text: we will suppose there exists constants C, r > d/2, and s > d such that ≤ (cid:107) (cid:107)− | | ≤ (cid:107) (cid:107) − ρˆ (b) C b 2r, ρ (p) C(1 + p ) s. (3.15) p p This, e.g., allows the application of our results to situations such that − dν = δ(x a )dx, (3.16) 0 in which all particles are initially located at the same position a and 0 in which, therefore, the only randomness that appears is in the choice of the initial momenta. We now show how to establish (3.4) for this class of measures. The constant C will depend on C, r, and s. For that purpose, it is in turn µ (cid:88) suﬃcient to establish the analog of (3.7), i.e., E (χ ) = χ˜ ν˜((cid:96))ρˆ (2π(cid:96)t), (3.17) I,t (cid:96) p (cid:90)∈Z (cid:96) d where ν˜((cid:96)) = exp(i2π(cid:96)x)dν. T d ∞ To that end, we consider a sequence χ(n) of C functions that ap- I proximate χ in the L2-norm I (cid:107) − (cid:107) lim χ(n) χ = 0. (3.18) n→+∞ I I 2 Writing χ˜(n) for their Fourier coeﬃcients, it follows that lim → ∞ χ˜(n) = (cid:96) n + (cid:96) χ˜(cid:96) and the smoothness of the∈χ(cid:88)(ITn) implies that the Fourier coeﬃcients de- cay fast in (cid:96). Then, for all x d, − χ(n)(x) = χ˜(n) exp( i2π(cid:96)x). I (cid:96) (cid:96) The convergence of this series is uniform in x because of the fast decay of ∈ R the Fourier coeﬃ(cid:90)cien(cid:18)ts(cid:90). Then, for all t , (cid:19) E (cid:90) (cid:18)(cid:90) (cid:16) (cid:17) (cid:19) (χ ) = χ (x + pt))ρ (p)dp dν(x) I,t I p T R d d (cid:90) (cid:18)(cid:90) − (cid:19) = χ (x + pt) χ(n)(x + pt) ρ (p)dp dν(x) T R I I p d d + χ(n)(x + pt))ρ (p)dp dν(x). T R I p d d (3.19) ∈ Z ∈ Z Let C , (cid:96) d be the family of cubes centered at (cid:96) d and of linear (cid:96) ∈ T size 1. Then we can write, for all x d, 9

text: we

text: will

text: suppose

text: there

text: exists

text: constants

text: C,

text: r

text: and

text: s

text: d

text: such

text: that

text: ρˆ

text: (b)

text: C

text: b

text: ρ

text: (p)

text: p

text: s.

text: p

text: p

text: This,

text: e.g.,

text: allows

text: the

text: application

text: of

text: our

text: results

text: to

text: situations

text: such

text: that

text: dν

text: δ(x

text: a

text: )dx,

text: in

text: which

text: all

text: particles

text: are

text: initially

text: located

text: at

text: the

text: same

text: position

text: a

text: and

text: in

text: which,

text: therefore,

text: the

text: only

text: randomness

text: that

text: appears

text: is

text: in

text: the

text: choice

text: of

text: the

text: initial

text: momenta.

text: We

text: now

text: show

text: how

text: to

text: establish

text: for

text: this

text: class

text: of

text: measures.

text: The

text: constant

text: C

text: will

text: depend

text: on

text: C,

text: r,

text: and

text: s.

text: For

text: that

text: purpose,

text: it

text: is

text: in

text: turn

text: µ

text: suﬃcient

text: to

text: establish

text: the

text: analog

text: of

text: i.e.,

text: E

text: (χ

text: χ˜

text: I,t

text: p

text: d

text: where

text: T

text: d

text: To

text: that

text: end,

text: we

text: consider

text: a

text: sequence

text: χ(n)

text: of

text: C

text: functions

text: that

text: ap-

text: I

text: proximate

text: χ

text: in

text: the

text: I

text: lim

text: χ(n)

text: χ

text: n→+∞

text: I

text: I

text: Writing

text: χ˜(n)

text: for

text: their

text: Fourier

text: coeﬃcients,

text: it

text: follows

text: that

text: lim

text: χ˜(n)

text: n

text: and

text: the

text: smoothness

text: of

text: implies

text: that

text: the

text: Fourier

text: coeﬃcients

text: de-

text: cay

text: fast

text: in

text: Then,

text: for

text: all

text: x

text: d,

text: χ(n)(x)

text: χ˜(n)

text: exp(

text: I

text: The

text: convergence

text: of

text: this

text: series

text: is

text: uniform

text: in

text: x

text: because

text: of

text: the

text: fast

text: decay

text: of

text: R

text: the

text: Fourier

text: Then,

text: for

text: all

text: t

text: E

text: (χ

text: χ

text: (x

text: pt))ρ

text: (p)dp

text: dν(x)

text: I,t

text: I

text: p

text: T

text: R

text: d

text: d

text: χ

text: (x

text: pt)

text: χ(n)(x

text: pt)

text: ρ

text: (p)dp

text: dν(x)

text: T

text: R

text: I

text: I

text: p

text: d

text: d

text: χ(n)(x

text: pt))ρ

text: (p)dp

text: dν(x).

text: T

text: R

text: I

text: p

text: d

text: d

text: Z

text: Z

text: Let

text: C

text: d

text: be

text: the

text: family

text: of

text: cubes

text: centered

text: at

text: d

text: and

text: of

text: linear

text: T

text: size

text: Then

text: we

text: can

text: write,

text: for

text: all

text: x

text: d,


text: (cid:90) (cid:16) (cid:17) | χ ((cid:88)x +(cid:90)pt)(cid:16)− χ(n)(x + pt) ρ (p)dp|(cid:17)= R I I p d t1d | χI(x +(cid:88)y) −(cid:18)χ(cid:90)(In)(x + y) ρ(cid:19)p(y/t)dy| (cid:96) C(cid:96) ≤ 1 (cid:107) − (cid:107) 1/2 χ χ(n) ρ2(y/t)dy td I I 2 p (cid:96) C(cid:96) ≤ (cid:107) − (cid:107) C˜ χ χ(n) . s,t I I 2 (cid:90) (cid:18)(cid:90) ∈ R (cid:19) Inserting this into (3.19), one ﬁnds, for all t , E(χ ) = lim χ(n)(x + pt)ρ (p)dp dν(x) I,t n→+∞ Td Rd(cid:88)I p Hence E(χ ) = lim χ˜(n)ν˜((cid:96))ρˆ (2π(cid:96)t). I,t n→+∞ ∈Z (cid:96) p (cid:96) d Using condition (3.15) and bounded convergence, one ﬁnds ﬁnally (3.17) and hence (3.4). The model we study here is a variation of one that was introduced in [Fr58], where a result equivalent to (3.9) is proven. In [Be10] (Theo- rem 3) a result similar to (3.10) is proven on a closely related model. Equa- ⊂ T tions (3.9)-(3.10) mean that for large enough N and t, and for all I d, | | the coarse-grained random variable f has a mean close to I and a very I { | ∈ { }} small variance. This implies that for any partition I α 1, . . . , L of α the conﬁguration space, the coarse-grained time-dependent density pro- | | ﬁle ρ (t) = Nf (X, P, t))/ I is, on average, and for suﬃciently large N α Iα α and t, close to being spatially homogeneous. This provides a weak sense of the approach to spatial homogeneity. Indeed, the idea is that when a random variable has a very small variance it is “almost constant”. In that | | sense, one may claim that f (X, P, t) is “typically” equal to I , provided Iα α N and t are large enough. Note however that (3.9)-(3.10) do not as such give any information about the approach of the coarse-grained observables | | ∈ f (X, P, t) to I for any particular initial condition (X, P) Γ , i.e., Iα α N about the evolution of individual macroscopic systems. Figure 2, on the other hand, illustrates for single trajectories of individual N-particle sys- tems, how typical ﬂuctuations in f about its equilibrium value decrease I with increasing particle number N, but do not decay to zero as a func- tion of time. In this ﬁgure, individual trajectories, after a relatively brief initial equilibration time that does not appear to depend on N, display values of f that ﬂuctuate about the expected mean value of 0.5. I − | | Note that, according to (3.9), the average of f I over many such I trajectories will approach zero with increasing t. On the other hand, as is evident from the data in Fig. 2, and as is conﬁrmed below in (3.13), the variance of that quantity at ﬁxed N approaches, for large times, a ﬁnite value, which can be estimated from the size of the temporal ﬂuctuations appearing in the ﬁgure. Numerical results evaluated at times very much 10 ##LTLine## ##LTLine##

label: equation

text: χ

text: χ(n)(x

text: pt)

text: ρ

text: R

text: I

text: I

text: p

text: d

text: χI(x

text: y)

text: χ

text: χ(n)

text: td

text: I

text: I

text: p

text: C˜

text: χ

text: χ(n)

text: s,t

text: I

text: I

label: paragraph

text: R

text: Inserting

text: this

text: into

text: one

text: ﬁnds,

text: for

text: all

text: t

text: E(χ

text: lim

text: χ(n)(x

text: pt)ρ

text: (p)dp

text: dν(x)

text: I,t

text: n→+∞

text: Td

text: p

text: Hence

text: E(χ

text: lim

text: I,t

text: n→+∞

text: ∈Z

text: p

text: d

text: Using

text: condition

text: and

text: bounded

text: convergence,

text: one

text: ﬁnds

text: ﬁnally

text: and

text: hence

text: The

text: model

text: we

text: study

text: here

text: is

text: a

text: variation

text: of

text: one

text: that

text: was

text: introduced

text: in

text: where

text: a

text: result

text: equivalent

text: to

text: is

text: proven.

text: In

text: (Theo-

text: rem

text: a

text: result

text: similar

text: to

text: is

text: proven

text: on

text: a

text: closely

text: related

text: model.

text: Equa-

text: T

text: tions

text: mean

text: that

text: for

text: large

text: enough

text: N

text: and

text: t,

text: and

text: for

text: all

text: I

text: d,

text: the

text: coarse-grained

text: random

text: variable

text: f

text: has

text: a

text: mean

text: close

text: to

text: I

text: and

text: a

text: very

text: I

text: small

text: variance.

text: This

text: implies

text: that

text: for

text: any

text: partition

text: I

text: α

text: L

text: of

text: α

text: the

text: conﬁguration

text: space,

text: the

text: coarse-grained

text: time-dependent

text: density

text: pro-

text: ﬁle

text: ρ

text: (t)

text: Nf

text: (X,

text: P,

text: t))/

text: I

text: is,

text: on

text: average,

text: and

text: for

text: suﬃciently

text: large

text: N

text: α

text: Iα

text: α

text: and

text: t,

text: close

text: to

text: being

text: spatially

text: homogeneous.

text: This

text: provides

text: a

text: weak

text: sense

text: of

text: the

text: approach

text: to

text: spatial

text: homogeneity.

text: Indeed,

text: the

text: idea

text: is

text: that

text: when

text: a

text: random

text: variable

text: has

text: a

text: very

text: small

text: variance

text: it

text: is

text: “almost

text: constant”.

text: In

text: that

text: sense,

text: one

text: may

text: claim

text: that

text: f

text: (X,

text: P,

text: t)

text: is

text: “typically”

text: equal

text: to

text: I

text: provided

text: Iα

text: α

text: N

text: and

text: t

text: are

text: large

text: enough.

text: Note

text: however

text: that

text: do

text: not

text: as

text: such

text: give

text: any

text: information

text: about

text: the

text: approach

text: of

text: the

text: coarse-grained

text: observables

text: f

text: (X,

text: P,

text: t)

text: to

text: I

text: for

text: any

text: particular

text: initial

text: condition

text: (X,

text: P)

text: Γ

text: i.e.,

text: Iα

text: α

text: N

text: about

text: the

text: evolution

text: of

text: individual

text: macroscopic

text: systems.

text: Figure

text: on

text: the

text: other

text: hand,

text: illustrates

text: for

text: single

text: trajectories

text: of

text: individual

text: N-particle

text: sys-

text: tems,

text: how

text: typical

text: ﬂuctuations

text: in

text: f

text: about

text: its

text: equilibrium

text: value

text: decrease

text: I

text: with

text: increasing

text: particle

text: number

text: N,

text: but

text: do

text: not

text: decay

text: to

text: zero

text: as

text: a

text: func-

text: tion

text: of

text: time.

text: In

text: this

text: ﬁgure,

text: individual

text: trajectories,

text: after

text: a

text: relatively

text: brief

text: initial

text: equilibration

text: time

text: that

text: does

text: not

text: appear

text: to

text: depend

text: on

text: N,

text: display

text: values

text: of

text: f

text: that

text: ﬂuctuate

text: about

text: the

text: expected

text: mean

text: value

text: of

text: I

text: Note

text: that,

text: according

text: to

text: the

text: average

text: of

text: f

text: I

text: over

text: many

text: such

text: I

text: trajectories

text: will

text: approach

text: zero

text: with

text: increasing

text: t.

text: On

text: the

text: other

text: hand,

text: as

text: is

text: evident

text: from

text: the

text: data

text: in

text: Fig.

text: and

text: as

text: is

text: conﬁrmed

text: below

text: in

text: the

text: variance

text: of

text: that

text: quantity

text: at

text: ﬁxed

text: N

text: approaches,

text: for

text: large

text: times,

text: a

text: ﬁnite

text: value,

text: which

text: can

text: be

text: estimated

text: from

text: the

text: size

text: of

text: the

text: temporal

text: ﬂuctuations

text: appearing

text: in

text: the

text: ﬁgure.

text: Numerical

text: results

text: evaluated

text: at

text: times

text: very

text: much

text: ##LTLine##

text: ##LTLine##


text: Random partitions of the plane via Poissonian coloring, and a self-similar process of coalescing planar partitions ∗ David J. Aldous May 29, 2020 Abstract Plant diﬀerently colored points in the plane; then let random points (“Poisson rain”) fall, and give each new point the color of the nearest existing point. Previous investigation and simulations strongly sug- gest that the colored regions converge (in some sense) to a random partition of the plane. We prove a weak version of this, showing that normalized empirical measures converge to Lebesgue measures on a random partition into measurable sets. Topological properties remain an open problem. In the course of the proof, which heavily exploits time-reversals, we encounter a novel self-similar process of coalescing planar partitions. In this process, sets A(z) in the partition are asso- ciated with Poisson random points z, and the dynamics are as follows. Points are deleted randomly at rate 1; when z is deleted, its set A(z) (cid:48) (cid:48) is adjoined to the set A(z ) of the nearest other point z . MSC 2010 subject classiﬁcations: 60D06, 60G57. Key words: random tessellation, Poisson point process, spatial tree, stochastic coalescence. ∗ Department of Statistics, 367 Evans Hall # 3860, U.C. Berkeley CA 94720; al- dous@stat.berkeley.edu; www.stat.berkeley.edu/users/aldous. Aldous’s research sup- ported by N.S.F Grant DMS-1504802. 1 ##LTLine##

text: Random

label: title

text: partitions

text: of

text: the

text: plane

text: via

text: Poissonian

text: coloring,

text: and

text: a

text: self-similar

text: process

text: of

text: coalescing

text: planar

text: partitions

label: author

text: David

text: J.

text: Aldous

text: May

text: Abstract

text: Plant

label: abstract

text: diﬀerently

text: colored

text: points

text: in

text: the

text: plane;

text: then

text: let

text: random

text: points

text: (“Poisson

text: rain”)

text: fall,

text: and

text: give

text: each

text: new

text: point

text: the

text: color

text: of

text: the

text: nearest

text: existing

text: point.

text: Previous

text: investigation

text: and

text: simulations

text: strongly

text: sug-

text: gest

text: that

text: the

text: colored

text: regions

text: converge

text: (in

text: some

text: sense)

text: to

text: a

text: random

text: partition

text: of

text: the

text: plane.

text: We

text: prove

text: a

text: weak

text: version

text: of

text: this,

text: showing

text: that

text: normalized

text: empirical

text: measures

text: converge

text: to

text: Lebesgue

text: measures

text: on

text: a

text: random

text: partition

text: into

text: measurable

text: sets.

text: Topological

text: properties

text: remain

text: an

text: open

text: problem.

text: In

text: the

text: course

text: of

text: the

text: proof,

text: which

text: heavily

text: exploits

text: time-reversals,

text: we

text: encounter

text: a

text: novel

text: self-similar

text: process

text: of

text: coalescing

text: planar

text: partitions.

text: In

text: this

text: process,

text: sets

text: A(z)

text: in

text: the

text: partition

text: are

text: asso-

text: ciated

text: with

text: Poisson

text: random

text: points

text: z,

text: and

text: the

text: dynamics

text: are

text: as

text: follows.

text: Points

text: are

text: deleted

text: randomly

text: at

text: rate

text: when

text: z

text: is

text: deleted,

text: its

text: set

text: A(z)

text: is

text: adjoined

text: to

text: the

text: set

text: A(z

text: of

text: the

text: nearest

text: other

text: point

text: z

text: MSC

text: subject

text: classiﬁcations:

text: Key

text: words:

text: random

text: tessellation,

text: Poisson

text: point

text: process,

text: spatial

text: tree,

text: stochastic

text: coalescence.

text: Department

text: of

text: Statistics,

text: Evans

text: Hall

text: U.C.

text: Berkeley

text: CA

text: al-

text: dous@stat.berkeley.edu;

text: Aldous’s

text: research

text: sup-

text: ported

text: by

text: N.S.F

text: Grant

text: ##LTLine##


text: 2.2 Derivation of an EA process We study lines of descent in the genealogical tree process. Consider a particle ξ present at time 0 at position z . From the thinning process rep- 0 − resentation, its arrival time T < 0 is such that T has Exponential(1) ≥ 0 0 distribution. For i 1 write (T , Z ) for the arrival time and position of its i i i’th generation ancestor, that is parent[i, ξ]. We will show how to represent this process in terms of a certain Markov process we will call the excluded area (EA) process. { } Conditional on T = t the particles present at t are distributed as the 0 0 0 PPP Ξ , and so parent[i, ξ] is the closest such point to z , at position Z <t0 { } 0 1 say. Conditional also on Z = z , we know there are no points of Ξ in 1|| −1 || <t0 the interior of C := disc(z , z z ). The arrival time T of Z has density ∝ 1 −∞ 0 1 0 − 1 1 function et on < t < t , implying that t T has Exponential(1) 0 0 1 distribution. Now given T = t and (T , Z ) = (t , z ), the information we have about 0 0 1 1 1 1 Ξ is precisely the fact that it has no points in C . So Z is the closest <t1 R \ 1 2 − point to z in a PPP of rate et1 on 2 C . And as before, t T has 1 1 1 2 Exponential(1) distribution. Now given T = t and (T , Z ) = (t , z ) and (T , Z ) = (t , z ), we have 0 0 1 1 ∪ 1 1 || − 2|| 2 2 2 built an “excluded region” C := C disc(z , z z ). The information we 2 1 1 2 1 have about Ξ is precisely that it is a PPP of rate et2 with no points in C , <t2 ≥ 2 and we can continue inductively to describe the entire process ((T , Z ), i 0). i i 2.3 Deﬁnition of the EA process Here we re-specify the process above in intrinsic terms. Working with ↓ −∞ time is rather counter-intuitive, so in the deﬁnition below it seems helpful to reverse the direction of time. C Consider the space of triples c = (C, z, τ) such that R ∈ ≤ ∞ C is a compact set in 2; z C; 0 τ < . (cid:101) ∈ C Giv(cid:101)en an element c = (C, z, τ) we can deﬁne a probability distribution C − R \ µ on as follows. Take a PPP Ξ of rate e τ on 2 C. Let ξ be the point c of Ξ closest to z. Set (cid:48) (cid:48) ∪ || − || (cid:48) (cid:101) z = ξ; C = C disc(z, ξ z ); τ = τ + θ where θ has Exponential(1) distribution independent of Ξ. Then let µ be (cid:48) (cid:48) (cid:48) c the distribution of (C , z , τ ). C ≤ Deﬁne the EA process to be the -valued Markov chain (C = (C , Z , τ ), 0 i i i i 11

label: section

text: Derivation

text: of

text: an

text: EA

text: process

text: We

text: study

text: lines

text: of

text: descent

text: in

text: the

text: genealogical

text: tree

text: process.

text: Consider

text: a

text: particle

text: ξ

text: present

text: at

text: time

text: at

text: position

text: z

text: From

text: the

text: thinning

text: process

text: rep-

text: resentation,

text: its

text: arrival

text: time

text: T

text: is

text: such

text: that

text: T

text: has

text: distribution.

text: For

text: i

text: write

text: (T

text: Z

text: for

text: the

text: arrival

text: time

text: and

text: position

text: of

text: its

text: i

text: i

text: i’th

text: generation

text: ancestor,

text: that

text: is

text: parent[i,

text: ξ].

text: We

text: will

text: show

text: how

text: to

text: represent

text: this

text: process

text: in

text: terms

text: of

text: a

text: certain

text: Markov

text: process

text: we

text: will

text: call

text: the

text: excluded

text: area

text: (EA)

text: process.

text: Conditional

text: on

text: T

text: t

text: the

text: particles

text: present

text: at

text: t

text: are

text: distributed

text: as

text: the

text: PPP

text: Ξ

text: and

text: so

text: parent[i,

text: ξ]

text: is

text: the

text: closest

text: such

text: point

text: to

text: z

text: at

text: position

text: Z

text: say.

text: Conditional

text: also

text: on

text: Z

text: z

text: we

text: know

text: there

text: are

text: no

text: points

text: of

text: Ξ

text: in

text: −1

text: the

text: interior

text: of

text: C

text: disc(z

text: z

text: z

text: The

text: arrival

text: time

text: T

text: of

text: Z

text: has

text: density

text: function

text: et

text: on

text: t

text: t

text: implying

text: that

text: t

text: T

text: has

text: distribution.

text: Now

text: given

text: T

text: t

text: and

text: (T

text: Z

text: (t

text: z

text: the

text: information

text: we

text: have

text: about

text: Ξ

text: is

text: precisely

text: the

text: fact

text: that

text: it

text: has

text: no

text: points

text: in

text: C

text: So

text: Z

text: is

text: the

text: closest

text: R

text: point

text: to

text: z

text: in

text: a

text: PPP

text: of

text: rate

text: on

text: C

text: And

text: as

text: before,

text: t

text: T

text: has

text: distribution.

text: Now

text: given

text: T

text: t

text: and

text: (T

text: Z

text: (t

text: z

text: and

text: (T

text: Z

text: (t

text: z

text: we

text: have

text: built

text: an

text: “excluded

text: region”

text: C

text: C

text: disc(z

text: z

text: z

text: The

text: information

text: we

text: have

text: about

text: Ξ

text: is

text: precisely

text: that

text: it

text: is

text: a

text: PPP

text: of

text: rate

text: with

text: no

text: points

text: in

text: C

text: and

text: we

text: can

text: continue

text: inductively

text: to

text: describe

text: the

text: entire

text: process

text: ((T

text: Z

text: i

text: i

text: i

text: Deﬁnition

text: of

text: the

text: EA

text: process

text: Here

text: we

text: re-specify

text: the

text: process

text: above

text: in

text: intrinsic

text: terms.

text: Working

text: with

text: time

text: is

text: rather

text: counter-intuitive,

text: so

text: in

text: the

text: deﬁnition

text: below

text: it

text: seems

text: helpful

text: to

text: reverse

text: the

text: direction

text: of

text: time.

text: C

text: Consider

text: the

text: space

text: of

text: triples

text: c

text: (C,

text: z,

text: τ)

text: such

text: that

text: R

text: C

text: is

text: a

text: compact

text: set

text: in

text: z

text: C;

text: τ

text: C

text: an

text: element

text: c

text: (C,

text: z,

text: τ)

text: we

text: can

text: deﬁne

text: a

text: probability

text: distribution

text: C

text: R

text: µ

text: on

text: as

text: follows.

text: Take

text: a

text: PPP

text: Ξ

text: of

text: rate

text: e

text: τ

text: on

text: C.

text: Let

text: ξ

text: be

text: the

text: point

text: c

text: of

text: Ξ

text: closest

text: to

text: z.

text: Set

text: z

text: ξ;

text: C

text: C

text: disc(z,

text: ξ

text: z

text: τ

text: τ

text: θ

text: where

text: θ

text: has

text: distribution

text: independent

text: of

text: Ξ.

text: Then

text: let

text: µ

text: be

text: c

text: the

text: distribution

text: of

text: (C

text: z

text: τ

text: C

text: Deﬁne

text: the

text: EA

text: process

text: to

text: be

text: the

text: -valued

text: Markov

text: chain

text: (C

text: (C

text: Z

text: τ

text: i

text: i

text: i

text: i


text: ∩ where H is the half-space containing y. Now area(D H) is a certain function ≥ of r and r r , and clearly this function is, for ﬁxed r , minimized at r = r , 0 0 0 0 and there its value is 1πr2. So 2 0 ∪ − ≥ area(C D) area(C) 1πr2 2 0 and combining with (8) gives (7). In proving (7) we assumed C was convex. For general C we can apply ∗ (7) to its convex hull C and then, noting ∗ ∗ ∪ − ∗ ≤ ∪ − diam(C ) = diam(C), area(C D) area(C ) area(C D) area(C) we see that (7) remains true for non-convex C. 2.5 Completing the proof of Proposition 4 Returning to the standard EA process C = (C , Z , τ ), we now have i i i i suﬃcient tools to study τ and i A := area(C ), D := diam(C ). i i i i From Lemma 7 we obtain a constructive representation of the distribution ≤ ∞ of ((A , τ ), 0 i < ), as follows. i i (cid:80) ≥ ∞ The process (τ , i 0) forms a Poisson process of rate 1 on (0, ). (9) i − ≥ A = i 1 eτjθ where (θ , j 0) are i.i.d. Exponential(1), i j=0 j j ≥ independent of (τ , i 0). (10) i (cid:18) (cid:113) (cid:113) (cid:19) Then from Lemma 8 we get the inequality ≤ − D max D + 2(Ai+1 Ai), 4Ai+1 . (11) i+1 i π π (cid:113) In this section we use only the weaker inequality ≤ D D + 4Ai+1 . (12) i+1 i π 15 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: where

text: H

text: is

text: the

text: half-space

text: containing

text: y.

text: Now

text: area(D

text: H)

text: is

text: a

text: certain

text: function

text: of

text: r

text: and

text: r

text: r

text: and

text: clearly

text: this

text: function

text: is,

text: for

text: ﬁxed

text: r

text: minimized

text: at

text: r

text: r

text: and

text: there

text: its

text: value

text: is

text: So

text: area(C

text: D)

text: area(C)

text: and

text: combining

text: with

text: gives

text: In

text: proving

text: we

text: assumed

text: C

text: was

text: convex.

text: For

text: general

text: C

text: we

text: can

text: apply

text: to

text: its

text: convex

text: hull

text: C

text: and

text: then,

text: noting

text: diam(C

text: diam(C),

text: area(C

text: D)

text: area(C

text: area(C

text: D)

text: area(C)

text: we

text: see

text: that

text: remains

text: true

text: for

text: non-convex

text: C.

text: Completing

text: the

text: proof

text: of

text: Proposition

text: Returning

text: to

text: the

text: standard

text: EA

text: process

text: C

text: (C

text: Z

text: τ

text: we

text: now

text: have

text: i

text: i

text: i

text: i

text: suﬃcient

text: tools

text: to

text: study

text: τ

text: and

text: i

text: A

text: area(C

text: D

text: diam(C

text: i

text: i

text: i

text: i

text: From

text: Lemma

text: we

text: obtain

text: a

text: constructive

text: representation

text: of

text: the

text: distribution

text: of

text: ((A

text: τ

text: i

text: as

text: follows.

text: i

text: i

text: The

text: process

text: (τ

text: i

text: forms

text: a

text: Poisson

text: process

text: of

text: rate

text: on

text: i

text: A

text: i

text: eτjθ

text: where

text: (θ

text: j

text: are

text: i.i.d.

text: i

text: j

text: j

text: independent

text: of

text: (τ

text: i

text: i

text: Then

text: from

text: Lemma

text: we

text: get

text: the

text: inequality

text: D

text: max

text: D

text: Ai),

text: i

text: π

text: π

text: In

text: this

text: section

text: we

text: use

text: only

text: the

text: weaker

text: inequality

text: D

text: D

text: i

text: π

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: ≥ where (η , u 1) are i.i.d. Exponential(1), independent of the Poisson u ≥ process (σ , u 1). So Proposition 4 holds for u P ∞ G(r) := (cid:82) (χ > r), 0 < r < (17) ∞ ∞ and it is easy to check that rG(r)dr < . 0 2.6 A large deviation bound for occupation times The following technical bound will enable us to bound the distance re- quired for two lines of descent to merge (Proposition 13 later). Proposition 9. For the standard EA process write ∪ B = { − }[τ − , τ ). i:e τi/2 diam(Ci)>b i 1 i ∞ Then for suﬃciently large b there exist A < and ρ > 0 such that P ∩ ≤ − ∞ (Leb(B [0, T ]) > T/3) A exp( ρT ), 0 < T < (18) where Leb denotes Lebesgue measure. We previously u(cid:113)sed ineq(cid:80)uality(cid:113)(12) to bound Di := diam(Ci) in terms of the areas A = area(A ). Here we will use a slightly diﬀerent bound. i i ≤ (cid:113) − − Lemma 10. D(cid:101) 4Ai + i 1 2(Aj+1 Aj). i π j=0 π (cid:18) − (cid:113) (cid:18)(cid:113) (cid:113) (cid:19) (cid:19) Proof. Setting D := D 4Ai , inequality (11) becomes (cid:101) i i(cid:101) π ≤ − − D (cid:113)max D(cid:113) + 2(Ai+1 Ai) + 4Ai 4Ai+1 , 0 . i+1 i π π(cid:101) π − But the term ( 4Aπi 4Aπi+1 ) is(cid:88)ne(cid:113)gative, and D0 = 0, so we ﬁnd (cid:101) − ≤ i 1 − D 2(Aj+1 Aj) i π j=0 establishing the asserted bound. 17 ##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: (η

text: u

text: are

text: i.i.d.

text: independent

text: of

text: the

text: Poisson

text: u

text: process

text: (σ

text: u

text: So

text: Proposition

text: holds

text: for

text: u

text: P

text: G(r)

text: (χ

text: r),

text: r

text: and

text: it

text: is

text: easy

text: to

text: check

text: that

text: rG(r)dr

text: A

text: large

text: deviation

text: bound

text: for

text: occupation

text: times

text: The

text: following

text: technical

text: bound

text: will

text: enable

text: us

text: to

text: bound

text: the

text: distance

text: re-

text: quired

text: for

text: two

text: lines

text: of

text: descent

text: to

text: merge

text: (Proposition

text: later).

text: Proposition

text: For

text: the

text: standard

text: EA

text: process

text: write

text: B

text: }[τ

text: τ

text: i:e

text: diam(Ci)>b

text: i

text: i

text: Then

text: for

text: suﬃciently

text: large

text: b

text: there

text: exist

text: A

text: and

text: ρ

text: such

text: that

text: P

text: (Leb(B

text: T

text: A

text: exp(

text: ρT

text: T

text: where

text: Leb

text: denotes

text: Lebesgue

text: measure.

text: We

text: previously

text: to

text: bound

text: Di

text: diam(Ci)

text: in

text: terms

text: of

text: the

text: areas

text: A

text: area(A

text: Here

text: we

text: will

text: use

text: a

text: slightly

text: diﬀerent

text: bound.

text: i

text: i

text: Lemma

text: i

text: Aj).

text: i

text: π

text: π

text: Proof.

text: Setting

text: D

text: D

text: inequality

text: becomes

text: i

text: π

text: D

text: Ai)

text: i

text: π

text: π

text: But

text: the

text: term

text: and

text: so

text: we

text: ﬁnd

text: i

text: D

text: Aj)

text: i

text: π

text: establishing

text: the

text: asserted

text: bound.

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: P || || − log ( Z > r) coal γ := lim (28) →∞ r log r || − || exists and does not depend on z1 z2 or t . This is closely related to 0 0 0 the “are boundaries fractal?” issue, as will be discussed in section . For ease of exposition we will present the proof of Proposition 13 in the case t = 0. The general case requires only minor modiﬁcations, noted below. 0 3.2 The coalescence step Here we will give conditions to ensure a non-vanishing probability of co- alescing at the next step. This requires only a simple geometric lemma. ∈ R Z Lemma 14. Let z , z 2, and let r, λ > 0. Let λ be a Poisson point 1 2 R process of rate λ on 2. Then the event ∈ Z the nearest point z λ to z is also the nearest point to z , and || − || || − || 1 2 min( z z , z z ) > r 1 2 has probability at least − || − || − || − || exp( λπ(r + z z )2) 2c λ1/2 z z (29) 2 1 0 2 1 for a certain absolute constant c . 0 || − || Proof. Write d = z z . Consider the events 2 1 Z (A): the distance from z to the nearest point of λ is at least r + d; 1 Z (B): the distances D(λ) and D(λ) from z to the nearest two points of λ − 1 ≤ 2 2 are such that D(λ) D(λ) 2d. 2 1 We assert that, in order that the event in Lemma 14 occurs, it is suﬃcient that the event (A) occurs and the event (B) does not occur. To prove this (cid:48) Z (cid:48) assertion, let z be the closest point of λ to z , and let z be the closest Z 1 (cid:48) (cid:54) (cid:48) 1 2 point of λ to z . Suppose z = z . By the triangle inequality 2 2 1 || − (cid:48) || ≤ || − (cid:48) || || − (cid:48) || ≤ || − (cid:48) || z z d + z z and z z d + z z . 1 2 2 2 2 1 1 1 (cid:48) Because z is the closest point to z 1 1 || − (cid:48) || || − (cid:48) || z z < z z . 1 1 1 2 25 ##LTLine##

text: P

text: log

text: Z

text: r)

text: coal

text: γ

text: lim

text: r

text: log

text: r

text: exists

text: and

text: does

text: not

text: depend

text: on

text: or

text: t

text: This

text: is

text: closely

text: related

text: to

text: the

text: “are

text: boundaries

text: fractal?”

text: issue,

text: as

text: will

text: be

text: discussed

text: in

text: section

text: For

text: ease

text: of

text: exposition

text: we

text: will

text: present

text: the

text: proof

text: of

text: Proposition

text: in

text: the

text: case

text: t

text: The

text: general

text: case

text: requires

text: only

text: minor

text: modiﬁcations,

text: noted

text: below.

text: The

text: coalescence

text: step

text: Here

text: we

text: will

text: give

text: conditions

text: to

text: ensure

text: a

text: non-vanishing

text: probability

text: of

text: co-

text: alescing

text: at

text: the

text: next

text: step.

text: This

text: requires

text: only

text: a

text: simple

text: geometric

text: lemma.

text: R

text: Z

text: Lemma

text: Let

text: z

text: z

text: and

text: let

text: r,

text: λ

text: Let

text: λ

text: be

text: a

text: Poisson

text: point

text: R

text: process

text: of

text: rate

text: λ

text: on

text: Then

text: the

text: event

text: Z

text: the

text: nearest

text: point

text: z

text: λ

text: to

text: z

text: is

text: also

text: the

text: nearest

text: point

text: to

text: z

text: and

text: min(

text: z

text: z

text: z

text: z

text: r

text: has

text: probability

text: at

text: least

text: exp(

text: λπ(r

text: z

text: z

text: z

text: z

text: for

text: a

text: certain

text: absolute

text: constant

text: c

text: Proof.

text: Write

text: d

text: z

text: z

text: Consider

text: the

text: events

text: Z

text: (A):

text: the

text: distance

text: from

text: z

text: to

text: the

text: nearest

text: point

text: of

text: λ

text: is

text: at

text: least

text: r

text: d;

text: Z

text: (B):

text: the

text: distances

text: D(λ)

text: and

text: D(λ)

text: from

text: z

text: to

text: the

text: nearest

text: two

text: points

text: of

text: λ

text: are

text: such

text: that

text: D(λ)

text: D(λ)

text: We

text: assert

text: that,

text: in

text: order

text: that

text: the

text: event

text: in

text: Lemma

text: occurs,

text: it

text: is

text: suﬃcient

text: that

text: the

text: event

text: (A)

text: occurs

text: and

text: the

text: event

text: (B)

text: does

text: not

text: occur.

text: To

text: prove

text: this

text: Z

text: assertion,

text: let

text: z

text: be

text: the

text: closest

text: point

text: of

text: λ

text: to

text: z

text: and

text: let

text: z

text: be

text: the

text: closest

text: Z

text: point

text: of

text: λ

text: to

text: z

text: Suppose

text: z

text: z

text: By

text: the

text: triangle

text: inequality

text: z

text: z

text: d

text: z

text: z

text: and

text: z

text: z

text: d

text: z

text: z

text: Because

text: z

text: is

text: the

text: closest

text: point

text: to

text: z

text: z

text: z

text: z

text: z

text: ##LTLine##


text: { ≥ } ≥ is a martingale. On the event L(n, p ) m, S > n we have M − − 0 n (1 p ) m and so 0 E ≥ E ≥ − − P ≥ 1 = M M 1 ≥ (1 p ) m (L(n, p ) m, S > n). n n (L(n,p0) m,S>n) 0 0 Because P P ≥ P (S > n) = (L(n, p ) m, S > n) + (L(n, p ) < m, S > n) 0 0 the result follows. We can now combine previous ingredients to prove Proposition 13. Sup- pose i is such that I > i and the conﬁguration (C1; C2) satisﬁes (31). coal i i Then by (30) the probability of coalescing on the next step is at least − || − || − − || − || − exp( 4π( z1 z2 e τi/2 + 2b)2) 2c z1 z2 e τi/2. 0 0 0 0 0 So there exist constants α > 0 and p > 0 (determined by b and c ) such F 0 0 that, for the natural ﬁltration ( ) of the coupled EA process, i P |F ≥ { || − || − ≤ } (I = i+1 ) p on I > i, z1 z2 e τi/2 α, (C1; C2) satisﬁes (31) . coal i 0 coal 0 0 i i (38) Appealing to Lemma 20, P ≤ − P (I > n) (1 p )m + (L < m, I > n) (39) coal 0 n coal where |{ ≤ ≤ − P |F ≥ }| L = i : 0 i n 1, (I = i + 1 ) p . n coal i 0 Recall the deﬁnition of N (t) in Proposition 16. Take a > 0 (to be speciﬁed b later) and consider some m < an. If || − || − ≤ (cid:98) (cid:99) − z1 z2 e τj/2 α for j = an m (40) 0 0 31 ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: is

text: a

text: martingale.

text: On

text: the

text: event

text: L(n,

text: p

text: m,

text: S

text: n

text: we

text: have

text: M

text: n

text: p

text: m

text: and

text: so

text: E

text: E

text: P

text: M

text: M

text: p

text: m

text: (L(n,

text: p

text: m,

text: S

text: n).

text: n

text: n

text: m,S>n)

text: Because

text: P

text: P

text: P

text: (S

text: n)

text: (L(n,

text: p

text: m,

text: S

text: n)

text: (L(n,

text: p

text: m,

text: S

text: n)

text: the

text: result

text: follows.

text: We

text: can

text: now

text: combine

text: previous

text: ingredients

text: to

text: prove

text: Proposition

text: Sup-

text: pose

text: i

text: is

text: such

text: that

text: I

text: i

text: and

text: the

text: conﬁguration

text: satisﬁes

text: coal

text: i

text: i

text: Then

text: by

text: the

text: probability

text: of

text: coalescing

text: on

text: the

text: next

text: step

text: is

text: at

text: least

text: exp(

text: e

text: e

text: So

text: there

text: exist

text: constants

text: α

text: and

text: p

text: (determined

text: by

text: b

text: and

text: c

text: such

text: F

text: that,

text: for

text: the

text: natural

text: ﬁltration

text: of

text: the

text: coupled

text: EA

text: process,

text: i

text: P

text: |F

text: (I

text: p

text: on

text: I

text: i,

text: e

text: α,

text: satisﬁes

text: coal

text: i

text: coal

text: i

text: i

text: Appealing

text: to

text: Lemma

text: P

text: P

text: (I

text: n)

text: p

text: )m

text: (L

text: m,

text: I

text: n)

text: coal

text: n

text: coal

text: where

text: P

text: |F

text: L

text: i

text: i

text: n

text: (I

text: i

text: p

text: n

text: coal

text: i

text: Recall

text: the

text: deﬁnition

text: of

text: N

text: (t)

text: in

text: Proposition

text: Take

text: a

text: (to

text: be

text: speciﬁed

text: b

text: later)

text: and

text: consider

text: some

text: m

text: an.

text: If

text: e

text: α

text: for

text: j

text: an

text: m

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: { } then, on the event I > n, N (t) > an , the events in (38) hold for at ≤ coal b ≥ least m values of i n, which implies L m. So if (40) holds then n P ≤ P ≤ (L < m, I > n) (I > n, N (n) an) n coal coal b and then using (39) we have P ≤ − P ≤ P (I > n) (1 p )m+ (I > n, N (n) an)+ (I > n, event (40) fails). coal 0 coal b coal Proposition 16 implies that for suﬃciently large b there exist constants a, ρ > ∞ 0 and K < such that P ≤ ≤ − (T > n/3, N (n) an) K exp( ρn), n = 1, 2, 3, . . . . coal b Using this choice of a above, P ≤ − (I > n) (1 p )m + O (n) coal 0 exp P ≤ P + (I > n, T n/3) + (I > n, event (40) fails) coal coal coal where O (n) denotes a “vanishing probability” sequence which is O(ρn) as → ∞ exp n for some ρ < 1. Now note that elementary large deviation bounds for the rate-2 PPP (τ ) show that i P ≤ P ≤ P ≤ (I > n, T n/3), (I n, T > n), (τ n/3) are all O (n). coal coal coal coal n exp (41) (cid:98) (cid:99) Choosing m = an/2 we ﬁnd P ≤ P (I > n) O (n) + ( event (40) fails), n = 1, 2, 3, . . . coal exp || − || where the O (n) term does not depend on z1 z2 . From the deﬁnition exp 0 0 of event (40) and the choice of m P( event (40) fails) ≤ P(τ ≤ 2 log ||z01−z02||) for j = (cid:98)an/2(cid:99) − 1. (42) j α From the ﬁnal term in (41) there exists a constant β such that P || − || ( event (40) fails) = O (n) for n > β log+ z1 z2 . exp 0 0 So now P ≤ || − || (I > n) O (n) for n > β log+ z1 z2 . coal exp 0 0 P ≤ P P ≤ Because (T > n) (I > n) + (I n, T > n) we have coal coal coal coal established Proposition 13. 32 ##LTLine##

text: then,

text: on

text: the

text: event

text: I

text: n,

text: N

text: (t)

text: an

text: the

text: events

text: in

text: hold

text: for

text: at

text: coal

text: b

text: least

text: m

text: values

text: of

text: i

text: n,

text: which

text: implies

text: L

text: m.

text: So

text: if

text: holds

text: then

text: n

text: P

text: P

text: (L

text: m,

text: I

text: n)

text: (I

text: n,

text: N

text: (n)

text: an)

text: n

text: coal

text: coal

text: b

text: and

text: then

text: using

text: we

text: have

text: P

text: P

text: P

text: (I

text: n)

text: p

text: )m+

text: (I

text: n,

text: N

text: (n)

text: an)+

text: (I

text: n,

text: event

text: fails).

text: coal

text: coal

text: b

text: coal

text: Proposition

text: implies

text: that

text: for

text: suﬃciently

text: large

text: b

text: there

text: exist

text: constants

text: a,

text: ρ

text: and

text: K

text: such

text: that

text: P

text: (T

text: N

text: (n)

text: an)

text: K

text: exp(

text: ρn),

text: n

text: coal

text: b

text: Using

text: this

text: choice

text: of

text: a

text: above,

text: P

text: (I

text: n)

text: p

text: )m

text: O

text: (n)

text: coal

text: exp

text: P

text: P

text: (I

text: n,

text: T

text: (I

text: n,

text: event

text: fails)

text: coal

text: coal

text: coal

text: where

text: O

text: (n)

text: denotes

text: a

text: “vanishing

text: probability”

text: sequence

text: which

text: is

text: O(ρn)

text: as

text: exp

text: n

text: for

text: some

text: ρ

text: Now

text: note

text: that

text: elementary

text: large

text: deviation

text: bounds

text: for

text: the

text: PPP

text: (τ

text: show

text: that

text: i

text: P

text: P

text: P

text: (I

text: n,

text: T

text: (I

text: n,

text: T

text: n),

text: (τ

text: are

text: all

text: O

text: (n).

text: coal

text: coal

text: coal

text: coal

text: n

text: exp

text: Choosing

text: m

text: we

text: ﬁnd

text: P

text: P

text: (I

text: n)

text: O

text: (n)

text: event

text: fails),

text: n

text: coal

text: exp

text: where

text: the

text: O

text: (n)

text: term

text: does

text: not

text: depend

text: on

text: From

text: the

text: deﬁnition

text: exp

text: of

text: event

text: and

text: the

text: choice

text: of

text: m

text: P(

text: event

text: fails)

text: P(τ

text: log

text: for

text: j

text: j

text: α

text: From

text: the

text: ﬁnal

text: term

text: in

text: there

text: exists

text: a

text: constant

text: β

text: such

text: that

text: P

text: event

text: fails)

text: O

text: (n)

text: for

text: n

text: β

text: log+

text: exp

text: So

text: now

text: P

text: (I

text: n)

text: O

text: (n)

text: for

text: n

text: β

text: log+

text: coal

text: exp

text: P

text: P

text: P

text: Because

text: (T

text: n)

text: (I

text: n)

text: (I

text: n,

text: T

text: n)

text: we

text: have

text: coal

text: coal

text: coal

text: coal

text: established

text: Proposition

text: ##LTLine##


text: 4 Proof of the main theorems 4.1 Notation In this section we use the preceding bounds to prove Theorems 1 and M R 2. Recall some deﬁnitions from section . ( 2) denotes the space of ﬁnite R measures on 2, equipped with the usual topology of weak convergence. For a particle ξ, the time-t ancestor is denoted ancestor(t, ξ), and Descend(t , t , ζ) 1 2 denotes the set of particles born before t whose time-t ancestor is ζ. And ≤ ∈ 2 1 for t t and ζ Ξ≤ 1 2 t1 − µ is the measure µ putting weight e t2 t1,t2,ζ on the position of each particle in Descend(t , t , ζ). (43) 1 2 ∈ Note that given Ξ≤ , the “marks” (µ , ζ Ξ≤ ) are still random Mt1 R t1,t2,ζ t elements of the space ( 2), whose distributions depend on all Ξ≤ and t1 are dependent as ζ varies. If we use Ξ≤ to deﬁne a translation-invariant marked PPP of the form { ∈t } (ξ, m+(ξ)), ξ Ξ≤ with non-negative real marks m+(ξ), then there is a t spatial average rate of mark values, which we will write as ∈ ave(m+(ξ) : ξ Ξ≤ ) t (cid:88) deﬁned as the value of a such that E × ⊂ R m+(ξ) = a area(B), B 2. ∈ ∈ ξ Ξ≤ , z B t ξ For instance we have, for t < t , 1 2 | | ∈ ave( µ : ξ Ξ≤ ) = 1 (44) t1,t2,ξ t1 | | where µ denotes the total mass of µ. The self-similarity property of the underlying space-time PPP Ξ allows us to write down exact self-similarity properties for our marked point processes. → − R In particular, the action of the scaling map z e t/2z on 2, applied to the { ∈ } distribution of (z , µ ), ξ Ξ≤ , gives a distribution which coincides ξ t1,t2,ξ t1 33

text: Proof

text: of

text: the

text: main

text: theorems

text: Notation

text: In

text: this

text: section

text: we

text: use

text: the

text: preceding

text: bounds

text: to

text: prove

text: Theorems

text: and

text: M

text: R

text: Recall

text: some

text: deﬁnitions

text: from

text: section

text: denotes

text: the

text: space

text: of

text: ﬁnite

text: R

text: measures

text: on

text: equipped

text: with

text: the

text: usual

text: topology

text: of

text: weak

text: convergence.

text: For

text: a

text: particle

text: ξ,

text: the

text: time-t

text: ancestor

text: is

text: denoted

text: ancestor(t,

text: ξ),

text: and

text: Descend(t

text: t

text: ζ)

text: denotes

text: the

text: set

text: of

text: particles

text: born

text: before

text: t

text: whose

text: time-t

text: ancestor

text: is

text: ζ.

text: And

text: for

text: t

text: t

text: and

text: ζ

text: Ξ≤

text: µ

text: is

text: the

text: measure

text: µ

text: putting

text: weight

text: e

text: on

text: the

text: position

text: of

text: each

text: particle

text: in

text: Descend(t

text: t

text: ζ).

text: Note

text: that

text: given

text: Ξ≤

text: the

text: “marks”

text: (µ

text: ζ

text: Ξ≤

text: are

text: still

text: random

text: R

text: t

text: elements

text: of

text: the

text: space

text: whose

text: distributions

text: depend

text: on

text: all

text: Ξ≤

text: and

text: are

text: dependent

text: as

text: ζ

text: varies.

text: If

text: we

text: use

text: Ξ≤

text: to

text: deﬁne

text: a

text: translation-invariant

text: marked

text: PPP

text: of

text: the

text: form

text: ∈t

text: (ξ,

text: m+(ξ)),

text: ξ

text: Ξ≤

text: with

text: non-negative

text: real

text: marks

text: m+(ξ),

text: then

text: there

text: is

text: a

text: t

text: spatial

text: average

text: rate

text: of

text: mark

text: values,

text: which

text: we

text: will

text: write

text: as

text: ave(m+(ξ)

text: ξ

text: Ξ≤

text: t

text: deﬁned

text: as

text: the

text: value

text: of

text: a

text: such

text: that

text: E

text: R

text: m+(ξ)

text: a

text: area(B),

text: B

text: ξ

text: Ξ≤

text: z

text: B

text: t

text: ξ

text: For

text: instance

text: we

text: have,

text: for

text: t

text: t

text: ave(

text: µ

text: ξ

text: Ξ≤

text: where

text: µ

text: denotes

text: the

text: total

text: mass

text: of

text: µ.

text: The

text: self-similarity

text: property

text: of

text: the

text: underlying

text: space-time

text: PPP

text: Ξ

text: allows

text: us

text: to

text: write

text: down

text: exact

text: self-similarity

text: properties

text: for

text: our

text: marked

text: point

text: processes.

text: R

text: In

text: particular,

text: the

text: action

text: of

text: the

text: scaling

text: map

text: z

text: e

text: on

text: applied

text: to

text: the

text: distribution

text: of

text: (z

text: µ

text: ξ

text: Ξ≤

text: gives

text: a

text: distribution

text: which

text: coincides

text: ξ


text: Deﬁne P ρ(t , ε) = lim inf lim inf (A(t , t , t , ε)). (46) 0 →∞ →∞ 0 1 2 t1 t2 We will show Lemma 26. lim →∞ ρ(t , ε) = 1 for each ε > 0. t0 0 Granted that, consider | | | | ∈ ≤ a(t , t ) := ave( min( µ , µ ) , ζ Ξ≤ ) 1. 1 2 0,t1,ζ 0,t2,ζ 0 R By averaging over area-t squares in 2, 0 ≥ − P a(t , t ) (1 ε) (A(t , t , t , ε)). 1 2 0 1 2 So Lemma 26 implies lim inf lim inf a(t , t ) = 1. (47) →∞ →∞ 1 2 t1 t2 By the triangle inequality and (44), | | | − | | | ∈ ≤ ave( µ µ : ζ Ξ≤ ) 2a(t , t ). 0,t1,ζ 0,t2,ζ 0 1 2 Then (47) and the Cauchy criterion imply there exist limits m for which 0,ζ | | | − | ∈ lim ave( µ m : ζ Ξ≤ ) = 0. →∞ 0,t,ζ 0,ζ 0 t Finally, Corollary 22 provides the “uniform integrability” condition needed ∈ to pass (44) to the limit to obtain ave(m : ζ Ξ≤ ) = 1. This establishes 0,ζ 0 Proposition 25. (cid:3) Proof. [of Lemma 26] Write Ξ≤(t0) for the restriction of Ξ≤ to particles within (cid:3) t t (cid:3) (t ). We can upper bound the mean number of pairs (ξ , ξ ) in (t ) with ∈0 (cid:88) (cid:88) 1 2 0 ξ Ξ≤ and with diﬀerent time-0 ancestors, as follows. Write i ti M(t , t , t ) := 1{ (cid:54) }. 0 1 2 ancestor(0,ξ1)=ancestor(0,ξ2) ξ1∈Ξ(cid:3)≤(tt10) ξ2∈Ξ(cid:3)≤(tt20) 36 ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Deﬁne

text: P

text: ρ(t

text: ε)

text: lim

text: inf

text: lim

text: inf

text: (A(t

text: t

text: t

text: ε)).

text: We

text: will

text: show

text: Lemma

text: lim

text: ρ(t

text: ε)

text: for

text: each

text: ε

text: Granted

text: that,

text: consider

text: a(t

text: t

text: ave(

text: min(

text: µ

text: µ

text: ζ

text: Ξ≤

text: R

text: By

text: averaging

text: over

text: area-t

text: squares

text: in

text: P

text: a(t

text: t

text: ε)

text: (A(t

text: t

text: t

text: ε)).

text: So

text: Lemma

text: implies

text: lim

text: inf

text: lim

text: inf

text: a(t

text: t

text: By

text: the

text: triangle

text: inequality

text: and

text: ave(

text: µ

text: µ

text: ζ

text: Ξ≤

text: t

text: Then

text: and

text: the

text: Cauchy

text: criterion

text: imply

text: there

text: exist

text: limits

text: m

text: for

text: which

text: lim

text: ave(

text: µ

text: m

text: ζ

text: Ξ≤

text: t

text: Finally,

text: Corollary

text: provides

text: the

text: “uniform

text: integrability”

text: condition

text: needed

text: to

text: pass

text: to

text: the

text: limit

text: to

text: obtain

text: ave(m

text: ζ

text: Ξ≤

text: This

text: establishes

text: Proposition

text: Proof.

text: [of

text: Lemma

text: Write

text: for

text: the

text: restriction

text: of

text: Ξ≤

text: to

text: particles

text: within

text: t

text: t

text: (t

text: We

text: can

text: upper

text: bound

text: the

text: mean

text: number

text: of

text: pairs

text: (ξ

text: ξ

text: in

text: (t

text: with

text: ∈0

text: ξ

text: Ξ≤

text: and

text: with

text: diﬀerent

text: ancestors,

text: as

text: follows.

text: Write

text: i

text: ti

text: M(t

text: t

text: t

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 5 Discussion 5.1 In what sense is this a tree process? We have used the language of ancestors and descendants, but otherwise have not really exploited the implicit tree structure of the colored point pro- cess construction. If we draw the process as a random tree in the plane, with edges drawn as line segments, it is clear from Figure 1 that edges sometimes cross, so we do not get a “tree” in the usual sense. This suggests that, in the opening “k colors in the unit square” model, in the limit partition into k colored regions, these regions are not necessarily connected. Figure 7 il- lustrates how this could happen. Simulations strongly indicate that in fact a typical region is not connected but that only a very small proportion of its area is outside its largest connected component. Figure 7: A possible realization of the tree in the unit square on the ﬁrst n = 11 arriving points. The edge from 1 to 2 is omitted, to show the k = 2 subtrees associated with the ﬁrst two vertices. At this stage the unit square is Voronoi-partitioned into 2 components according to whether the nearest ◦ • ◦ vertex is or , and the component is not connected. We expect this → ∞ disconnection to persist in the n limit. 11 10 6 1 9 7 5 3 8 4 2 43 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Discussion

text: In

text: what

text: sense

text: is

text: this

text: a

text: tree

text: process?

text: We

text: have

text: used

text: the

text: language

text: of

text: ancestors

text: and

text: descendants,

text: but

text: otherwise

text: have

text: not

text: really

text: exploited

text: the

text: implicit

text: tree

text: structure

text: of

text: the

text: colored

text: point

text: pro-

text: cess

text: construction.

text: If

text: we

text: draw

text: the

text: process

text: as

text: a

text: random

text: tree

text: in

text: the

text: plane,

text: with

text: edges

text: drawn

text: as

text: line

text: segments,

text: it

text: is

text: clear

text: from

text: Figure

text: that

text: edges

text: sometimes

text: cross,

text: so

text: we

text: do

text: not

text: get

text: a

text: “tree”

text: in

text: the

text: usual

text: sense.

text: This

text: suggests

text: that,

text: in

text: the

text: opening

text: “k

text: colors

text: in

text: the

text: unit

text: square”

text: model,

text: in

text: the

text: limit

text: partition

text: into

text: k

text: colored

text: regions,

text: these

text: regions

text: are

text: not

text: necessarily

text: connected.

text: Figure

text: il-

text: lustrates

text: how

text: this

text: could

text: happen.

text: Simulations

text: strongly

text: indicate

text: that

text: in

text: fact

text: a

text: typical

text: region

text: is

text: not

text: connected

text: but

text: that

text: only

text: a

text: very

text: small

text: proportion

text: of

text: its

text: area

text: is

text: outside

text: its

text: largest

text: connected

text: component.

text: Figure

text: A

text: possible

text: realization

text: of

text: the

text: tree

text: in

text: the

text: unit

text: square

text: on

text: the

text: ﬁrst

text: n

text: arriving

text: points.

text: The

text: edge

text: from

text: to

text: is

text: omitted,

text: to

text: show

text: the

text: k

text: subtrees

text: associated

text: with

text: the

text: ﬁrst

text: two

text: vertices.

text: At

text: this

text: stage

text: the

text: unit

text: square

text: is

text: Voronoi-partitioned

text: into

text: components

text: according

text: to

text: whether

text: the

text: nearest

text: vertex

text: is

text: or

text: and

text: the

text: component

text: is

text: not

text: connected.

text: We

text: expect

text: this

text: disconnection

text: to

text: persist

text: in

text: the

text: n

text: limit.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: R Figure 8: Bond percolation clusters as a partition of 2. A B C D 5.3 Heuristic arguments Arguments in this section are heuristics, only parts of which seem easily formalized. We conjectured at (28) that the tail behavior of the “meeting || || distance” random variable Z is of the form coal P || || (cid:16) − → ∞ ( Z > r) r γ as r (58) coal for some exponent γ. This is heuristically related to the issue of fractal ∈ dimension of the boundaries of the regions (A(0, ξ), ξ Ξ≤ ), as follows. 0 Consider the boundaries within the unit square. Saying this has fractal 45 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: R

label: caption

text: Figure

text: Bond

text: percolation

text: clusters

text: as

text: a

text: partition

text: of

text: A

text: B

text: C

text: D

text: Heuristic

text: arguments

text: Arguments

text: in

text: this

text: section

text: are

text: heuristics,

text: only

text: parts

text: of

text: which

text: seem

text: easily

text: formalized.

text: We

text: conjectured

text: at

text: that

text: the

text: tail

text: behavior

text: of

text: the

text: “meeting

text: distance”

text: random

text: variable

text: Z

text: is

text: of

text: the

text: form

text: coal

text: P

text: Z

text: r)

text: r

text: γ

text: as

text: r

text: coal

text: for

text: some

text: exponent

text: γ.

text: This

text: is

text: heuristically

text: related

text: to

text: the

text: issue

text: of

text: fractal

text: dimension

text: of

text: the

text: boundaries

text: of

text: the

text: regions

text: ξ),

text: ξ

text: Ξ≤

text: as

text: follows.

text: Consider

text: the

text: boundaries

text: within

text: the

text: unit

text: square.

text: Saying

text: this

text: has

text: fractal

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text: − dimension d is saying that for small x > 0 we need order x d radius-x discs to cover these boundaries. Consider a uniform random point z in the square 1 and another random point z uniform on disc(z , x). The chance that z and 2 1 1 z are in diﬀerent components is the same order as the chance they are in the 2 − same covering disc, which chance is order x2 d. But the former chance is the same order as the chance that the meeting distance M between their lines P x P (cid:16) − of descent is at least 1, that is (M > 1). So we expect (M > 1) x2 d ↓ x x as x 0. Now by self-similarity M = xM . So setting r = 1/x x d 1 P (cid:16) − − (M > r) r (2 d) (59) 1 and we heuristically identify the fractal dimension as − d = 2 γ. 5.3.1 Heuristic 1: the boundary has fractal dimension 1 For large t consider the Voronoi regions associated with the diﬀerent sets 0 of particles Descend(0, t , ζ) as ζ varies. The particles in Ξ≤ are separated − 0 t0 by distance of order e t0/2. Consider three points (z , z , z ) on the boundary − 1 2 3 at time t at distances of (say) 5e t0/2 apart. As t increases, particles arriving 0 nearby move the boundary near these three points. The ﬁrst such move is − over a distance of order e t0/2 and subsequent moves decrease geometrically. → ∞ So in the t limit the positions of the boundaries near (z , z , z ) should − − − 1 2 3 become (z + D e t0/2, z + D e t0/2, z + D e t0/2) for some (D , D , D ) 1 1 2 2 → ∞ 3 3 1 2 3 with a non-vanishing limit as t . But this is saying that in the limit ∈ 0 || − || partition (A(0, ζ), ζ Ξ≤ ), on every scale σ = y y , for y and y on 0 1 3 1 3 the boundary, the distance from the midpoint (y + y )/2 to the boundary 1 3 is of the form Dσ for some random D > 0. This is a hallmark of “fractal dimension = 1”. 5.3.2 Heuristic 2: the boundary has fractal dimension 1 The genealogical tree deﬁnes a “line of descent” for each particle in Ξ; R these particle positions are dense in 2, so let us suppose that in the contin- R uum limit there is such a “line of descent” from almost all points z of 2 to R inﬁnity. Draw the tree via line segments in 2. Consider a point (x, 0) on the x-axis. The route from there to inﬁnity ﬁrst crosses the y = 1 line at some C random point (c(x), 1). Consider the random set of all such values c(x) as x varies. This is stationary (translation-invariant) and so has some intensity 46

text: dimension

text: d

text: is

text: saying

text: that

text: for

text: small

text: x

text: we

text: need

text: order

text: x

text: d

text: radius-x

text: discs

text: to

text: cover

text: these

text: boundaries.

text: Consider

text: a

text: uniform

text: random

text: point

text: z

text: in

text: the

text: square

text: and

text: another

text: random

text: point

text: z

text: uniform

text: on

text: disc(z

text: x).

text: The

text: chance

text: that

text: z

text: and

text: z

text: are

text: in

text: diﬀerent

text: components

text: is

text: the

text: same

text: order

text: as

text: the

text: chance

text: they

text: are

text: in

text: the

text: same

text: covering

text: disc,

text: which

text: chance

text: is

text: order

text: d.

text: But

text: the

text: former

text: chance

text: is

text: the

text: same

text: order

text: as

text: the

text: chance

text: that

text: the

text: meeting

text: distance

text: M

text: between

text: their

text: lines

text: P

text: x

text: P

text: of

text: descent

text: is

text: at

text: least

text: that

text: is

text: (M

text: So

text: we

text: expect

text: (M

text: d

text: x

text: x

text: as

text: x

text: Now

text: by

text: self-similarity

text: M

text: xM

text: So

text: setting

text: r

text: x

text: d

text: P

text: (M

text: r)

text: r

text: d)

text: and

text: we

text: heuristically

text: identify

text: the

text: fractal

text: dimension

text: as

text: d

text: γ.

text: Heuristic

text: the

text: boundary

text: has

text: fractal

text: dimension

text: For

text: large

text: t

text: consider

text: the

text: Voronoi

text: regions

text: associated

text: with

text: the

text: diﬀerent

text: sets

text: of

text: particles

text: t

text: ζ)

text: as

text: ζ

text: varies.

text: The

text: particles

text: in

text: Ξ≤

text: are

text: separated

text: by

text: distance

text: of

text: order

text: e

text: Consider

text: three

text: points

text: (z

text: z

text: z

text: on

text: the

text: boundary

text: at

text: time

text: t

text: at

text: distances

text: of

text: (say)

text: apart.

text: As

text: t

text: increases,

text: particles

text: arriving

text: nearby

text: move

text: the

text: boundary

text: near

text: these

text: three

text: points.

text: The

text: ﬁrst

text: such

text: move

text: is

text: over

text: a

text: distance

text: of

text: order

text: e

text: and

text: subsequent

text: moves

text: decrease

text: geometrically.

text: So

text: in

text: the

text: t

text: limit

text: the

text: positions

text: of

text: the

text: boundaries

text: near

text: (z

text: z

text: z

text: should

text: become

text: (z

text: D

text: e

text: z

text: D

text: e

text: z

text: D

text: e

text: for

text: some

text: (D

text: D

text: D

text: with

text: a

text: non-vanishing

text: limit

text: as

text: t

text: But

text: this

text: is

text: saying

text: that

text: in

text: the

text: limit

text: partition

text: ζ),

text: ζ

text: Ξ≤

text: on

text: every

text: scale

text: σ

text: y

text: y

text: for

text: y

text: and

text: y

text: on

text: the

text: boundary,

text: the

text: distance

text: from

text: the

text: midpoint

text: (y

text: y

text: to

text: the

text: boundary

text: is

text: of

text: the

text: form

text: Dσ

text: for

text: some

text: random

text: D

text: This

text: is

text: a

text: hallmark

text: of

text: “fractal

text: dimension

text: Heuristic

text: the

text: boundary

text: has

text: fractal

text: dimension

text: The

text: genealogical

text: tree

text: deﬁnes

text: a

text: “line

text: of

text: descent”

text: for

text: each

text: particle

text: in

text: Ξ;

text: R

text: these

text: particle

text: positions

text: are

text: dense

text: in

text: so

text: let

text: us

text: suppose

text: that

text: in

text: the

text: contin-

text: R

text: uum

text: limit

text: there

text: is

text: such

text: a

text: “line

text: of

text: descent”

text: from

text: almost

text: all

text: points

text: z

text: of

text: to

text: R

text: inﬁnity.

text: Draw

text: the

text: tree

text: via

text: line

text: segments

text: in

text: Consider

text: a

text: point

text: (x,

text: on

text: the

text: x-axis.

text: The

text: route

text: from

text: there

text: to

text: inﬁnity

text: ﬁrst

text: crosses

text: the

text: y

text: line

text: at

text: some

text: C

text: random

text: point

text: (c(x),

text: Consider

text: the

text: random

text: set

text: of

text: all

text: such

text: values

text: c(x)

text: as

text: x

text: varies.

text: This

text: is

text: stationary

text: (translation-invariant)

text: and

text: so

text: has

text: some

text: intensity


text: This is the set of particles born before t whose time-t ancestor in the 2 1 line of descent was ζ. In the coloring story, this is “the set of particles at time t which have inherited the same color as ζ, if we gave all the particles 2 ≤ ∈ at t diﬀerent colors”. Then, still for t t and ζ Ξ≤ , deﬁne 1 1 2 t1 − µ is the measure µ putting weight e t2 t1,t2,ζ on the position of each particle in Descend(t , t , ζ). (3) 1 2 M R So µ is a random element of the space ( 2) of ﬁnite measures R t1,t2,ζ on 2, equipped with the usual topology of weak convergence. To obtain the limit theorem we ﬁrst show (Proposition 24 in section ) that there exist → ∞ M R t limits in probability (as ( 2)-valued random variables); that is, 2 { ∈ } there exist random measures µ ∞ : ζ Ξ≤ such that t1, ,ζ t1 → → ∞ ∀ ∈ µ µ ∞ in probability as t , ( ζ Ξ≤ ). (4) t1,t2,ζ t1, ,ζ 2 t1 The proof essentially relies on Proposition 4 and self-similarity. We then use Proposition 13 to show that a limit µ ∞ is in fact Lebesgue measure t1, ,ζ { restricted to some random set A(t , ζ), implying that the collection A(t , ζ) : ∈ } 1 R 1 ζ Ξ≤ is necessarily a partition of 2. t1 For ﬁxed t we can regard { ∈ } Z(t) = (z , A(t, ξ)) : ξ Ξ≤ ξ t −∞ ∞ as a marked point process. As t increases, the process (Z(t), < t < ) evolves in a way one can describe qualitatively: new points arrive randomly at rate et per unit area per unit time; when a point ξ arrives at time t, the region A(t, ζ) associated with the closest existing point ζ is split into two regions A(t + dt, ζ) and A(t + dt, ξ). But the probability distribution over possible splits depends on Z(t) in some complicated way which we are unable to describe explicitly. However, the key feature of this process is that as t decreases the regions merge according to the simple rule stated earlier. To summarize: 6

text: This

text: is

text: the

text: set

text: of

text: particles

text: born

text: before

text: t

text: whose

text: time-t

text: ancestor

text: in

text: the

text: line

text: of

text: descent

text: was

text: ζ.

text: In

text: the

text: coloring

text: story,

text: this

text: is

text: “the

text: set

text: of

text: particles

text: at

text: time

text: t

text: which

text: have

text: inherited

text: the

text: same

text: color

text: as

text: ζ,

text: if

text: we

text: gave

text: all

text: the

text: particles

text: at

text: t

text: diﬀerent

text: colors”.

text: Then,

text: still

text: for

text: t

text: t

text: and

text: ζ

text: Ξ≤

text: deﬁne

text: µ

text: is

text: the

text: measure

text: µ

text: putting

text: weight

text: e

text: on

text: the

text: position

text: of

text: each

text: particle

text: in

text: Descend(t

text: t

text: ζ).

text: M

text: R

text: So

text: µ

text: is

text: a

text: random

text: element

text: of

text: the

text: space

text: of

text: ﬁnite

text: measures

text: R

text: on

text: equipped

text: with

text: the

text: usual

text: topology

text: of

text: weak

text: convergence.

text: To

text: obtain

text: the

text: limit

text: theorem

text: we

text: ﬁrst

text: show

text: (Proposition

text: in

text: section

text: that

text: there

text: exist

text: M

text: R

text: t

text: limits

text: in

text: probability

text: (as

text: random

text: variables);

text: that

text: is,

text: there

text: exist

text: random

text: measures

text: µ

text: ζ

text: Ξ≤

text: such

text: that

text: ,ζ

text: µ

text: µ

text: in

text: probability

text: as

text: t

text: ζ

text: Ξ≤

text: ,ζ

text: The

text: proof

text: essentially

text: relies

text: on

text: Proposition

text: and

text: self-similarity.

text: We

text: then

text: use

text: Proposition

text: to

text: show

text: that

text: a

text: limit

text: µ

text: is

text: in

text: fact

text: Lebesgue

text: measure

text: ,ζ

text: restricted

text: to

text: some

text: random

text: set

text: A(t

text: ζ),

text: implying

text: that

text: the

text: collection

text: A(t

text: ζ)

text: R

text: ζ

text: Ξ≤

text: is

text: necessarily

text: a

text: partition

text: of

text: For

text: ﬁxed

text: t

text: we

text: can

text: regard

text: Z(t)

text: (z

text: A(t,

text: ξ))

text: ξ

text: Ξ≤

text: ξ

text: t

text: as

text: a

text: marked

text: point

text: process.

text: As

text: t

text: increases,

text: the

text: process

text: (Z(t),

text: t

text: evolves

text: in

text: a

text: way

text: one

text: can

text: describe

text: qualitatively:

text: new

text: points

text: arrive

text: randomly

text: at

text: rate

text: et

text: per

text: unit

text: area

text: per

text: unit

text: time;

text: when

text: a

text: point

text: ξ

text: arrives

text: at

text: time

text: t,

text: the

text: region

text: A(t,

text: ζ)

text: associated

text: with

text: the

text: closest

text: existing

text: point

text: ζ

text: is

text: split

text: into

text: two

text: regions

text: A(t

text: dt,

text: ζ)

text: and

text: A(t

text: dt,

text: ξ).

text: But

text: the

text: probability

text: distribution

text: over

text: possible

text: splits

text: depends

text: on

text: Z(t)

text: in

text: some

text: complicated

text: way

text: which

text: we

text: are

text: unable

text: to

text: describe

text: explicitly.

text: However,

text: the

text: key

text: feature

text: of

text: this

text: process

text: is

text: that

text: as

text: t

text: decreases

text: the

text: regions

text: merge

text: according

text: to

text: the

text: simple

text: rule

text: stated

text: earlier.

text: To

text: summarize:


text: 14 0.5 2.0 0.0 ()qt - ()/()δ0ttMM 1.5 δ 0.5 1.0 - 1.0 0 10 20 / 30 40 50 10 20 / 30 40 50 t Gyr t Gyr (cid:18) (cid:20) (cid:114) (cid:21)(cid:19) FIG. 12: The evolution of the deceleration param- FIG. 14: The evolution of the ratio of the matter (cid:18) (cid:20) (cid:114) (cid:21)(cid:19) eter q(t) with cosmic time for the model f(T) = perturbation parameter δ (t) to its current value M − − T δ (t ) with cosmic time for the model f(T) = αT 1 exp p for p = 2 and 5 and the M 0 0 T − − T 0 αT 1 exp p for p = 2 and 5 and the ΛCDM model. All models exhibit the same sta−rting 0 T0 value of 0.5 and late times limiting value of 1.0. ΛCDM model. Both models mimic the ΛCDM Each model transitions to an accele≈rating universe (solid) behaviour; each start at a large value at early at approximately the same time (t 8 Gyr). The times, decay to approximately 1 at around today’s p = 5 (dashed) model greatly mimics the ΛCDM time and then keeps on decreasing until reaching a (solid) evolution whilst the p = 2 (dotted) model, al- constant value, meaning that the perturbation is sta- though having a similar behaviour, diﬀers from the ble but persists. The diﬀerence lies in the values, latter. but with the major diﬀerence occurring for p = 2 ≈ (dotted) [for ΛCDM, δ(t)/δ (t ) 0.85 whilst for ≈ 0 6 p = 2, δ(t≈)/δ (t0) 0.70 and for p = 5 (dashed), δ(t)/δ (t ) 0.80]. 0 5 4 perturbations are stable. For the Hubble pertur- ()/()δδtt0 3 b(paotiwoenr,-liatwis) eoirthbeerindgeciadyeinntgicatollyzezreoroat(dlaet-Seittitmere)s. 2 On the other hand, the matter perturbation ei- 1 ther decays to zero at late times (power-law) or is constant (de-Sitter) meaning that such perturba- 0 tions persist. Although such solutions are stable, / 0 10 20 30 40 50 a problem arises due to tt-Friedmann equation t Gyr Eq. (12) since this restricts the possible functions FIG.(cid:18)13: Th(cid:20)e ev(cid:114)olutio(cid:21)n(cid:19) of the ratio of the Hub- of f(T) where power-law and de-Sitter evolution ble perturbation parameter δ(t) to its current value can be considered. In fact, we have shown that δ (t ) with cosmic time for the model f(T) = 0 the only functions which allow these types of evo- − − T αT 1 exp p for p = 2 and 5 and the lutions are TEGR and rescaled TEGR. Nonethe- 0 T 0 less, since the universe is composed of a mix of ΛCDM model. In this case, the models exhibit the ﬂuids and hence the history of the universe is not same behaviour as ΛCDM (solid), starting from large values, decreasing and approaching approximately 1 described by either a power-law or exponential at at around present time and then decay to zero at all times (these serve as a good approximation at late times, meaning these perturbations are stable certain eras), we instead have set our focus on and do not persist. The p = 5 (dashed) model is non-trivial f(T) functions which mimic ΛCDM, very close to the ΛCDM model behaviour whilst the which is a much more accurate description of the p = 2 (dotted) model diﬀers from the other two mod- history of the universe. els at earlier times. The ﬁrst model considered in this paper is the power-law model. By examining the exotic ﬂuid’s the Hubble parameter, which ultimately deter- EoS parameter and deceleration parameter at mines whether such solutions are stable. This current times, it was concluded that two possible − was analysed under power-law and de-Sitter evo- non-trivial models can be obtained, n = 1 and − lutions, as well as under two separate f(T) mod- n = 2. Such models were found to be stable, els. and mimic the behaviour of ΛCDM. The Hubble For power-law and de-Sitter evolutions, the perturbation decays with cosmic time and ap- ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

font: default

text: ()qt

text: δ

text: t

text: Gyr

text: t

text: Gyr

text: FIG.

text: The

text: evolution

text: of

text: the

text: deceleration

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text: FIG.

text: The

text: evolution

text: of

text: the

text: ratio

text: of

text: the

text: matter

text: eter

text: q(t)

text: with

text: cosmic

text: time

text: for

text: the

text: model

text: f(T)

text: perturbation

text: parameter

text: δ

text: (t)

text: to

text: its

text: current

text: value

text: M

text: T

text: δ

text: (t

text: with

text: cosmic

text: time

text: for

text: the

text: model

text: f(T)

text: αT

text: exp

text: p

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text: p

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text: the

text: M

text: T

text: T

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text: p

text: for

text: p

text: and

text: and

text: the

text: ΛCDM

text: model.

text: All

text: models

text: exhibit

text: the

text: same

text: sta−rting

text: value

text: of

text: and

text: late

text: times

text: limiting

text: value

text: of

text: ΛCDM

text: model.

text: Both

text: models

text: mimic

text: the

text: ΛCDM

text: Each

text: model

text: transitions

text: to

text: an

text: accele≈rating

text: universe

text: (solid)

text: behaviour;

text: each

text: start

text: at

text: a

text: large

text: value

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text: early

text: at

text: approximately

text: the

text: same

text: time

text: (t

text: Gyr).

text: The

text: times,

text: decay

text: to

text: approximately

text: at

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text: p

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text: the

text: ΛCDM

text: time

text: and

text: then

text: keeps

text: on

text: decreasing

text: until

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text: a

text: (solid)

text: evolution

text: whilst

text: the

text: p

text: (dotted)

text: model,

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text: constant

text: value,

text: meaning

text: that

text: the

text: perturbation

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text: a

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text: from

text: the

text: ble

text: but

text: persists.

text: The

text: diﬀerence

text: lies

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text: the

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text: latter.

text: but

text: with

text: the

text: major

text: diﬀerence

text: occurring

text: for

text: p

text: (dotted)

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text: ΛCDM,

text: δ(t)/δ

text: (t

text: whilst

text: for

text: p

text: δ(t≈)/δ

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text: for

text: p

text: (dashed),

text: δ(t)/δ

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text: stable.

text: For

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text: b(paotiwoenr,-liatwis)

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text: the

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text: hand,

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text: ther

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text: zero

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text: late

text: times

text: (power-law)

text: or

text: is

text: constant

text: (de-Sitter)

text: meaning

text: that

text: such

text: perturba-

text: tions

text: persist.

text: Although

text: such

text: solutions

text: are

text: stable,

text: a

text: problem

text: arises

text: due

text: to

text: tt-Friedmann

text: equation

text: t

text: Gyr

text: Eq.

text: since

text: this

text: restricts

text: the

text: possible

text: functions

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text: the

text: ratio

text: of

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text: of

text: f(T)

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text: perturbation

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text: be

text: considered.

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text: fact,

text: we

text: have

text: shown

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text: δ

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text: T

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text: T

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text: the

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text: composed

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text: ΛCDM

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text: this

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text: the

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text: ﬂuids

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text: the

text: history

text: of

text: the

text: universe

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text: not

text: same

text: behaviour

text: as

text: ΛCDM

text: (solid),

text: starting

text: from

text: large

text: values,

text: decreasing

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text: approaching

text: approximately

text: described

text: by

text: either

text: a

text: power-law

text: or

text: exponential

text: at

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text: around

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text: time

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text: decay

text: to

text: zero

text: at

text: all

text: times

text: (these

text: serve

text: as

text: a

text: good

text: approximation

text: at

text: late

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text: meaning

text: these

text: perturbations

text: are

text: stable

text: certain

text: eras),

text: we

text: instead

text: have

text: set

text: our

text: focus

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text: and

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text: The

text: p

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text: non-trivial

text: f(T)

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text: mimic

text: ΛCDM,

text: very

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text: the

text: ΛCDM

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text: the

text: which

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text: history

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text: els

text: at

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text: times.

text: The

text: ﬁrst

text: model

text: considered

text: in

text: this

text: paper

text: is

text: the

text: power-law

text: model.

text: By

text: examining

text: the

text: exotic

text: ﬂuid’s

text: the

text: Hubble

text: parameter,

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text: EoS

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text: deceleration

text: parameter

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text: mines

text: whether

text: such

text: solutions

text: are

text: stable.

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text: current

text: times,

text: it

text: was

text: concluded

text: that

text: two

text: possible

text: was

text: analysed

text: under

text: power-law

text: and

text: de-Sitter

text: evo-

text: non-trivial

text: models

text: can

text: be

text: obtained,

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text: and

text: lutions,

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text: well

text: as

text: under

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text: f(T)

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text: n

text: Such

text: models

text: were

text: found

text: to

text: be

text: stable,

text: els.

text: and

text: mimic

text: the

text: behaviour

text: of

text: ΛCDM.

text: The

text: Hubble

text: For

text: power-law

text: and

text: de-Sitter

text: evolutions,

text: the

text: perturbation

text: decays

text: with

text: cosmic

text: time

text: and

text: ap-

text: ##LTFigure##

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