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