v

text: meaning

text: that

text: the

text: rightmost

text: particle

text: of

text: the

text: c

text: subtree

text: moves

text: at

text: the

text: velocity

text: v

text: c

text: In

text: the

text: particular

text: case

text: where√f(v)

text: the

text: solution

text: of

text: is

text: y

text: (v

text: (where

text: v

text: and

text: this

text: leads

text: to

text: the

text: second

text: line

text: of

text: c

text: c

text: The

text: N-BBM

text: In

text: the

text: N-BBM,

text: the

text: picture

text: is

text: rather

text: similar

text: and

text: one

text: has

text: to

text: estimate

text: the

text: probability

text: p

text: that

text: a

text: subtree

text: will

text: kill

text: a

text: red

text: particle.

text: To

text: do

text: so

text: one

text: needs

text: the

text: red

text: particle

text: to

text: slow

text: down

text: so

text: that

text: the

text: subtree

text: produces

text: N

text: particles

text: ahead

text: of

text: the

text: red

text: particle

text: to

text: eliminate

text: it.

text: The

text: probability

text: that

text: the

text: red

text: particle

text: moves

text: a

text: distance

text: x

text: during

text: time

text: s

text: is

text: still

text: given

text: by

text: We

text: now

text: need

text: to

text: estimate

text: the

text: probability

text: Q(x,

text: s)

text: that

text: the

text: subtree

text: produces,

text: at

text: time

text: s,

text: N

text: particles

text: on

text: the

text: right

text: of

text: postion

text: x.

text: We

text: do

text: not

text: have

text: an

text: expression

text: for

text: Q(x,

text: s)

text: (see

text: the

text: discusion

text: in

text: the

text: conclusion

text: for

text: a

text: conjecture).

text: One

text: can

text: however

text: obtain

text: an

text: easy

text: upper

text: bound

text: (using

text: the

text: Markov

text: inequality)

text: (x,

text: s)

text: Q(x,

text: s)

text: N

text: N

text: where

text: (x,

text: s)

text: is

text: the

text: number

text: of

text: particles

text: of

text: a

text: subtree

text: (of

text: age

text: s)

text: on

text: the

text: right

text: of

text: position

text: x.

text: One

text: has

text: (x,

text: s)

text: exp

text: s

text: so

text: that

text: Q(x,

text: s)

text: min

text: which,

text: as

text: for

text: the

text: L-BBM,

text: we

text: can

text: write

text: for

text: more

text: generality

text: Q(x,

text: s)

text: min

text: es

text: lnN

text: sf(xs)

text: to

text: treat

text: the

text: case

text: of

text: an

text: arbitrary

text: N-BBM.

text: Now

text: we

text: need

text: to

text: ﬁnd

text: a

text: bound

text: for

text: p

text: given

text: by

text: and

text: the

text: discussion

text: is

text: very

text: similar

text: to

text: what

text: we

text: did

text: for

text: the

text: L-BBM:

text: If

text: dominates

text: in

text: then

text: x

text: ys

text: where

text: s

text: and

text: y

text: are

text: related

text: by

text: s

text: ln

text: N

text: sf(y)

text: The

text: optimisation

text: of

text: under

text: the

text: constraint

text: leads

text: to

text: p

text: es[

text: f(y)+f(v)+(y

text: v)f

text: (v)]

text: where

text: y

text: is

text: solution

text: of

text: f(y)

text: yf

text: (y)

text: f(v)

text: vf

text: (v)

text: f

text: (y)

text: f

text: (v)

text: [A

text: solution

text: y

text: v

text: exists

text: for

text: v

text: v

text: for

text: the

text: same

text: reason

text: as

text: in

text: One

text: gets

text: after

text: c

text: some

text: algebra

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: May 28, 2020 10:49 ws-procs961x669 WSPC Proceedings - 9.61in x 6.69in ws-procs961x669 page 3 3 The action (3) can be reformulated in terms of geometrical quantities of the con- gruences of uµ, see13. The operators in this action are multiplied by the parameters ≡ ≡ ≡ − ≡ c c + c + 3c , c c + c , c c c , c c + c , (6) θ 1 3 2 σ 1 3 ω 1 3 a 1 4 which we will use henceforth. This is convenient because the predictions of the khronometric theory can be found from the Einstein-aether calculations in the limit → ∞ c . (7) ω 3. Degrees of freedom and matter ﬁelds The conﬁguration g = η and uµ = δµ is a solution of the equations of motion µν µν 0 derived from the action (3). The spectrum of perturbations around it consists of a tensor graviton mode, a vector and a scalar modes characterised by the velocities − − 1 c + c c c (c + 2c )(1 c /2) v2 = − , v2 = σ ω − σ ω , v2 = θ − σ a . (8) t 1 c v 2c (1 c ) s 3c (1 c )(1 + c /2) σ σ σ a σ θ Notice that the graviton’s velocity is modiﬁed. These velocities should all be real to avoid gradient instabilities. The absence of gravitational Cherenkov radiation requires that they are close to or greater than the speed of light. In the khronometric limit (7), the vector modes disappear (they become inﬁnitely heavy). Before discussing the phenomenological constraints in the previous parameters, it is worth mentioning that the most serious challenges of LV theories come from the matter sector. In fact, the ﬁeld u can couple to the ﬁelds in the standard model µ of particle physics (SM). This generates LV eﬀects whose study has produced very strong bounds on the possible couplings5,14. The consequences of these bounds for gravitation are not universal. If LV aﬀects identically the SM and gravitation the smallness of the parameters c will be beyond any foreseeable test. However, there i are mechanisms that separate the LV eﬀects in these sectors and make the SM tests irrelevant to constrain c . This may happen due to (softly broken) supersymmetry, i suppression mechanisms or by a dynamical emergence of LI at low energies in the SM (see5 for review). We will assume that the ﬁelds in the SM do not interact directly with the ﬁeld u . µ 4. Solar System constraints The LV parameters appearing in (3) are eﬃciently constrained by Solar System observations. For the latter, it is enough to consider the post-Newtonian (PN) limit of the equations of motion15. Unlike in GR, one needs to specify the velocity of the sources (the Sun in this case) with respect to the preferred frame represented by uµ (in particular its PN order). The most common assumption is that the preferred frame is basically aligned with the cosmic microwave background (CMB) frame. This is in agreement with requiring isotropy of the Universe at large scales. Under these conditions the Newton’s constant appearing in Newton’s law reads ≡ 1 G − , (9) N 4πM2 (2 c ) 0 a ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: May

text: WSPC

text: Proceedings

text: x

text: page

text: The

text: action

text: can

text: be

text: reformulated

text: in

text: terms

text: of

text: geometrical

text: quantities

text: of

text: the

text: con-

text: gruences

text: of

text: uµ,

text: The

text: operators

text: in

text: this

text: action

text: are

text: multiplied

text: by

text: the

text: parameters

text: c

text: c

text: c

text: c

text: c

text: c

text: c

text: c

text: c

text: c

text: c

text: c

text: θ

text: σ

text: ω

text: a

text: which

text: we

text: will

text: use

text: henceforth.

text: This

text: is

text: convenient

text: because

text: the

text: predictions

text: of

text: the

text: khronometric

text: theory

text: can

text: be

text: found

text: from

text: the

text: Einstein-aether

text: calculations

text: in

text: the

text: limit

text: c

text: ω

text: Degrees

text: of

text: freedom

text: and

text: matter

text: ﬁelds

text: The

text: conﬁguration

text: g

text: η

text: and

text: uµ

text: δµ

text: is

text: a

text: solution

text: of

text: the

text: equations

text: of

text: motion

text: µν

text: µν

text: derived

text: from

text: the

text: action

text: The

text: spectrum

text: of

text: perturbations

text: around

text: it

text: consists

text: of

text: a

text: tensor

text: graviton

text: mode,

text: a

text: vector

text: and

text: a

text: scalar

text: modes

text: characterised

text: by

text: the

text: velocities

text: c

text: c

text: c

text: c

text: (c

text: c

text: σ

text: ω

text: σ

text: ω

text: θ

text: σ

text: a

text: t

text: c

text: v

text: c

text: s

text: c

text: c

text: σ

text: σ

text: σ

text: a

text: σ

text: θ

text: Notice

text: that

text: the

text: graviton’s

text: velocity

text: is

text: modiﬁed.

text: These

text: velocities

text: should

text: all

text: be

text: real

text: to

text: avoid

text: gradient

text: instabilities.

text: The

text: absence

text: of

text: gravitational

text: Cherenkov

text: radiation

text: requires

text: that

text: they

text: are

text: close

text: to

text: or

text: greater

text: than

text: the

text: speed

text: of

text: light.

text: In

text: the

text: khronometric

text: limit

text: the

text: vector

text: modes

text: disappear

text: (they

text: become

text: inﬁnitely

text: heavy).

text: Before

text: discussing

text: the

text: phenomenological

text: constraints

text: in

text: the

text: previous

text: parameters,

text: it

text: is

text: worth

text: mentioning

text: that

text: the

text: most

text: serious

text: challenges

text: of

text: LV

text: theories

text: come

text: from

text: the

text: matter

text: sector.

text: In

text: fact,

text: the

text: ﬁeld

text: u

text: can

text: couple

text: to

text: the

text: ﬁelds

text: in

text: the

text: standard

text: model

text: µ

text: of

text: particle

text: physics

text: (SM).

text: This

text: generates

text: LV

text: eﬀects

text: whose

text: study

text: has

text: produced

text: very

text: strong

text: bounds

text: on

text: the

text: possible

text: The

text: consequences

text: of

text: these

text: bounds

text: for

text: gravitation

text: are

text: not

text: universal.

text: If

text: LV

text: aﬀects

text: identically

text: the

text: SM

text: and

text: gravitation

text: the

text: smallness

text: of

text: the

text: parameters

text: c

text: will

text: be

text: beyond

text: any

text: foreseeable

text: test.

text: However,

text: there

text: i

text: are

text: mechanisms

text: that

text: separate

text: the

text: LV

text: eﬀects

text: in

text: these

text: sectors

text: and

text: make

text: the

text: SM

text: tests

text: irrelevant

text: to

text: constrain

text: c

text: This

text: may

text: happen

text: due

text: to

text: (softly

text: broken)

text: supersymmetry,

text: i

text: suppression

text: mechanisms

text: or

text: by

text: a

text: dynamical

text: emergence

text: of

text: LI

text: at

text: low

text: energies

text: in

text: the

text: SM

text: for

text: review).

text: We

text: will

text: assume

text: that

text: the

text: ﬁelds

text: in

text: the

text: SM

text: do

text: not

text: interact

text: directly

text: with

text: the

text: ﬁeld

text: u

text: µ

text: Solar

text: System

text: constraints

text: The

text: LV

text: parameters

text: appearing

text: in

text: are

text: eﬃciently

text: constrained

text: by

text: Solar

text: System

text: observations.

text: For

text: the

text: latter,

text: it

text: is

text: enough

text: to

text: consider

text: the

text: post-Newtonian

text: (PN)

text: limit

text: of

text: the

text: equations

text: of

text: Unlike

text: in

text: GR,

text: one

text: needs

text: to

text: specify

text: the

text: velocity

text: of

text: the

text: sources

text: (the

text: Sun

text: in

text: this

text: case)

text: with

text: respect

text: to

text: the

text: preferred

text: frame

text: represented

text: by

text: uµ

text: (in

text: particular

text: its

text: PN

text: order).

text: The

text: most

text: common

text: assumption

text: is

text: that

text: the

text: preferred

text: frame

text: is

text: basically

text: aligned

text: with

text: the

text: cosmic

text: microwave

text: background

text: (CMB)

text: frame.

text: This

text: is

text: in

text: agreement

text: with

text: requiring

text: isotropy

text: of

text: the

text: Universe

text: at

text: large

text: scales.

text: Under

text: these

text: conditions

text: the

text: Newton’s

text: constant

text: appearing

text: in

text: Newton’s

text: law

text: reads

text: G

text: N

text: c

text: a

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: Title of your IAU Symposium (cid:13) Proceedings IAU Symposium No. xxx, 2008 c 2008 International Astronomical Union A.C. Editor, B.D. Editor & C.E. Editor, eds. DOI: 00.0000/X000000000000000X The Binggeli eﬀect Biernacka M.1, Panko E.2, God(cid:32)lowski W.3, Bajan K.4, Flin P.1 1Institute of Physics, Jan Kochanowski University, Kielce, Poland email: bmonika@ujk.edu.pl 2Kalinenkov Observatory, Nikolaev National University, Nikolaev, Ukraine 3Institute of Physics, Opole University, Opole, Poland 4Mt. Suhora Observatory, Cracow Pedagogical University, Krakow, Poland Abstract. We found the alignement of elongated clusters of BM type I and III (the excess − of small values of the ∆θ angles is observed), having range till about 60h 1Mpc. The ﬁrst one is probably connected with the origin of supergiant galaxy, while the second one with environmental eﬀects in clusters, originated on the long ﬁlament or plane. Keywords. galaxy cluster, alignment 1. Introduction Binggeli (1982) was the ﬁrst who found that galaxy clusters tend to be aligned pointing each other. Later on the existence of this eﬀect was discussed by several authors and usually the signiﬁcant alignment was reported. The distance between clusters for which − − the eﬀect was detected changed from 10h 1Mpc till 150h 1Mpc and the strength of the eﬀect diminished with distance (Struble & Peebles (1985), Flin (1987), Rhee & Katgert (1987), Ulmer et al. (1989), Plionis (1994), Chambers et al. (2002)). These investigation employed both optical and X-ray data, as well as clusters assigned or not to superclusters. Nowadays it is accepted that the eﬀect is real, not due to selection eﬀects, and it distance − − scale is between (10 60)h 1Mpc. Better understanding of physical processes leading to cluster alignment revealed numerical simulations. This was done in the framework of cold dark matter (CDM) model regarded now as being correct for large scale structure fomation (Onuora & Thomas (2000)) using large - scale simulation found that in LCDM − cosmological model for distance up to 30h 1Mpc, while in tCDM models the range of eﬀects is twice smaller. In SCDM and OCDM models, where smaller scale simuations were performed some alignment eﬀect could be noted. 2. Observational data The catalogue of 6188 galaxy clusters (hereafter PF, Panko & Flin (2006)) served as observational basis of our studies. It is prepared applying Voronoy tessellation technique (Ramella et al. 1999, 2001) to Muenster Red Sky Survey (MRSS, Ungruhe et al. (2003)). The PF catalogue is statistically complete till r = 18.3m. Our cluster contains at least 10 F galaxies in the brightnest range m , m +3m (m is the magnitude of the third brightest 3 3 3 galaxy). The distance to clusters were determined using the dependence between the tenth brightest galaxy m and redshift (Panko et al. (2009)). Clusters morphological 10 types for 1056 PF objects were taken from ACO (Abell et al. (1989)). 3. Method and results Method used to determine galaxy clusters shape (ellipticity and position angle) was covariance ellipse method (Carter & Metcalfe (1980)). The method is based on the ﬁrst ﬁve moments of the observed distribution of galaxy coordinates x , y . The existence i i 1 ##LTLine##

text: Title

text: of

text: your

text: IAU

text: Symposium

text: Proceedings

text: IAU

text: Symposium

text: No.

text: xxx,

text: c

text: International

text: Astronomical

text: Union

text: A.C.

text: Editor,

text: B.D.

text: Editor

text: C.E.

text: Editor,

text: eds.

text: DOI:

text: The

text: Binggeli

text: eﬀect

text: Biernacka

text: Panko

text: Bajan

text: Flin

text: of

text: Physics,

text: Jan

text: Kochanowski

text: University,

text: Kielce,

text: Poland

text: email:

text: bmonika@ujk.edu.pl

text: Observatory,

text: Nikolaev

text: National

text: University,

text: Nikolaev,

text: Ukraine

text: of

text: Physics,

text: Opole

text: University,

text: Opole,

text: Poland

text: Suhora

text: Observatory,

text: Cracow

text: Pedagogical

text: University,

text: Krakow,

text: Poland

text: Abstract.

text: We

text: found

text: the

text: alignement

text: of

text: elongated

text: clusters

text: of

text: BM

text: type

text: I

text: and

text: III

text: (the

text: excess

text: of

text: small

text: values

text: of

text: the

text: ∆θ

text: angles

text: is

text: observed),

text: having

text: range

text: till

text: about

text: The

text: ﬁrst

text: one

text: is

text: probably

text: connected

text: with

text: the

text: origin

text: of

text: supergiant

text: galaxy,

text: while

text: the

text: second

text: one

text: with

text: environmental

text: eﬀects

text: in

text: clusters,

text: originated

text: on

text: the

text: long

text: ﬁlament

text: or

text: plane.

text: Keywords.

text: galaxy

text: cluster,

text: alignment

text: Introduction

text: Binggeli

text: was

text: the

text: ﬁrst

text: who

text: found

text: that

text: galaxy

text: clusters

text: tend

text: to

text: be

text: aligned

text: pointing

text: each

text: other.

text: Later

text: on

text: the

text: existence

text: of

text: this

text: eﬀect

text: was

text: discussed

text: by

text: several

text: authors

text: and

text: usually

text: the

text: signiﬁcant

text: alignment

text: was

text: reported.

text: The

text: distance

text: between

text: clusters

text: for

text: which

text: the

text: eﬀect

text: was

text: detected

text: changed

text: from

text: till

text: and

text: the

text: strength

text: of

text: the

text: eﬀect

text: diminished

text: with

text: distance

text: (Struble

text: Peebles

text: Flin

text: Rhee

text: Katgert

text: Ulmer

text: et

text: al.

text: Plionis

text: Chambers

text: et

text: al.

text: These

text: investigation

text: employed

text: both

text: optical

text: and

text: X-ray

text: data,

text: as

text: well

text: as

text: clusters

text: assigned

text: or

text: not

text: to

text: superclusters.

text: Nowadays

text: it

text: is

text: accepted

text: that

text: the

text: eﬀect

text: is

text: real,

text: not

text: due

text: to

text: selection

text: eﬀects,

text: and

text: it

text: distance

text: scale

text: is

text: between

text: Better

text: understanding

text: of

text: physical

text: processes

text: leading

text: to

text: cluster

text: alignment

text: revealed

text: numerical

text: simulations.

text: This

text: was

text: done

text: in

text: the

text: framework

text: of

text: cold

text: dark

text: matter

text: (CDM)

text: model

text: regarded

text: now

text: as

text: being

text: correct

text: for

text: large

text: scale

text: structure

text: fomation

text: (Onuora

text: Thomas

text: using

text: large

text: scale

text: simulation

text: found

text: that

text: in

text: LCDM

text: cosmological

text: model

text: for

text: distance

text: up

text: to

text: while

text: in

text: tCDM

text: models

text: the

text: range

text: of

text: eﬀects

text: is

text: twice

text: smaller.

text: In

text: SCDM

text: and

text: OCDM

text: models,

text: where

text: smaller

text: scale

text: simuations

text: were

text: performed

text: some

text: alignment

text: eﬀect

text: could

text: be

text: noted.

text: Observational

text: data

text: The

text: catalogue

text: of

text: galaxy

text: clusters

text: (hereafter

text: PF,

text: Panko

text: Flin

text: served

text: as

text: observational

text: basis

text: of

text: our

text: studies.

text: It

text: is

text: prepared

text: applying

text: Voronoy

text: tessellation

text: technique

text: (Ramella

text: et

text: al.

text: to

text: Muenster

text: Red

text: Sky

text: Survey

text: (MRSS,

text: Ungruhe

text: et

text: al.

text: The

text: PF

text: catalogue

text: is

text: statistically

text: complete

text: till

text: r

text: Our

text: cluster

text: contains

text: at

text: least

text: F

text: galaxies

text: in

text: the

text: brightnest

text: range

text: m

text: m

text: (m

text: is

text: the

text: magnitude

text: of

text: the

text: third

text: brightest

text: galaxy).

text: The

text: distance

text: to

text: clusters

text: were

text: determined

text: using

text: the

text: dependence

text: between

text: the

text: tenth

text: brightest

text: galaxy

text: m

text: and

text: redshift

text: (Panko

text: et

text: al.

text: Clusters

text: morphological

text: types

text: for

text: PF

text: objects

text: were

text: taken

text: from

text: ACO

text: (Abell

text: et

text: al.

text: Method

text: and

text: results

text: Method

text: used

text: to

text: determine

text: galaxy

text: clusters

text: shape

text: (ellipticity

text: and

text: position

text: angle)

text: was

text: covariance

text: ellipse

text: method

text: (Carter

text: Metcalfe

text: The

text: method

text: is

text: based

text: on

text: the

text: ﬁrst

text: ﬁve

text: moments

text: of

text: the

text: observed

text: distribution

text: of

text: galaxy

text: coordinates

text: x

text: y

text: The

text: existence

text: i

text: i

text: ##LTLine##


text: 8 − † − Appendix A: Equations of motion on the lattice tte U(cid:50)(t, i, j, −yˆ) becomes the matr−ix produc†t U (t, i, j 1, +yˆ)U(t, i, j 1, +xˆ)U(t, i + 1, j 1, +yˆ)U (t, i, j, +xˆ). We solved the equations of motion on a two-dimensional The lattice energy functional is (cid:88) spacial lattice whose lattice spacing we denote by “a”. The matter ﬁeld and its conjugate momentum are repre- × H H H H sented by 3 3 complex matrices Φ and Π with one color E = ( + + + ), lattice magnetic electric potential kinetic and one ﬂavor index which live on the lattice sites. The × ij gauge ﬁelds are 3 3 unitary matrices U with two color (cid:18) (cid:19) (A1) indices living on the links, and their time derivative is en- where coded in the electric ﬁeld E which is a Hermitian matrix. In what follows, i and j are discretized versions of spatial H 6 (cid:2) − 1(cid:60) (cid:3) = 1 Tr [U (t, i, j, +xˆ)] , (A2) coordinates x and y, and t represents time. U(t, i, j, +µˆ) magnetic g2a2 3 (cid:50) represents a gauge link at time t that emanates from the (cid:2) (cid:3) H 1 site (i, j) along the µˆ direction. We perform our calcu- = Tr E(t, i, j)2 , (A3) electric 2 (cid:2) (cid:3) lcaotniostnrsucint athpelatqeumepttoeraalt gtaimugeet,, Adet(nto,tie,dj)b=y U0(cid:50). (tO, in,ej,cµˆa)n, Hpotential = Tr −m2Φ(t, i, j)†Φ(t, i, j) (A4) † by starting at site (i, j) and going in µˆ direction. After +Tr λ (Φ(t, i, j) Φ(t, i, j))2 2 each step, the orientation is changed by a 90-degree turn (cid:2) (cid:3) † 3m4 to the left, such that the plaquette closes after four steps. +λ (Tr[Φ(t, i, j) Φ(t, i, j)])2 + , 1 4λ At any given time t, starting at site (i, j), one can thus H † construct four diﬀerent elementary plaquettes, depending kinetic = Tr Π(t, i, j) Π(t, i, j) . (A5) on the direction of the initial step. The plaquette then corresponds to the product of the four link variables in The lattice equations of motion derived from the above the order of their appearance. For example, the plaque- energy functional are (cid:2) (cid:3) ∇ − † − † Π˙ (t, i, j) = 2Φ(t, i, j) + m2Φ(t, i, j) 2λ Tr Φ(t, i, j) Φ(t, i, j) Φ(t, i, j) 2λ Φ(t, i, j)Φ(t, i, j) Φ(t, i, j), (A6) 1 2 (cid:0) (cid:1) where ∇ 1 (cid:0) † − − − (cid:1) 2Φ(t, i, j) = U(t, i, j, +xˆ)Φ(t, i + 1, j) + U (t, i 1, j, +xˆ)Φ(t, i 1, j) 2Φ(t, i, j) a2 1 † − − − (cid:18) (cid:20) +(cid:18) U(t, i, j, +yˆ)Φ(t, i, j + 1) + U (t, i, j 1, +yˆ)Φ(t, i, j 1) 2Φ(t, i, j) , (cid:19)(cid:21)(cid:19) (A7) a2 (cid:0) (cid:1) 2 − − 2g † E˙ α(t, i, j) = Im Tr Tα (U (t, i, j, +xˆ) U (t, i, j, yˆ)) + U(t, i, j, +xˆ)Φ(t, i + 1, j)Φ (t, i, j) , (A8) x (cid:18) (cid:20) (cid:18)ga (cid:50) (cid:50) a (cid:19)(cid:21)(cid:19) (cid:0) (cid:1) 2 − 2g † E˙ α(t, i, j) = Im Tr Tα (U (t, i, j, +yˆ) U (t, i, j, +xˆ)) + U(t, i, j, +yˆ)Φ(t, i, j + 1)Φ (t, i, j) , (A9) y ga (cid:50) (cid:50)(cid:2) a (cid:3) − U(t + δt, i, j, +µˆ) = exp iga δt Eα(t + δt, i, j)Tα U(t, i, j, +µˆ), (A10) µ where Tα’s are the SU(3) generators. We choose the normalization Tr[TαTβ] = δαβ. Φ(t + δt, i, j) = Φ(t, i, j) + δt Π(t + δt, i, j). (A11) In order to obtain the initial superﬂuid vortex conﬁg- conﬁguration and use it as the input conﬁguration for uration on the lattice, we use the Langevin evolution our numerical studies on stability. In the Langevin im- method. The continuum proﬁle of the superﬂuid vor- plementation, equations (A10) and (A11) are modiﬁed as tex is ﬁrst evolved using the Langevin approach, with follows : temperature set to zero. We then take the ﬁnal relaxed (cid:104) (cid:16) (cid:16) (cid:17)(cid:17) (cid:105) − − U(t + δt, i, j, +µˆ) = exp iga δt Eα(t, i, j) + δt E˙ α(t, i, j) ηEα(t, i, j) + ζα(t, i, j) Tα U(t, i, j, +µˆ), (A12) µ µ µ ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Appendix

text: A:

text: Equations

text: of

text: motion

text: on

text: the

text: lattice

text: tte

text: i,

text: j,

text: −yˆ)

text: becomes

text: the

text: matr−ix

text: produc†t

text: U

text: (t,

text: i,

text: j

text: +yˆ)U(t,

text: i,

text: j

text: +xˆ)U(t,

text: i

text: j

text: +yˆ)U

text: (t,

text: i,

text: j,

text: +xˆ).

text: We

text: solved

text: the

text: equations

text: of

text: motion

text: on

text: a

text: two-dimensional

text: The

text: lattice

text: energy

text: functional

text: is

text: spacial

text: lattice

text: whose

text: lattice

text: spacing

text: we

text: denote

text: by

text: “a”.

text: The

text: matter

text: ﬁeld

text: and

text: its

text: conjugate

text: momentum

text: are

text: repre-

text: H

text: H

text: H

text: H

text: sented

text: by

text: complex

text: matrices

text: Φ

text: and

text: Π

text: with

text: one

text: color

text: E

text: lattice

text: magnetic

text: electric

text: potential

text: kinetic

text: and

text: one

text: ﬂavor

text: index

text: which

text: live

text: on

text: the

text: lattice

text: sites.

text: The

text: ij

text: gauge

text: ﬁelds

text: are

text: unitary

text: matrices

text: U

text: with

text: two

text: color

text: indices

text: living

text: on

text: the

text: links,

text: and

text: their

text: time

text: derivative

text: is

text: en-

text: where

text: coded

text: in

text: the

text: electric

text: ﬁeld

text: E

text: which

text: is

text: a

text: Hermitian

text: matrix.

text: In

text: what

text: follows,

text: i

text: and

text: j

text: are

text: discretized

text: versions

text: of

text: spatial

text: H

text: Tr

text: [U

text: (t,

text: i,

text: j,

text: +xˆ)]

text: coordinates

text: x

text: and

text: y,

text: and

text: t

text: represents

text: time.

text: U(t,

text: i,

text: j,

text: +µˆ)

text: magnetic

text: represents

text: a

text: gauge

text: link

text: at

text: time

text: t

text: that

text: emanates

text: from

text: the

text: H

text: site

text: (i,

text: j)

text: along

text: the

text: µˆ

text: direction.

text: We

text: perform

text: our

text: calcu-

text: Tr

text: E(t,

text: i,

text: electric

text: lcaotniostnrsucint

text: athpelatqeumepttoeraalt

text: gtaimugeet,,

text: Adet(nto,tie,dj)b=y

text: (tO,

text: in,ej,cµˆa)n,

text: Hpotential

text: Tr

text: i,

text: j)†Φ(t,

text: i,

text: j)

text: by

text: starting

text: at

text: site

text: (i,

text: j)

text: and

text: going

text: in

text: µˆ

text: direction.

text: After

text: +Tr

text: λ

text: (Φ(t,

text: i,

text: j)

text: Φ(t,

text: i,

text: each

text: step,

text: the

text: orientation

text: is

text: changed

text: by

text: a

text: turn

text: to

text: the

text: left,

text: such

text: that

text: the

text: plaquette

text: closes

text: after

text: four

text: steps.

text: +λ

text: (Tr[Φ(t,

text: i,

text: j)

text: Φ(t,

text: i,

text: 4λ

text: At

text: any

text: given

text: time

text: t,

text: starting

text: at

text: site

text: (i,

text: j),

text: one

text: can

text: thus

text: H

text: construct

text: four

text: diﬀerent

text: elementary

text: plaquettes,

text: depending

text: kinetic

text: Tr

text: Π(t,

text: i,

text: j)

text: Π(t,

text: i,

text: j)

text: on

text: the

text: direction

text: of

text: the

text: initial

text: step.

text: The

text: plaquette

text: then

text: corresponds

text: to

text: the

text: product

text: of

text: the

text: four

text: link

text: variables

text: in

text: The

text: lattice

text: equations

text: of

text: motion

text: derived

text: from

text: the

text: above

text: the

text: order

text: of

text: their

text: appearance.

text: For

text: example,

text: the

text: plaque-

text: energy

text: functional

text: are

text: Π˙

text: (t,

text: i,

text: j)

text: i,

text: j)

text: i,

text: j)

text: 2λ

text: Tr

text: Φ(t,

text: i,

text: j)

text: Φ(t,

text: i,

text: j)

text: Φ(t,

text: i,

text: j)

text: 2λ

text: Φ(t,

text: i,

text: j)Φ(t,

text: i,

text: j)

text: Φ(t,

text: i,

text: j),

text: where

text: i,

text: j)

text: U(t,

text: i,

text: j,

text: +xˆ)Φ(t,

text: i

text: j)

text: U

text: (t,

text: i

text: j,

text: +xˆ)Φ(t,

text: i

text: j)

text: i,

text: j)

text: U(t,

text: i,

text: j,

text: +yˆ)Φ(t,

text: i,

text: j

text: U

text: (t,

text: i,

text: j

text: +yˆ)Φ(t,

text: i,

text: j

text: i,

text: j)

text: E˙

text: α(t,

text: i,

text: j)

text: Im

text: Tr

text: Tα

text: (U

text: (t,

text: i,

text: j,

text: +xˆ)

text: U

text: (t,

text: i,

text: j,

text: yˆ))

text: U(t,

text: i,

text: j,

text: +xˆ)Φ(t,

text: i

text: j)Φ

text: (t,

text: i,

text: j)

text: x

text: a

text: E˙

text: α(t,

text: i,

text: j)

text: Im

text: Tr

text: Tα

text: (U

text: (t,

text: i,

text: j,

text: +yˆ)

text: U

text: (t,

text: i,

text: j,

text: +xˆ))

text: U(t,

text: i,

text: j,

text: +yˆ)Φ(t,

text: i,

text: j

text: (t,

text: i,

text: j)

text: y

text: ga

text: a

text: U(t

text: δt,

text: i,

text: j,

text: +µˆ)

text: exp

text: iga

text: δt

text: Eα(t

text: δt,

text: i,

text: j)Tα

text: U(t,

text: i,

text: j,

text: +µˆ),

text: µ

text: where

text: Tα’s

text: are

text: the

text: generators.

text: We

text: choose

text: the

text: normalization

text: Tr[TαTβ]

text: δαβ.

text: Φ(t

text: δt,

text: i,

text: j)

text: Φ(t,

text: i,

text: j)

text: δt

text: Π(t

text: δt,

text: i,

text: j).

text: In

text: order

text: to

text: obtain

text: the

text: initial

text: superﬂuid

text: vortex

text: conﬁg-

text: conﬁguration

text: and

text: use

text: it

text: as

text: the

text: input

text: conﬁguration

text: for

text: uration

text: on

text: the

text: lattice,

text: we

text: use

text: the

text: Langevin

text: evolution

text: our

text: numerical

text: studies

text: on

text: stability.

text: In

text: the

text: Langevin

text: im-

text: method.

text: The

text: continuum

text: proﬁle

text: of

text: the

text: superﬂuid

text: vor-

text: plementation,

text: equations

text: and

text: are

text: modiﬁed

text: as

text: tex

text: is

text: ﬁrst

text: evolved

text: using

text: the

text: Langevin

text: approach,

text: with

text: follows

text: temperature

text: set

text: to

text: zero.

text: We

text: then

text: take

text: the

text: ﬁnal

text: relaxed

text: U(t

text: δt,

text: i,

text: j,

text: +µˆ)

text: exp

text: iga

text: δt

text: Eα(t,

text: i,

text: j)

text: δt

text: E˙

text: α(t,

text: i,

text: j)

text: ηEα(t,

text: i,

text: j)

text: ζα(t,

text: i,

text: j)

text: Tα

text: U(t,

text: i,

text: j,

text: +µˆ),

text: µ

text: µ

text: µ

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