Eviden e for a ontinuum limit in ausal set dynami s

D. P. Rideout

�

and R. D. Sorkin

y

Department of Physi s, Syra use University

Syra use, NY, 13244-1130, U.S.A.

November 1, 2003

Abstra t

We �nd eviden e for a ontinuum limit of a parti ular ausal set dynami s whi h depends on only

a single \ oupling onstant" p and is easy to simulate on a omputer. The model in question is a

sto hasti pro ess that an also be interpreted as 1-dimensional dire ted per olation, or in terms of

random graphs.

1 Introdu tion

In an earlier paper [1℄ we investigated a type of ausal set dynami s that an be des ribed as a ( lassi ally)

sto hasti pro ess of growth or \a retion". In a language natural to that dynami s, the passage of time

onsists in the ontinual birth of new elements of the ausal set and the history of a sequen e of su h

births an be represented as an upward path through a poset of all �nite ausal sets. We alled su h a

sto hasti pro ess a sequential growth dynami s be ause the elements arise singly, rather than in pairs or

larger multiplets.

A sequential des ription of this sort is advantageous in representing the future as developing out of

the past, but on the other hand it ould seem to rely on an external parameter time (the \time" in

whi h the growth o urs), thereby violating the prin iple that physi al time is en oded in the intrinsi

order-relation of the ausal set and nothing else. If physi ally real, su h a parameter time would yield a

distinguished labeling of the elements and thereby a notion of \absolute simultaneity", in ontradi tion to

the lessons of both spe ial and general relativity. To avoid su h a onsequen e, we postulated a prin iple

of dis rete general ovarian e, a ording to whi h no probability of the theory an depend on | and no

physi ally meaningful question an refer to | the imputed order of births, ex ept insofar as that order

re e ts the intrinsi pre eden e relation of the ausal set itself.

To dis rete general ovarian e, we added two other prin iples that we alled Bell ausality and internal

temporality. The �rst is a dis rete analog of the ondition that no in uen e an propagate faster than

light, and the se ond simply requires that no element be born to the past of any existing element.

1

These

prin iples led us almost uniquely to a family of dynami al laws (sto hasti pro esses) parameterized by a

ountable sequen e of oupling onstants q

n

. In addition to this generi family, there are some ex eptional

families of solutions, but we onje ture that they are all singular limits of the generi family. We have

he ked in parti ular that \originary per olation" (see se tion 2) is su h a limit.

2

�

rideout�physi s.syr.edu

y

sorkin�physi s.syr.edu

1

This last ondition guarantees that the \parameter time" of our sto hasti pro ess is ompatible with physi al temporality,

as re orded in the order relation � that gives the ausal set its stru ture. In a broader sense, general ovarian e itself is also

an aspe t of internal temporality, sin e it guarantees that the parameter time adds nothing to the relation �.

2

In the notation of [1℄, it is the A!1 limit of the dynami s given by t

0

= 1, t

n

= At

n

, n = 1; 2; 3; : : :.

1

Now among these dynami al laws, the one resulting from the hoi e q

n

= q

n

is one of the easiest

to work with, both on eptually and for purposes of omputer simulation. De�ned by a single real

parameter q 2 [0; 1℄, it is des ribed in more detail in Se tion 2 below. In [1℄, we referred to it as transitive

per olation be ause it an be interpreted in terms of a random \turning on" of nonlo al bonds (with

probability p = 1� q) in a one-dimensional latti e. Another thing making it an attra tive spe ial ase to

work with is the availability in the mathemati s literature of a number of results governing the asymptoti

behavior of posets generated in this manner [2, 3℄.

Aside from its onvenien e, this per olation dynami s, as we will all it, possesses other distinguishing

features, in luding an underlying time-reversal invarian e and a spe ial relevan e to ausal set osmology,

as we des ribe brie y below. In this paper, we sear h for eviden e of a ontinuum limit of per olation

dynami s.

One might question whether a ontinuum limit is even desirable in a fundamentally dis rete theory,

but a ontinuum approximation in a suitable regime is ertainly ne essary if the theory is to reprodu e

known physi s. Given this, it seems only a small step to a rigorous ontinuum limit, and onversely,

the existen e of su h a limit would en ourage the belief that the theory is apable of yielding ontinuum

physi s with suÆ ient a ura y.

Perhaps an analogy with kineti theory an guide us here. In quantum gravity, the dis reteness s ale

is set, presumably, by the Plan k length l = (��h)

1=2

(where � = 8�G), whose vanishing therefore signals

a ontinuum limit. In kineti theory, the dis reteness s ales are set by the mean free path � and the mean

free time � , both of whi h must go to zero for a des ription by partial di�erential equations to be ome

exa t. Corresponding to these two independent length and time s ales are two \ oupling onstants":

the di�usion onstant D and the speed of sound

sound

. Just as the value of the gravitational oupling

onstant G�h re e ts (presumably) the magnitude of the fundamental spa etime dis reteness s ale, so the

values of D and

sound

re e t the magnitudes of the mi ros opi parameters � and � a ording to the

relations

D �

�

2

�

;

sound

�

�

�

or onversely

� �

D

sound

; � �

D

2

sound

:

In a ontinuum limit of kineti theory, therefore, we must have either D ! 0 or

sound

!1. In the former

ase, we an hold

sound

�xed, but we get a purely me hani al ma ros opi world, without di�usion or

vis osity. In the latter ase, we an hold D �xed, but we get a \purely di�usive" world with me hani al

for es propagating at in�nite speed. In ea h ase we get a well de�ned | but defe tive | ontinuum

physi s, la king some features of the true, atomisti world.

If we an trust this analogy, then something very similar must hold in quantum gravity. To send l to

zero, we must make either G or �h vanish. In the former ase, we would expe t to obtain a quantum world

with the metri de oupled from non-gravitational matter; that is, we would expe t to get a theory of

quantum �eld theory in a purely lassi al ba kground spa etime solving the sour e-free Einstein equations.

In the latter ase, we would expe t to obtain lassi al general relativity. Thus, there might be two distin t

ontinuum limits of quantum gravity, ea h physi ally defe tive in its own way, but nonetheless well de�ned.

For our purposes in this paper, the important point is that, although we would not expe t quantum

gravity to exist as a ontinuum theory, it ould have limits whi h do, and one of these limits might be

lassi al general relativity. It is thus sensible to inquire whether one of the lassi al ausal set dynami s

we have de�ned des ribes lassi al spa etimes. In the following, we make a beginning on this question

by asking whether the spe ial ase of \per olated ausal sets", as we will all them, admits a ontinuum

limit at all.

Of ourse, the physi al ontent of any ontinuum limit we might �nd will depend on what we hold

�xed in passing to the limit, and this in turn is intimately linked to how we hoose the oarse-graining

2

pro edure that de�nes the e�e tive ma ros opi theory whose existen e the ontinuum limit signi�es.

Obviously, we will want to send N ! 1 for any ontinuum limit, but it is less evident how we should

oarse-grain and what oarse grained parameters we want to hold �xed in taking the limit. Indeed, the

appropriate hoi es will depend on whether the ma ros opi spa etime region we have in mind is, to take

some naturally arising examples, (i) a �xed bounded portion of Minkowski spa e of some dimension,

or (ii) an entire y le of a Friedmann universe from initial expansion to �nal re ollapse, or (iii) an N -

dependent portion of an unbounded spa etime M that expands to en ompass all of M as N ! 1. In

the sequel, we will have in mind primarily the �rst of the three examples just listed. Without attempting

an de�nitive analysis of the oarse-graining question, we will simply adopt the simplest de�nitions that

seem to us to be suited to this example. More spe i� ally, we will oarse-grain by randomly sele ting a

sub- ausal-set of a �xed number of elements, and we will hoose to hold �xed some onvenient invariants

of that sub- ausal-set, one of whi h an be interpreted

3

as the dimension of the spa etime region it

onstitutes. As we will see, the resulting s heme has mu h in ommon with the kind of oarse-graining

that goes into the de�nition of renormalizability in quantum �eld theory. For this reason, we believe it

an serve also as an instru tive \laboratory" in whi h this on ept, and related on epts like \running

oupling onstant" and \non-trivial �xed point", an be onsidered from a fresh perspe tive.

In the remaining se tions of this paper we: de�ne transitive per olation dynami s more pre isely;

spe ify the oarse-graining pro edure we have used; report on the simulations we have run looking for a

ontinuum limit in the sense thereby de�ned; and o�er some on luding omments.

1.1 De�nitions used in the sequel

Causal set theory postulates that spa etime, at its most fundamental level, is dis rete, and that its

ma ros opi geometri al properties re e t a deep stru ture whi h is purely order theoreti in nature.

This deep stru ture is taken to be a partial order and alled a ausal set (or \ auset" for short). For an

introdu tion to ausal set theory, see [4, 5, 6, 7℄. In this se tion, we merely re all some de�nitions whi h

we will be using in the sequel.

A (partial) order or poset is a set S endowed with a relation � whi h is:

transitive 8x; y; z 2 S x � y and y � z ) x � z

a y li 8x; y 2 S x � y ) y 6� x

irre exive 8x 2 S x 6� x

(Irre exivity is merely a onvention; with it, a y li ity is a tually redundant.) For example, the events

of Minkowski spa e (in any dimension) form a poset whose order relation is the usual ausal order. In an

order S, the interval int(x; y) is de�ned to be

int(x; y) = fz 2 Sjx � z � yg :

An order is said to be lo ally �nite if all its intervals are �nite (have �nite ardinality). A ausal set is a

lo ally �nite order.

It will be helpful to have names for some small ausal sets. Figure 1.1 provides su h names for the

ausal sets with three or fewer elements.

2 The dynami s of transitive per olation

Regarded as a sequential growth dynami s of the sort derived in [1℄, transitive per olation is des ribed

by one free parameter q su h that q

n

= q

n

. This is equivalent (at stage N of the growth pro ess) to using

the following \per olation" algorithm to generate a random auset.

3

This interpretation is stri tly orre t only if the ausal set forms an interval or \Alexandrov neighborhood" within the

spa etime.

3

1-chain

2-chain 2-antichain

3-chain "V" "L" 3-antichainΛ""

Figure 1: Names for small ausets

1. Start with N elements labeled 0; 1; 2; : : : ; N � 1.

2. With a �xed probability p (= 1 � q), introdu e a relation i � j between every pair of elements

labeled i and j, where i 2 f0 � � �N � 2g and j 2 fi+ 1 � � �N � 1g.

3. Form the transitive losure of these relations (e.g. if 2 � 5 and 5 � 8 then enfor e that 2 � 8.)

Given the simpli ity of this dynami al model, both on eptually and from an algorithmi standpoint, it

o�ers a \stepping stone" allowing us to look into some general features of ausal set dynami s. (The

name \per olation" omes from thinking of a relation i � j as a \bond" or \ hannel" between i and j.)

There exists another model whi h is very similar to transitive per olation, alled \originary transitive

per olation". The rule for randomly generating a auset is the same as for transitive per olation, ex ept

that ea h new element is required to be related to at least one existing element. Algorithmi ally, we

generate potential elements one by one, exa tly as for plain per olation, but dis ard any su h element

whi h would be unrelated to all previous elements. Causets formed with this dynami s always have a

single minimal element, an \origin".

Re ent work by Dou [8℄ suggests that originary per olation might have an important role to play in

osmology. Noti e �rst that, if a given osmologi al \ y le" ends with the auset ollapsing down to a

single element, then the ensuing re-expansion is ne essarily given by an originary auset. Now, in the

limited ontext of per olation dynami s, Alon et al. have proved rigorously [3℄ that su h osmologi al

\boun es" (whi h they all posts) o ur with probability 1 (if p > 0), from whi h it follows that there are

in�nitely many osmologi al y les, ea h y le but the �rst having the dynami s of originary per olation.

For more general hoi es of the dynami al parameters q

n

of [1℄, posts an again o ur, but now the q

n

take on new e�e tive values in ea h y le, related to the old ones by the a tion of a sort of \ osmologi al

renormalization group"; and Dou [8℄ has found eviden e that originary per olation is a \stable �xed

point" of this a tion, meaning that the universe would tend to evolve toward this behavior, no matter

what dynami s it began with.

It would thus be of interest to investigate the ontinuum limit of originary per olation as well as plain

per olation. In the present paper, however, we limit ourselves to the latter type, whi h we believe is

more appropriate (albeit not fully appropriate for reasons dis ussed in the on lusion) in the ontext of

spa etime regions of sub- osmologi al s ale.

4

3 The riti al point at p = 0, N =1

In the previous se tion we have introdu ed a model of random ausets, whi h depends on two parameters,

p 2 [0; 1℄ and N 2 N . For a given p, the model de�nes a probability distribution on the set of N -element

ausets.

4

For p = 0, the only auset with nonzero probability, obviously, is the N -anti hain. Now let

p > 0. With a little thought, one an onvin e oneself that for N ! 1, the auset will look very mu h

like a hain. Indeed it has been proved [9℄ (see also [10℄) that, as N !1 with p �xed at some (arbitrarily

small) positive number, r ! 1 in probability, where

r �

R

N(N � 1)=2

=

R

�

N

2

�

;

R being the number of relations in the auset, i.e. the number of pairs of auset elements x, y su h that

x � y or y � x. Note that the N - hain has the greatest possible number

�

N

2

�

of relations, so r ! 1 gives

a pre ise meaning to \looking like a hain". We all r the ordering fra tion of the ausal set, following

[11℄.

We see that for N ! 1, there is a hange in the qualitative nature of the auset as p varies away

from zero, and the point p = 0; N =1 (or p = 1=N = 0) is in this sense a riti al point of the model. It

is the behavior of the model near this riti al point whi h will on ern us in this paper.

4 Coarse graining

An advantageous feature of ausal sets is that there exists for them a simple yet pre ise notion of oarse

graining. A oarse grained approximation to a auset C an be formed by sele ting a sub- auset C

0

at random, with equal sele tion probability for ea h element, and with the ausal order of C

0

inherited

dire tly from that of C (i.e. x � y in C

0

if and only if x � y in C.)

For example, let us start with the 20 element auset C shown in Figure 2. (whi h was per olated

using p = 0:25), and su essively oarse grain it down to ausets of 10, 5 and 3 elements. We see that,

at the largest s ale shown (i.e. the smallest number of remaining elements), C has oarse-grained in this

instan e to the 3-element \V" auset. Of ourse, oarse graining itself is a random pro ess, so from a

single auset of N elements, it gives us in general, not another single auset, but a probability distribution

on the ausets of m < N elements.

A noteworthy feature of this de�nition of oarse graining, whi h in some ways is similar to what is

often alled \de imation" in the ontext of spin systems, is the random sele tion of a subset. In the

absen e of any ba kground latti e stru ture to refer to, no other possibility for sele ting a sub- auset is

evident. Random sele tion is also re ommended strongly by onsiderations of Lorentz invarian e [12℄.

The fa t that a oarse grained auset is automati ally another auset will make it easy for us to formulate

pre ise notions of ontinuum limit, running of the oupling onstant p, et . In this respe t, we believe

that this model ombines pre ision with novelty in su h a manner as to furnish an instru tive illustration

of on epts related to renormalizability, independently of its appli ation to quantum gravity. We remark

in this onne tion, that transitive per olation is readily embedded in a \two-temperature" statisti al

me hani s model, and as su h, happens also to be exa tly soluble in the sense that the partition fun tion

an be omputed exa tly [13, 14℄.

4

Stri tly speaking this distribution has gauge-invariant meaning only in the limit N !1 (p �xed); for it is only insofar

as the growth pro ess \runs to ompletion" that generally ovariant questions an be asked. Noti e that this limit is inherent

in ausal set dynami s itself, and has nothing to do with the ontinuum limit we are on erned with herein, whi h sends p

to zero as N !1.

5

AGAIN:

or

Figure 2: Three su essive oarse grainings of a 20-element auset

5 The large s ale e�e tive theory

In se tion 2 we des ribed a \mi ros opi " dynami s for ausal sets (that of transitive per olation) and

in se tion 4 we de�ned a pre ise notion of oarse graining (that of random sele tion of a sub- ausal-set).

On this basis, we an produ e an e�e tive \ma ros opi " dynami s by imagining that a auset C is �rst

per olated with N elements and then oarse-grained down to m < N elements. This two-step pro ess

onstitutes an e�e tive random pro edure for generating m element ausets depending (in addition to m)

on the parameters N and p. In ausal set theory, number of elements orresponds to spa etime volume,

so we an interpret N=m as the fa tor by whi h the \observation s ale" has been in reased by the oarse

graining. If, then, V

0

is the ma ros opi volume of the spa etime region onstituted by our auset, and if

we take V

0

to be �xed as N !1, then our pro edure for generating ausets of m elements provides the

e�e tive dynami s at volume-s ale V

0

=m (i.e. length s ale (V

0

=m)

1=d

for a spa etime of dimension d).

What does it mean for our e�e tive theory to have a ontinuum limit in this ontext? Our sto hasti

mi ros opi dynami s gives, for ea h hoi e of p, a probability distribution on the set of ausal sets C

with N elements, and by hoosing m, we determine at whi h s ale we wish to examine the orresponding

e�e tive theory. This e�e tive theory is itself just a probability distribution f

m

on the set of m-element

ausets, and so our dynami s will have a well de�ned ontinuum limit if there exists, as N ! 1, a

traje tory p = p(N) along whi h the orresponding probability distributions f

m

on oarse grained ausets

approa h �xed limiting distributions f

1

m

for all m. The limiting theory in this sense is then a sequen e of

e�e tive theories, one for ea hm, all �tting together onsistently. (Thanks to the asso iative (semi-group)

hara ter of our oarse-graining pro edure, the existen e of a limiting distribution for any given m implies

its existen e for all smallerm. Thus it suÆ es that a limiting distribution f

m

exist form arbitrarily large.)

In general there will exist not just a single su h traje tory p = p(N), but a one-parameter family of them

( orresponding to the one real parameter p that hara terizes the mi ros opi dynami s at any �xed N),

6

and one may wonder whether all the traje tories will take on the same asymptoti form as they approa h

the riti al point p = 1=N = 0.

Consider �rst the simplest nontrivial ase, m = 2. Sin e there are only two ausal sets of size two,

the 2- hain and the 2-anti hain, the distribution f

2

that gives the \large s ale physi s" in this ase is

des ribed by a single number whi h we an take to be f

2

( r

r

), the probability of obtaining a 2- hain

rather than a 2-anti hain. (The other probability, f

2

( r r), is of ourse not independent, sin e lassi al

probabilities must add up to unity.)

Interestingly enough, the number f

2

( r

r

) has a dire t physi al interpretation in terms of the Myrheim-

Meyer dimension of the �ne-grained auset C. Indeed, it is easy to see that f

2

( r

r

) is nothing but the

expe tation value of what was alled above the \ordering fra tion" of C. But the ordering fra tion, in turn,

determines the Myrheim-Meyer dimension d that indi ates the dimension of the Minkowski spa etime M

d

(if any) in whi h C would embed faithfully as an interval [15, 11℄. Thus, by oarse graining down to two

elements, we are e�e tively measuring a ertain kind of spa etime dimensionality of C. In pra ti e, we

would not expe t C to embed faithfully without some degree of oarse-graining, but the original r would

still provide a good dimension estimate sin e it is, on average, oarse-graining invariant.

As we begin to onsider oarse-graining to sizes m > 2, the degree of ompli ation grows rapidly,

simply be ause the number of partial orders de�ned on m elements grows rapidly with m. For m = 3

there are �ve possible ausal sets: r

r

r

, r

r r

A

A

�

�

, r r

r

,

r

r r�

�

A

A

, and

r r r

. Thus the e�e tive dynami s at this \s ale"

is given by �ve probabilities (so four free parameters). For m = 4 there are sixteen probabilities, for

m = 5 there are sixty three, and for m = 6, 7 and 8, the number of probabilities is respe tively 318, 2045

and 16999.

6 Eviden e from simulations

In this se tion, we report on some omputer simulations that address dire tly the question whether

transitive per olation possesses a ontinuum limit in the sense de�ned above. In a subsequent paper,

we will report on simulations addressing the subsidiary question of a possible s aling behavior in the

ontinuum limit.

In order that a ontinuum limit exist, it must be possible to hoose a traje tory for p as a fun tion of

N so that the resulting oarse-grained probability distributions, f

1

, f

2

, f

3

, . . . , have well de�ned limits as

N !1. To study this question numeri ally, one an simulate transitive per olation using the algorithm

des ribed in Se tion 2, while hoosing p so as to hold onstant (say) the m = 2 distribution f

2

(f

1

being

trivial). Be ause of the way transitive per olation is de�ned, it is intuitively obvious that p an be hosen

to a hieve this, and that in doing so, one leaves p with no further freedom. The de isive question then is

whether, along the traje tory thereby de�ned, the higher distribution fun tions, f

3

, f

4

, et . all approa h

nontrivial limits.

As we have already mentioned, holding f

2

�xed is the same thing as holding �xed the expe tation

value < r > of ordering fra tion r = R=

�

N

2

�

. To see in more detail why this is so, onsider the oarse-

graining that takes us from the original auset C

N

of N elements to a auset C

2

of two elements. Sin e

oarse-graining is just random sele tion, the probability f

2

( r

r

) that C

2

turns out to be a 2- hain is just

the probability that two elements of C

N

sele ted at random form a 2- hain rather than a 2-anti hain. In

other words, it is just the probability that two elements of C

N

sele ted at random are ausally related.

Plainly, this is the same as the fra tion of pairs of elements of C

N

su h that the two members of the pair

form a relation x � y or y � x. Therefore, the ordering fra tion r equals the probability of getting a

2- hain when oarse graining C

N

down to two elements; and f

2

( r

r

) =<r>, as laimed.

This reasoning illustrates, in fa t, how one an in prin iple determine any one of the distributions f

m

by answering the question, \What is the probability of getting this parti ular m-element auset from this

parti ular N -element auset if you oarse grain down to m elements?" To ompute the answer to su h

7

a question starting with any given auset C

N

, one examines every possible ombination of m elements,

ounts the number of times that the ombination forms the parti ular auset being looked for, and divides

the total by

�

N

m

�

. The ensemble mean of the resulting abundan e, as we will refer to it, is then f

m

(�),

where � is the auset being looked for. In pra ti e, of ourse, we would normally use a more eÆ ient

ounting algorithm than simply examining individually all

�

N

m

�

subsets of C

N

.

6.1 Histograms of 2- hain and 4- hain abundan es

number of relations

0

75

150

225

300

375

450

525

15,260 causets

6,630,255 6,822,375mean=6,722,782

skewness=-0.027kurtosis=2.993

Figure 3: Distribution of number of relations for N = 4096, p = 0:01155

As explained in the previous subse tion, the main omputational problem, on e the random auset

has been generated, is determining the number of sub ausets of di�erent sizes and types. To get a feel for

how some of the resulting \abundan es" are distributed, we start by presenting a ouple of histograms.

Figure 6.1 shows the number R of relations obtained from a simulation in whi h 15,260 ausal sets were

generated by transitive per olation with p = 0:01155, N = 4096. Visually, the distribution is Gaussian,

in agreement with the fa t that its \kurtosis"

(x� x)

4

�

(x� x)

2

2

of 2.993 is very nearly equal to its Gaussian value of 3 (the over-bar denotes sample mean). In these

simulations, p was hosen so that the number of 3- hains was equal on average to half the total number

possible, i.e. the \abundan e of 3- hains", (number of 3- hains)=

�

N

3

�

, was equal to 1=2 on average. The

pi ture is qualitatively identi al if one ounts 4- hains rather than 2- hains, as exhibited in Fig. 4.

(One may wonder whether it was to be expe ted that these distributions would appear to be so

normal. If the variable in question, here the number of 2- hains R or the number of 4- hains (C

4

, say),

8

number of 4-chains

0

75

150

225

300

375

450

525

600

15,260 causets

skewness=0.031kurtosis=2.99

2,476,985,149,915 3,062,430,629,438mean=2,745,459,887,579

Figure 4: Distribution of number of 4- hains for N = 4096, p = 0:01155

an be expressed as a sum of independent random variables, then the entral limit theorem provides an

explanation. So onsider the variables x

ij

whi h are 1 if i � j and zero otherwise. Then R is easily

expressed as a sum of these variables:

R =

X

i<j

x

ij

However, the x

ij

are not independent, due to transitivity. Apparently, this dependen e is not large enough

to interfere mu h with the normality of their sum. The number of 4- hains C

4

an be expressed in a

similar manner

C

4

=

X

i<j<k<l

x

ij

x

jk

x

kl

:

and similar remarks apply.)

Let us mention that for values of p suÆ iently lose to 0 or 1, these distributions will appear skew.

This o urs simply be ause the numbers under onsideration (e.g. the number of m- hains) are bounded

between zero and

�

N

m

�

and must deviate from normality if their mean gets too lose to a boundary relative

to the size of their standard deviation. Whenever we draw an error bar in the following, we will ignore

any deviation from normality in the orresponding distribution.

Noti e in identally that the total number of 4- hains possible is

�

4096

4

�

= 11; 710; 951; 848; 960. Con-

sequently, the mean 4- hain abundan e

5

in our simulation is only

2;745;459;887;579

11;710;951;848;960

= 0:234, a onsiderably

smaller value than the 2- hain abundan e of r =

6;722;782

(

4096

2

)

= 0:802. This was to be expe ted, onsidering

that the 2- hain is one of only two possible ausets of its size, while the 4- hain is one out 16 possibilities.

5

From this point on we will usually write simply \abundan e", in pla e of \mean abundan e", assuming the average is

obvious from ontext.

9

(Noti e also that 4- hains are ne essarily less probable than 2- hains, be ause every oarse-graining of a

4- hain is a 2- hain, whereas the 2- hain an ome from every 4-element auset save the 4-anti hain.)

6.2 Traje tories of p versus N

The question we are exploring is whether there exist, for N !1, traje tories p = p(N) along whi h the

mean abundan es of all �nite ausets tend to de�nite limits. To seek su h traje tories numeri ally, we will

sele t some �nite \referen e auset" and determine, for a range of N , those values of p whi h maintain

its abundan e at some target value. If a ontinuum limit does exist, then it should not matter in the end

whi h auset we sele t as our referen e, sin e any other hoi e (together with a mat hing hoi e of target

abundan e) should produ e the same traje tory asymptoti ally. We would also anti ipate that all the

traje tories would behave similarly for large N , and that, in parti ular, either all would lead to ontinuum

limits or all would not. In prin iple it ould happen that only a ertain subset led to ontinuum limits,

but we know of no reason to expe t su h an eventuality. In the simulations reported here, we have hosen

as our referen e ausets the 2-, 3- and 5- hains. We have omputed six traje tories, holding the 2- hain

abundan e �xed at 1/2, 1/3, and 1/10, the 3- hain abundan e �xed at 1/2 and .0814837, and the 5- hain

abundan e �xed at 1/2. For N , we have used as large a range as our omputers would allow.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07

orde

ring

frac

tion

= <

r>

p

cpfit = e

1

c = 1.348g = .001144

(1- ) g

Figure 5: Ordering fra tions as a fun tion of p for N = 2048

Before dis ussing the traje tories as su h, let us have a look at how the mean 2- hain abundan e <r>

(i.e. the mean ordering fra tion) varies with p for a �xed N of 2048, as exhibited in Figure 5. (Verti al

error bars are displayed in the �gure but are so small that they just look like horizontal lines. The plotted

points were obtained from an exa t expression for the ensemble average <r>, so the errors ome only

from oating point roundo�. The �tting fun tion used in Figure 5 will be dis ussed in a subsequent

paper [14℄, where we examine s aling behavior; see also [2℄.) As one an see, <r> starts at 0 for p = 0,

10

rises rapidly to near 1 and then asymptotes to 1 at p = 1 (not shown). Of ourse, it was evident a priori

that <r> would in rease monotoni ally from 0 to 1 as p varied between these same two values, but it is

perhaps noteworthy that its graph betrays no sign of dis ontinuity or non-analyti ity (no sign of a \phase

transition"). To this extent, it strengthens the expe tation that the traje tories we �nd will all share the

same qualitative behavior as N !1.

-16

-14

-12

-10

-8

-6

-4

-2

0

0 5 10 15 20

log

p

log N

5-chains 1/23-chains 1/23-chains .08152-chains 1/22-chains 1/32-chains 1/10

2

2

Figure 6: Flow of the \ oupling onstant" p as N !1 (six traje tories)

The six traje tories we have simulated are depi ted in Fig. 6.

6

A higher abundan e of m- hains

for �xed m leads to a traje tory with higher p. Also note that, as observed above, the longer hains

require larger values of p to attain the same mean abundan e, hen e a hoi e of mean abundan e = 1/2

orresponds in ea h ase to a di�erent traje tory. The traje tories with < r > held to lower values are

\higher dimensional" in the sense that < r >= 1=2 orresponds to a Myrheim-Meyer dimension of 2,

while < r >= 1=10 orresponds to a Myrheim-Meyer dimension of 4. Observe that the plots give the

impression of be oming straight lines with a ommon slope at large N . This tends to orroborate the

expe tation that they will exhibit some form of s aling with a ommon exponent, a behavior reminis ent

of that found with ontinuum limits in many other ontexts. This is further suggested by the fa t that

two distin t traje tories (f

2

( r

r

) = 1=2 and f

3

( r

r

r

) = :0814837), obtained by holding di�erent abundan es

�xed, seem to onverge for large N .

By taking the abs issa to be 1=N rather than log

2

N , we an bring the riti al point to the origin,

as in Fig. 7. The lines whi h pass through the data points there are just splines drawn to aid the eye

in following the traje tories. Note that the urves tend to asymptote to the p-axis, suggesting that p

falls o� more slowly than 1=N . This suggestion is orroborated by more detailed analysis of the s aling

behavior of these traje tories, as will be dis ussed in [14℄.

6

Noti e that the error bars are shown rotated in the legend. This will be the ase for all subsequent legends as well.

11

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

p

1/N

3-chains .08153-chains 1/2

2-chains 1/22-chains 1/3

5-chains 1/2

2-chains 1/10

Figure 7: Six traje tories approa hing the riti al point at p = 0, N =1

6.3 Flow of the oarse-grained theory along a traje tory

We ome �nally to a dire t test of whether the oarse-grained theory onverges to a limit as N ! 1.

Independently of s aling or any other indi ator, this is by de�nition the riterion for a ontinuum limit to

exist. We have examined this question by means of simulations ondu ted for �ve of the six traje tories

mentioned above. In ea h simulation we pro eeded as follows. For ea h hosen N , we experimentally

found a p suÆ iently lose to the desired traje tory. Having determined p, we then generated a large

number of ausets by the per olation algorithm des ribed in Se tion 2. (The number generated varied from

64 to 40,000.) For ea h su h random auset, we omputed the abundan es of the di�erent m-element

(sub) ausets under onsideration (2- hain, 3- hain, 3-anti hain, et ), and we ombined the results to

obtain the mean abundan es we have plotted here, together with their standard errors. (The errors

shown do not in lude any ontribution from the slight ina ura y in the value of p used. Ex ept for the

3- and 5- hain traje tories these errors are negligibly small.)

To ompute the abundan es of the 2-, 3-, and 4-orders for a given auset, we randomly sampled its

four-element sub ausets, ounting the number of times ea h of the sixteen possible 4-orders arose, and

dividing ea h of these ounts by the number of samples taken to get the orresponding abundan e. As an

aid in identifying to whi h 4-order a sampled sub auset belonged we used the following invariant, whi h

distinguishes all of the sixteen 4-orders, save two pairs.

I(S) =

Y

x2S

(2 + jpast(x)j)

Here, past(x) = fy 2 Sjy � xg is the ex lusive past of the element x and jpast(x)j is its ardinality.

Thus, we asso iate to ea h element of the auset, a number whi h is two more than the ardinality of its

ex lusive past, and we form the produ t of these numbers (four, in this ase) to get our invariant. (For

example, this invariant is 90 for the \diamond" poset, r

r

r

r

��

HH

.)

12

The number of samples taken from an N element auset was hosen to be

q

2

�

N

4

�

, on the grounds

that the probability to get the same four element subset twi e be omes appre iable with more than this

many samples. Numeri al tests on�rmed that this rule of thumb tends to minimize the sampling error,

as seen in Figure 8.

N4( )2

p = 0.1abundance = 0.140

N = 256

N4( )

( )

0.0001

0.001

0.01

0.1

1 10 100 1000 10000 100000 1e+06 1e+07 1e+08 1e+09

Err

or in

mea

n ab

unda

nce

of d

iam

onds

Number of samples

Figure 8: Redu tion of error in estimated diamond abundan e with in reasing number of samples

On e one has the abundan es of all the 4-orders, the abundan es of the smaller ausets an be found

by further oarse graining. By expli itly arrying out this oarse graining, one easily dedu es the following

relationships:

f

3

( r

r

r

) = f

4

( r

r

r

r

) +

1

2

�

f

4

( r

r

r r

��

) + f

4

( r

r

r

rHH )

�

+

1

4

f

4

( r

r

r

r) +

1

4

�

f

4

( r

r

r

r

��

) + f

4

( r

r

r

rHH

)

�

+

1

2

f

4

( r

r

r

r

��

HH

)

f

3

( r

r r

A

A

�

�

) =

1

2

f

4

( r

r

r r

��

) +

1

2

f

4

( r

r

r

r

��

) +

1

4

f

4

( r

r

r

r

��

HH

) +

3

4

f

4

(

r r r

rA

A

�

�

) +

1

4

f

4

(

r r

r rA

A

�

�

) +

1

4

f

4

( r

r

r

r

�

�

) +

1

2

f

4

( r

r

r

r

�

�

�

�

)

f

3

( r r

r

) =

3

4

f

4

( r

r

r

r) +

1

4

�

f

4

( r

r

r

r

��

) + f

4

( r

r

r

rHH

)

�

+

1

2

�

f

4

(

r r

r rA

A

�

�

) + f

4

(

r

rr r�

�

A

A

)

�

+ f

4

( r

r

r

r

) +

1

2

f

4

( r r r

r

) +

1

2

f

4

( r

r

r

r

�

�

)

f

3

(

r

r r�

�

A

A

) =

1

2

f

4

( r

r

r

rHH ) +

1

2

f

4

( r

r

r

rHH

) +

1

4

f

4

( r

r

r

r

��

HH

) +

3

4

f

4

( r r r

r

�

�

A

A

) +

1

4

f

4

(

r

rr r�

�

A

A

) +

1

4

f

4

( r

r

r

r

�

�

) +

1

2

f

4

( r

r

r

r

�

�

�

�

)

f

3

(

r r r

) =

1

4

�

f

4

(

r r r

rA

A

�

�

) + f

4

( r r r

r

�

�

A

A

)

�

+

1

4

�

f

4

(

r r

r rA

A

�

�

) + f

4

(

r

rr r�

�

A

A

)

�

+

1

2

f

4

( r r r

r

) + f

4

(

r r r r

)

f

2

( r

r

) = f

3

( r

r

r

) +

2

3

�

f

3

( r

r r

A

A

�

�

) + f

3

(

r

r r�

�

A

A

)

�

+

1

3

f

3

( r r

r

)

f

2

( r r) = 1� f

2

( r

r

)

In the �rst six equations, the oeÆ ient before ea h term on the right is the fra tion of oarse-grainings

of that auset whi h yield the auset on the left.

In Figures 9, 10, and 11, we exhibit how the oarse-grained probabilities of all possible 2, 3, and

4 element ausets vary as we follow the traje tory along whi h the oarse-grained 2- hain probability

f

2

( r

r

) = r is held at 1=2. By design, the oarse-grained probability for the 2- hain remains at at 50%,

13

0.492

0.494

0.496

0.498

0.5

0.502

0.504

0.506

0.508

0 2 4 6 8 10 12 14 16

Abu

ndan

ce

log N2

Figure 9: Flow of the oarse-grained probabilities f

m

for m = 2. The 2- hain probability is held at 1/2.

2

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0 2 4 6 8 10 12 14 16

Abu

ndan

ce

log N

Figure 10: Flow of the oarse-grained probabilities f

m

for m = 3. The 2- hain probability is held at 1/2.

14

2

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0 2 4 6 8 10 12 14 16

log N

Abu

ndan

ce

Figure 11: Flow of the oarse-grained probabilities f

m

for m = 4. The 2- hain probability is held at 1/2.

so Figure 9 simply shows the a ura y with whi h this was a hieved. (Observe the s ale on the verti al

axis.) Noti e that, sin e f

2

( r

r

) and f

2

( r r) must sum to 1, their error bars are ne essarily equal. (The

standard deviation in the abundan es de reases with in reasing N . The \blip" around log

2

N = 9 o urs

simply be ause we generated fewer ausets at that and larger values of N to redu e omputational osts.)

The ru ial question is whether the probabilities for the three and four element ausets tend to de�nite

limits as N tends to in�nity. Several features of the diagrams indi ate that this is indeed o urring. Most

obviously, all the urves, ex ept possibly a ouple in Figure 11, appear to be leveling o� at large N . But

we an bolster this on lusion by observing in whi h dire tion the urves are moving, and onsidering

their interrelationships.

For the moment let us fo us our attention on �gure 10. A priori there are �ve oarse-grained proba-

bilities to be followed. That they must add up to unity redu es the degrees of freedom to four. This is

redu ed further to three by the observation that, due to the time-reversal symmetry of the per olation

dynami s, we must have f

3

( r

r r

A

A

�

�

) = f

3

(

r

r r�

�

A

A

), as duly manifested in their graphs. Moreover, all �ve of the

urves appear to be monotoni , with the urves for

r

r r�

�

A

A

, r

r r

A

A

�

�

and

r r r

rising, and the urves for r

r

r

and r r

r

falling. If we a ept this indi ation of monotoni ity from the diagram, then �rst of all, every probability

f

3

(�) must onverge to some limiting value, be ause monotoni bounded fun tions always do; and some

of these limits must be nonzero, be ause the probabilities must add up to 1. Indeed, sin e f

3

( r

r r

A

A

�

�

) and

f

3

(

r

r r�

�

A

A

) are rising, they must onverge to some nonzero value, and this value must lie below 1/2 in order

that the total probability not ex eed unity. In onsequen e, the rising urve f

3

(

r r r

) must also onverge to

a nontrivial probability (one whi h is neither 0 nor 1). Taken all in all, then, it looks very mu h like the

m = 3 oarse-grained theory has a nontrivial N !1 limit, with at least three out of its �ve probabilities

onverging to nontrivial values.

Although the \rearrangement" of the oarse-grained probabilities appears mu h more dramati in

Figure 11, similar arguments an be made. Ex epting initial \transients", it seems reasonable to on lude

15

from the data that monotoni ity will be maintained. From this, it would follow that the probabilities for

r r r

rA

A

�

�

and r r r

r

�

�

A

A

(whi h must be equal by time-reversal symmetry) and the other rising probabilities, r

r

r

r

�

�

�

�

,

r r r r

, and r

r

r

r

��

HH

, all approa h nontrivial limits. The oarse-graining to 4 elements, therefore, would also

admit a ontinuum limit with a minimum of 4 out of the 11 independent probabilities being nontrivial.

To the extent that the m = 2 and m = 3 ases are indi ative, then, it is reasonable to on lude that

per olation dynami s admits a ontinuum limit whi h is non-trivial at all \s ales" m.

2

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0 2 4 6 8 10 12 14 16

Abu

ndan

ce

log N

Figure 12: Flow of the oarse-grained probabilities f

m

for m = 2. The 3- hain probability is held at

0.0814837.

The question suggests itself, whether the ow of the oarse-grained probabilities would di�er qualita-

tively if we held �xed some abundan e other than that of the 2- hain. In Figures 12, 13, and 14, we display

results obtained by �xing the 3- hain abundan e (its value having been hosen to make the abundan e of

2- hains be 1/2 when N = 2

16

). Noti e in Figure 12 that the abundan e of 2- hains varies onsiderably

along this traje tory, whilst that of the 3- hain (in �gure 13) of ourse remains onstant. On e again,

the �gures suggest strongly that the traje tory is approa hing a ontinuum limit, with nontrivial values

for the oarse-grained probabilities of at least the 3- hain, the \V" and the \�" (and in onsequen e,

nontrivial values for the 2- hain and 2-anti hain as well).

All the traje tories dis ussed so far produ e ausets with an ordering fra tion r lose to 1/2 for large

N . As mentioned earlier, r = 1=2 orresponds to a Myrheim-Meyer dimension of two. Figures 15 and 16

show the results of a simulation along the \four dimensional" traje tory de�ned by r = 1=10. (The value

r = 1=10 orresponds to a Myrheim-Meyer dimension of 4.) Here the appearan e of the ow is mu h less

elaborate, with the urves arrayed simply in order of in reasing ordering fra tion,

r r r

and

r r r r

being at

the top and r

r

r

and (imper eptibly) r

r

r

r

at the bottom. As before, all the urves are monotone as far as an

be seen. Aside from the intrinsi interest of the ase d = 4, these results indi ate that our on lusions

drawn for d near 2 will hold good for all larger d as well.

Figure 17 displays the ow of the oarse-grained probabilities from a simulation in the opposite

situation where the ordering fra tion is mu h greater than 1/2 (the Myrheim-Meyer dimension is down

near 1.) Shown are the results of oarse-graining to three element ausets along the traje tory whi h

16

2

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0 2 4 6 8 10 12 14 16

Abu

ndan

ce

log N

Figure 13: Flow of the oarse-grained probabilities f

m

for m = 3. The 3- hain probability is held at

0.0814837.

2

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0 2 4 6 8 10 12 14 16

Abu

ndan

ce

log N

Figure 14: Flow of the oarse-grained probabilities f

m

for m = 4. The 3- hain probability is held at

0.0814837.

17

2

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 2 4 6 8 10 12 14

Abu

ndan

ce

log N

Figure 15: Flow of the oarse-grained probabilities f

m

for m = 3. The 2- hain probability is held at 1/10.

2

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 2 4 6 8 10 12 14

Abu

ndan

ce

log N

Figure 16: Flow of the oarse-grained probabilities f

m

for m = 4. The 2- hain probability is held at 1/10.

Only those urves lying high enough to be seen distin tly have been labeled.

18

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1 2 3 4 5 6 7 8 9 10 11

2-chain3-chain

VLambda

L3-antichain

log N

Abu

ndan

ce

2

Figure 17: Flow of the oarse-grained probabilities f

m

for m = 3. The 3- hain probability is held at 1/2.

holds the 3- hain probability to 1/2. Also shown is the 2- hain probability. The behavior is similar to

that of Figure 15, ex ept that here the oarse-grained probability rises with the ordering fra tion instead

of falling. This o urs be ause onstraining f

3

( r

r

r

) to be 1/2 generates rather hain-like ausets whose

Myrheim-Meyer dimension is in the neighborhood of 1.34, as follows from the approximate limiting value

f

2

( r

r

) � 0:8. The slow, monotoni , variation of the probabilities at large N , along with the appearan e

of onvergen e to non-zero values in ea h ase, suggests the presen e of a nontrivial ontinuum limit for

r near unity as well.

Figures 18 and 19 present the results of a �nal set of simulations, the only ones we have arried out

whi h examined the abundan es of ausets ontaining more than four elements. In these simulations, the

mean 5- hain abundan e f

5

(5- hain) was held at 1/2, produ ing ausets that were even more hain-like

than before (Myrheim-Meyer dimension � 1:1). Figure 18 tra ks the resulting abundan es of all k- hains

for k between 2 and 7, in lusive. (We limited ourselves to hains, be ause their abundan es are relatively

easy to determine omputationally.) As in Figure 17, all the oarse-grained probabilities appear to be

tending monotoni ally to limits at large N . In fa t, they look amazingly onstant over the whole range

of N , from 5 to 2

15

. One may also observe that, as one might expe t, the oarse-grained probability of a

hain de reases markedly with its length (and almost linearly over the range examined!). It appears also

that the k- hain urves for k 6= 5 are \expanding away" from the 5- hain urve, but only very slightly.

Figure 19 tra ks the abundan es of all the four-element ausets. It is qualitatively similar to Figures 15{

17, with very at probability urves, and here with a strong preferen e for ausets having many relations

over those having few.

Comparing Figures 19 and 16 with Figures 14 and 11, one an observe that traje tories whi h generate

ausets that are rather hain-like or anti hain-like seem to produ e distributions whi h onverge more

rapidly than those along whi h the ordering fra tion takes values lose to 1/2.

In the way of further simulations, it would be extremely interesting to look for ontinuum limits

19

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

2 4 6 8 10 12 14 16

abun

danc

e

log2 N

2-chain3-chain4-chain5-chain6-chain7-chain

Figure 18: Flow of the oarse-grained probabilities f

m

(m� hain) for m = 2 to 7. The 5- hain probability

is held at 1/2.

2

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

2 3 4 5 6 7 8 9 10

Abu

ndan

ce

log N

Figure 19: Flow of the oarse-grained probabilities f

m

for m = 4. The 5- hain probability is held at 1/2.

20

of some of the more general dynami al laws dis ussed in x4.5 of Referen e [1℄. In doing so, however,

one would no longer have available (as one does have for transitive per olation) a very fast (yet easily

oded) algorithm that generates ausets randomly in a ord with the underlying dynami al law. Sin e

the sequential growth dynami s of [1℄ is produ ed by a sto hasti pro ess de�ned re ursively on the ausal

set, it is easily mimi ked algorithmi ally; but the most obvious algorithms that do so are too slow to

generate eÆ iently ausets of the size we have dis ussed in this paper. Hen e, one would either have to

devise better algorithms for generating ausets \one o�", or one would have to use an entirely di�erent

method to obtain the mean abundan es, like Monte Carlo simulation of the random auset.

7 Con luding Comments

Transitive per olation is a dis rete dynami al theory hara terized by a single parameter p lying between

0 and 1. Regarded as a sto hasti pro ess, it des ribes the steady growth of a ausal set by the ontinual

birth or \a retion" of new elements. If we limit ourselves to that portion of the auset omprising the

elements born between step N

0

and step N

1

of the sto hasti pro ess, we obtain a model of random

posets ontaining N = N

1

�N

0

elements. This is the model we have studied in this paper.

Be ause the underlying pro ess is homogeneous, this model does not depend on N

0

or N

1

separately,

but only on their di�eren e. It is therefore hara terized by just two parameters p and N . One should

be aware that this trun ation to a �nite model is not onsistent with dis rete general ovarian e, be ause

it is the subset of elements with ertain labels that has been sele ted out of the larger auset, rather

than a subset hara terized by any dire tly physi al ondition. Thus, we have introdu ed an \element of

gauge" and we hope that we are justi�ed in having negle ted it. That is, we hope that the random ausets

produ ed by the model we have a tually studied are representative of the type of suborder that one would

obtain by per olating a mu h larger (eventually in�nite) auset and then using a label-invariant riterion

to sele t a subset of N elements.

Leaving this question aside for now, let us imagine that our model represents an interval (say) in a

auset C underlying some ma ros opi spa etime manifold. With this image in mind, it is natural to

interpret a ontinuum limit as one in whi h N ! 1 while the oarse-grained features of the interval

in question remain onstant. We have made this notion pre ise by de�ning oarse-graining as random

sele tion of a suborder whose ardinalitymmeasures the \ oarseness" of our approximation. A ontinuum

limit then is de�ned to be one in whi h N tends to1 su h that, for ea h �nitem, the indu ed probability

distribution f

m

on the set of m-element posets onverges to a de�nite limit, the physi al meaning being

that the dynami s at the orresponding length-s ale is well de�ned. Now, how ould our model fail to

admit su h a limit?

In a �eld-theoreti setting, failure of a ontinuum limit to exist typi ally means that the oarse-

grained theory loses parameters as the uto� length goes to zero. For example, ��

4

s alar �eld theory

in 4 dimensions depends on two parameters, the mass � and the oupling onstant �. In the ontinuum

limit, � is lost, although one an arrange for � to survive. (At least this is what most workers believe

o urs.) Stri tly speaking, one should not say that a ontinuum limit fails to exist altogether, but only

that the limiting theory is poorer in oupling onstants than it was before the limit was taken. Now in

our ase, we have only one parameter to start with, and we have seen that it does survive as N ! 1

sin e we an, for example, hoose freely the m = 2 oarse-grained probability distribution f

2

. Hen e, we

need not fear su h a loss of parameters in our ase.

What about the opposite possibility? Could the oarse-grained theory gain parameters in the N !1

limit, as might o ur if the distributions f

m

were sensitive to the �ne details of the traje tory along

whi h N and p approa hed the \ riti al point" p = 0, N =1?

7

Our simulations showed no sign of su h

7

Su h an in rease of the parameter set through a limiting pro ess seems logi ally possible, although we know of no

example of it from �eld theory or statisti al me hani s, unless one ounts the extra global parameters that ome in with

21

sensitivity, although we did not look for it spe i� ally. (Compare, for example, Figure 10 with Figure 13

and 11 with 14.)

A third way the ontinuum limit ould fail might perhaps be viewed as an extreme form of the

se ond. It might happen that, no matter how one hose the traje tory p = p(N), some of the oarse-

grained probabilities f

m

(�) os illated inde�nitely as N !1, without ever settling down to �xed values.

Our simulations leave little room for this kind of breakdown, sin e they manifest the exa t opposite kind

of behavior, namely monotone variation of all the oarse-grained probabilities we \measured".

Finally, a ontinuum limit ould exist in the te hni al sense, but it still ould be e�e tively trivial

(on e again reminis ent of the ��

4

ase | if you are to regard a free �eld theory as trivial.) Here

triviality would mean that all | or almost all | of the oarse-grained probabilities f

m

(�) onverged

either to 0 or to 1. Plainly, we an avoid this for at least some of the f

m

(�). For example, we ould

hoose anm and hold either f

m

(m- hain) or f

m

(m-anti hain) �xed at any desired value. (Proof: as p! 1,

f

m

(m- hain) ! 1 and f

m

(m-anti hain) ! 0; as p ! 0, the opposite o urs.) However, in prin iple, it

ould still happen that all the other f

m

besides these two went to 0 in the limit. (Clearly, they ould not

go to 1, the other trivial value.) On e again, our simulations show the opposite behavior. For example,

we saw that f

3

( r

r r

A

A

�

�

) in reased monotoni ally along the traje tory of Figure 10.

Moreover, even without referen e to the simulations, we an make this hypotheti al \ hain-anti hain

degenera y" appear very implausible by onsidering a \typi al" auset C generated by per olation for

N >> 1 with p on the traje tory that, for some hosen m, holds f

m

(m- hain) �xed at a value a stri tly

between 0 and 1. Then our degenera y would insist that f

m

(m-anti hain) = 1� a and f

m

(�) = 0 for all

other �. But this would mean that, in a manner of speaking, \every" oarse-graining of C to m elements

would be either a hain or an anti hain. In parti ular the auset r r

r

ould not o ur as a sub auset of C;

when e, sin e r r

r

is a sub auset of every m-element auset ex ept the hain and the anti hain, C itself

would have to be either an anti hain or a hain. But it is absurd that per olation for any parameter

value p other than 0 and 1 would produ e a \bimodal" distribution su h that C would have to be either

a hain or an anti hain, but nothing in between. (It seems likely that similar arguments ould be devised

against the possibility of similar, but slightly less trivial trivial ontinuum limits, for example a limit in

whi h f

m

(�) would vanish unless � were a disjoint union of hains and anti hains.)

Putting all this together, we have persuasive eviden e that the per olation model does admit a ontin-

uum limit, with the limiting model being nontrivial and des ribed by a single \renormalized" parameter

or \ oupling onstant". Furthermore, the asso iated s aling behavior one might anti ipate in su h a ase

is also present, as we will dis uss further in [14℄.

But is the word \ ontinuum" here just a metaphor, or an it be taken more literally? This depends, of

ourse, on the extent to whi h the ausets yielded by per olation dynami s resemble genuine spa etimes.

Based on the meager eviden e available at the present time, we an only answer \it is possible". On one

hand, we know [1℄ that any spa etime produ ed by per olation would have to be homogeneous, like de

Sitter spa e or Minkowski spa e. We also know, from simulations in progress, that two very di�erent

dimension estimators seem to agree on per olated ausets, whi h one might not expe t, were there no

a tual dimensions for them to be estimating. Certain other indi ators tend to behave poorly, on the

other hand, but they are just the ones that are not invariant under oarse-graining (they are not \RG

invariants"), so their poor behavior is onsistent with the expe tation that the ausal set will not be

manifold-like at the smallest s ales (\foam"), but only after some degree of oarse-graining.

Finally, there is the ubiquitous issue of \�ne tuning" or \large numbers". In any ontinuum situation,

a large number is being manifested (an a tual in�nity in the ase of a true ontinuum) and one may

wonder where it ame from. In our ase, the large numbers were p

�1

and N . For N , there is no mystery:

unless the birth pro ess eases, N is guaranteed to grow as large as desired. But why should p be so

small? Here, perhaps, we an appeal to the preliminary results of Dou mentioned in the introdu tion. If

\spontaneous symmetry breaking".

22

| osmologi ally onsidered | the auset that is our universe has y led through one or more phases of

expansion and re ollapse, then its dynami s will have been �ltered through a kind of \temporal oarse-

graining" or \RG transformation" that tends to drive it toward transitive per olation. But what we

didn't mention earlier was that the parameter p of this e�e tive dynami s s ales like N

�1=2

0

, where N

0

is the number of elements of the auset pre eding the most re ent \boun e". Sin e this is sure to be

an enormous number if one waits long enough, p is sure to be ome arbitrarily small if suÆ iently many

y les o ur. The reason for the near atness of spa etime | or if you like for the large diameter of the

ontemporary universe | would then be just that the underlying ausal set is very old | old enough to

have a umulated, let us say, 10

480

elements in earlier y les of expansion, ontra tion and re-expansion.

It is a pleasure to thank Alan Daughton, Chris Stephens, Henri Waelbroe k and Denjoe

�

OConnor for

extensive dis ussions on the subje t of this paper. The resear h reported here was supported in part by

NSF grants PHY-9600620 and INT-9908763 and by a grant from the OÆ e of Resear h and Computing

of Syra use University.

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24