intellecton/papers/project_paper_1_relativity/references/Loomis2018.txt

arXiv:1709.00064v2  [gr-qc]  25 Sep 2017

September 2017

Suppression of non-manifold-like sets
in the causal set path integral

S. P. Loomis∗ and S. Carlip†

Department of Physics
University of California
Davis, CA 95616
USA

Abstract

While it is possible to build causal sets that approximate spacetime
manifolds, most causal sets are not at all manifold-like. We show that a
Lorentzian path integral with the Einstein-Hilbert action has a phase
in which one large class of non-manifold-like causal sets is strongly
suppressed, and suggest a direction for generalization to other classes.
While we cannot yet show our argument holds for all non-manifold-like
sets, our results make it plausible that the path integral might lead to
emergent manifold-like behavior with no need for further conditions.

∗email: sloomis@ucdavis.edu
†email: carlip@physics.ucdavis.edu


1.
Introduction

The causal set program offers a simple, elegant picture of spacetime as a discrete set of points,
characterized solely by their causal relations. For all its elegance, though, causal set theory has
a potentially fatal flaw.
We know how to construct causal sets that approximate spacetime
manifolds, by starting with a manifold and extracting a Poisson “sprinkling” of points. But such
manifold-like sets are highly atypical; almost all causal sets do not look like any manifold at all.
If causal sets are fundamental, and manifold-like behavior is emergent, a dynamical process must
somehow suppress almost all typical causal sets, leaving only the rare manifold-like ones. Finding
such a process—especially one that has not been artificially constructed merely to achieve this
goal—is not easy.
In this paper, we show that the ordinary path integral with the causal set version of the
(Lorentzian) Einstein-Hilbert action has a phase in which one large class of non-manifold-like
causal sets is strongly suppressed. The class for which we can rigorously show this suppression,
the two-level orders, is itself not “typical”—we certainly do not claim to show that all non-
manifold-like sets are suppressed. But the two-level orders form a fairly large class, one much
larger than the class of manifold-like causal sets. As we discuss in the conclusion, there are
also hints that our methods may extend to more general classes. Our results thus make it more
plausible that the ordinary path integral, with no additional assumptions, may be enough to lead
to emergent manifold-like behavior.
A numerical analysis of two-dimensional causal sets has shown a similar transition between
a phase dominated by non-manifold-like causal sets and one dominated by manifold-like sets [1].
In one way, that result is stronger than ours, since it accounts for all non-manifold-like sets. On
the other hand, our results are analytic, hold in any dimension, and use the Lorentzian path
integral rather than analytically continuing to Riemannian signature.

2.
Non-manifold-like causal sets

The procedure for constructing a manifold-like causal set is well understood [2]. One starts
with a finite-volume region of a manifold with a Lorentzian metric, “sprinkles” points randomly
by a Poisson process, determines the causal relations among these points from the causal structure
of the manifold, and then “forgets” the manifold, keeping only the points and their causal
relations. For a dense enough sprinkling of points, the resulting causal set retains the fundamental
properties of the original manifold: the Alexandrov neighborhoods determine the topology, the
causal relations determine the conformal class of the metric, and the density of points determines
the conformal factor [3,4].
But such manifold-like causal sets are highly atypical. The “typical” causal set is a Kleitman-
Rothschild (KR) order, a three-level causal set with approximately n/4 points in the “bottom”
and “top” layers and n/2 points in the “middle” layer [5]. In fact, as n → ∞, the proportion of
n-element causal sets that are KR orders goes to one.
Many other non-manifold-like causal sets also occur frequently. There is, in fact, a hierarchy
of classes of non-manifold-like causal sets [6–9]. Each class is characterized by a parameter p

1


that is the proportion of possible relations that are actualized, and is dominated by causal sets
with a particular number of levels. The dependence of the size of the class on p is not smooth,
but is described by an piecewise continuous function with infinitely many “phase transitions”
characterized by either the creation of new layers or changes in the relative sizes of the layers.
The intricacies of these classes are beyond the scope of this paper—see [8] for details—but it is
sufficient to point out that the class of non-manifold-like causal sets is dominated by three-level
orders, primarily the KR orders, followed by two-level orders and then four-level orders.
In this paper we will focus on the simplest case of two-level orders. Though these are not
as dominant as the three-level orders, they still form a significant part of the collection of non-
manifold like causal sets.

3.
Causal set path integrals

To define a path integral for causal sets, we need two ingredients: an appropriate generaliza-
tion of the Einstein-Hilbert action and a discrete version of an integration measure. The action
we shall use, the Benincasa-Dowker action, was introduced in [10]. For a causal set C with n
elements, it takes the general form [11,12]

1
ℏS(C) = µ

�

n +

kmax
�

k=0
λkNk

�

(3.1)

where µ and λk are appropriately chosen parameters and Nk denotes the number of pairs of
elements {x, y} ⊂ C such that the cardinality of the set {z ∈ C : x ≺ z ≺ y} is equal to k. The
upper limit kmax can be finite or infinite, though it has a lower bound of ⌊2 + d

2⌋, where d is the
target spacetime dimension.
Eq. (3.1) replicates the Einstein-Hilbert action in the following sense. Suppose we construct
a causal set by Poisson sprinkling points into a manifold of the target dimension. Then for a
high enough sprinkling density and the correct choices of µ and λk, S(C) is equal to the Einstein-
Hilbert action on average. The specific definitions of µ and λk are complicated, but for d = 4
and kmax = 3 we have

1
ℏS(C) =
� l

lp

�2
(n − N0 + 9N1 − 16N2 + 8N3)
(3.2)

where lp is the Planck length and l is a length scale determined by the sprinkling density of
events into the spacetime.
For our “integration measure” we shall simply sum over causal sets. As in causal dynamical
triangulations [13], we should perhaps include a combinatorial weight to avoid overcounting
causal sets with special symmetries, but that will not affect our conclusions. The Lorentzian
partition function over any particular class C of causal sets is then

Z[µ, λ0] =
�

C∈C
exp
� i

ℏS(C)
�
=
�

C∈C
exp

�

iµ

�

n +

kmax
�

k=0
λkNk

��

(3.3)

2


We will be interested in the large n behavior of this quantity; for a manifold-like causal set with
a fixed sprinkling density, this is the large volume limit.

4.
Suppression of two-level orders

For this paper we focus on two-level orders, that is, causal sets C of size n such that there
are no three distinct elements x, y, z ∈ C satisfying x ≺ y ≺ z. This means that Nk = 0 for
k > 0. Intuitively, such sets have only two “moments of time,” and clearly do not resemble
manifolds. As we have mentioned, while they are less common than the three-level KR orders,
two-level orders are still much more common than manifold-like causal sets, and they threaten
to dominate the path integral.
For any n-element causal set, N0 can be no larger than Nmax = n(n−1)

2
. We classify such sets
by the proportion 0 ≤ p ≤ 1 of relations, given by N0 = pNmax. For fixed n, p is a discrete
parameter, but in the limit of large n we can approximate it as continuous. The utility of this
classification is that the Benincasa-Dowker action is constant over the class of two-level sets with
a fixed p. Denoting such a class by Cp,n, we can write the partition function over two-level orders
of size n as

Z[µ, λ0] =
�
dp |Cp,n|eiS(p)/ℏ = eiµn
� 1

0
dp |Cp,n| exp
�1

2iµλ0pn2 + o(n2)
�
(4.1)

where |Cp,n| is the cardinality of the class Cp,n. Here we have written Nmax = 1

2n2 + o(n2), where
o(n2) denotes terms subleading to n2, which will be negligible in the large n limit.
To calculate |Cp,n| we consider a decomposition into classes Cq,p,n where we put qn of the
elements in the “top” level and (1−q)n in the “bottom” level. Let us denote α = q(1−q), where
α ≤ 1

4 since 0 ≤ q ≤ 1. From the structure of the system, there can be at most αn2 relations—the
maximum occurs when every “bottom” element is related to every “top” element—so from the
definition of p, we have α ≥ 1

2p. This in turn implies that p ≤ 1

2.
The number of ways to choose pNmax = 1

2pn(n − 1) pairs from the possible αn2 relations is

|Cq,p,n| =
�
αn2

1
2pn(n − 1)

�
(4.2)

With both arguments large, we can expand the binomial as

ln |Cq,p,n| =αn2 ln(αn2) − 1

2pn2 ln
�1

2pn2
�
−
�
α − 1

2p
�
n2 ln
��
α − 1

2p
�
n2
�
+ o(n2)

=
�
α ln α − 1

2p ln
�1

2p
�
−
�
α − 1

2p
�
ln
�
α − 1

2p
��
n2 + o(n2)
(4.3)

For 1

2p ≤ α ≤ 1

4, this is is a monotonically increasing function of α. This means that |Cq,p,n| is
maximized for q = 1

2. Now, |Cp,n| is bounded by
���C 1

2,p,n
��� ≤ |Cp,n| ≤
�

q
|Cq,p,n|
(4.4)

3


In the large n limit, the upper bound is dominated by the maximal value of q, so

ln |Cp,n| = ln |C 1

2 ,p,n| + o(n2) = 1

4h(2p)n2 + o(n2)
�
p ≤ 1

2

�
(4.5)

where
h(x) = −x ln x − (1 − x) ln(1 − x)
(4.6)

is the entropy function. (As we saw above, p ≤ 1

2 for two-level sets, so |Cp,n| = 0 for p > 1

2.)
Using (4.5), we can write the partition function as

Z[µ, λ0] = eiµn
� 1/2

0
dp exp
�1

2iµλ0pn2 + 1

4h(2p)n2 + o(n2)
�
(4.7)

To simplify notation, we define

− µλ0

2
= β,
2p = x
(4.8)

Note that 0 ≤ x ≤ 1 and that, from (3.2), β > 0. The exponent in (4.7) is then n2

4 E(x), with

E(x) = −2iβx + h(x)
(4.9)

We will evaluate the integral by the method of steepest descents.∗ Here we sketch the method
and results; details are given in the appendix. We first find the saddle point:

E′(x) = 0 = −2iβ − ln x + ln(1 − x)
(4.10)

⇒ x0 =
e−iβ

2 cos β = 1

2(1 − i tan β),
1 − x0 =
eiβ

2 cos β = 1

2(1 + i tan β)

The second derivative at x = x0 is

E′′(x0) = − 1

x0
−
1

1 − x0
= −4 cos2 β
(4.11)

so the direction of steepest descent is x − x0 real. At the saddle point,

h(x0) = − e−iβ

2 cos β ln
� e−iβ

2 cos β

�
−
eiβ

2 cos β ln
�
eiβ

2 cos β

�
= β tan β + ln(2 cos β)

E(x0) = −2iβx0 + h(x0) = −iβ + ln(2 cos β)
(4.12)

Remembering that the exponent is n2

4 E(x), we have a saddle point contribution of

Z[µ, λ0] ∼ eiµn

n

�
π

2|E′′(x0)| exp
�n2

4 E(x0)
�
=
�π

8
eiµn

n cos β exp
�n2

4 [−iβ + ln(2 cos β)]
�
(4.13)

If | cos β| < 1

2, the real part of the exponent is negative, and the path integral is exponentially
suppressed.

∗An earlier attempt to determine the integral in a quadratic approximation failed; we thank Lisa Glaser for
pointing out an algebraic error that invalidated our first approach.

4


•
•

•
x−

0
1

C−
1
C−
2

C−
3

Figure 1: Deformed contour through the saddle point at x0 with tan β > 0

This is not quite the whole story. The method of steepest descent requires a contour defor-
mation, and we must check that the rest of the contour does not spoil the result. For tan β > 0,
the saddle point is in the lower half plane, and the contour is shown in figure 1.
We show
in the appendix that the remaining pieces of the contour, C−
1 and C−
2 , are also exponentially
suppressed. If, on the other hand, tan β < 0, we must deform the contour into the upper half
plane, and the remaining pieces are not suppressed. We thus conclude that the path integral for
two-level orders is exponentially suppressed at large volume provided that

tan
�
−µλ0

2

�
> 0
and
����cos µλ0

2

���� < 1

2
⇒ tan
�
−µλ0

2

�
>
√

3
(4.14)

We can also carry the analysis one step further. The saddle point approximation (4.13) is
not exact, and one might worry about the higher order terms in the exponent. In the appendix,
we give a rigorous bound: exponential suppression is guaranteed provided that

tan
�
−µλ0

2

�
>
�27

4 e−1/2 − 1
�1/2
≈ 1.759
(4.15)

This gives a minimum value of |µλ0| ≈ 2.108, or a scale ℓ ≈ 1.452ℓp in (3.2).

5.
Discussion

The program we have described can be summarized as follows:

1. Identify a class of causal sets that can be divided into subclasses characterized by some
parameters pi such that the action is constant over each subclass.

2. Count how large each subclass is, to leading order in the size n of the set, as a function of
the parameters pi.

3. Analytically evaluate the partition function as an integral over pi, and study how it depends
on the parameters µ and λi in the action.

We have carried this out for a particularly simple case, in which the division into easily
countable subclasses was fairly straightforward.
But there are hints that our results can be

5


generalized. Once we move beyond two-level orders, the action (3.2) will include contributions
from N1, N2, and N3, greatly complicating the counting. But for sets with only a few levels,
these contributions may be strongly suppressed.
Consider, for example, a KR order, which has approximately n/4 points in a “bottom” level,
n/2 in a “middle” level, and n/4 in a “top” level.† Pick a “bottom” point x and a “top” point
y. Typically, x will link to approximately n/4 points in the middle level. Imagine coloring these
points red, and the remaining points blue. For {x, y} to contribute to N1, y must then link to
exactly one red point in the middle level, along with approximately n/4 blue points. It is easy
to see that the probability of such a pattern goes as a polynomial in n times 2−n/2. The same is
true for contributions to N2 (for which y must link to exactly two red points) and N3 (for which
y must link to exactly three red points). Similar arguments should hold whenever the number
of levels is small.
This suppression should reduce the analysis of KR orders, and perhaps similar few-level
sets, to the form we have already considered, in which only N0 is important. This is still a
preliminary argument, of course. The N0 combinatorics will be different for different orders,
and one must check that the “atypical” few-level causal sets—three-level sets with different
distributions of points or relations from the KR orders, for instance—remain subdominant. Here
the combinatoric results of [8] may prove useful, but much more work is needed.
There are
also subtleties involving the difference between labeled and unlabeled causal sets that require
careful attention [9]. Numerical exploration of distributions of causal sets and relations may shed
additional light on these problems.

Acknowledgments

We are very grateful to Lisa Glaser for pointing out a crucial error in an early version of this
work. We also thank David Rideout for helpful conversations. This work was supported in part
by U.S. Department of Energy grant DE-FG02-91ER40674.

Appendix.
Steepest descent details

In this appendix we describe some of the details involved in the steepest descent calculation
of section 4.

Contours

The integral (4.7) is over the interval 0 < x < 1. For the method of steepest descent, we
must first deform the contour to go through the saddle point in the direction of steepest descent.
The saddle point is x0 = 1

2(1 − i tan β) and the direction of steepest descent is x − x0 real, so the
contours are those of figure 2, where the lower branch is applicable for tan β > 0 and the upper
for tan β < 0.

†More precisely [5], a KR order has between n/4 − n1/2 ln n and n/4 + n1/2 ln n points in the bottom and top
levels, and between n/2 − ln n and n/2 + ln n points in the middle level.

6


•
•

•
x−

0
1

C−
1
C−
2

C−
3

•
x+

C+
1
C+
2

C+
3

Figure 2: Deformed contours through the saddle point at x0

Let us first exclude the contour in the upper half plane. Consider C+
1 . We can write

x = +iw,
0 < w < 1

2| tan β|
(A.1)

from which, with the branch cuts shown in figure 2,

ln x = πi

2 + ln w,
ln(1 − x) = ln
√

1 + w2 − i tan−1 w
(A.2)

with the inverse tangent lying between 0 and π

2. Hence

h(x) = −x ln x − (1 − x) ln(1 − x)

= −iw
�πi

2 + ln w
�
− (1 − iw)
�
ln
√

1 + w2 − i tan−1 w
�

= π

2 w + w tan−1 w − ln
√

1 + w2 + imaginary part
(A.3)

For ℑx > 0, the contribution from the term −2iβx in E(x) is positive, and

ℜ E =
�π

2 + tan−1 w + 2β
�
w − ln
√

1 + w2
(A.4)

For positive real w, this is always positive, so the integral acquires an exponentially large con-
tribution from C+
1 . This rules out the saddle point approximation for this contour.
Next consider the contour in the lower half plane. On C−
1 , we can write

x = −iw,
0 ≤ w ≤ 1

2 tan β

ln(−iw) = −πi

2 + ln w,
ln(1 + iw) = ln
√

1 + w2 + i tan−1 w
(A.5)

7


with the inverse tangent again lying between 0 and π

2. Then

h(x) = iw
�
−πi

2 + ln w
�
− (1 + iw)
�
ln
√

1 + w2 + i tan−1 w
�

= π

2 w + w tan−1 w − ln
√

1 + w2 + imaginary part
(A.6)

and thus
ℜ E =
�π

2 + tan−1 w − 2β
�
w − ln
√

1 + w2
(A.7)

For β > π

2, this is always negative, and the contribution from C−
1 is exponentially suppressed.
For 0 < β < π

2, the requirement that | cos β| < 1

2 limits us to the range π

3 < β < π

2. To proceed,
let us determine the maximum value of ℜ E in this range.
Note first that at w = 0, ℜ E = 0 and the derivative

d(ℜ E)

dw
= π

2 + tan−1 w − 2β
(A.8)

is negative, so ℜ E < 0 for small w. The turning point occurs at

π
2 + tan−1 w − 2β = 0 ⇒ w = − cot 2β = 1

2(tan β − cot β)
(A.9)

For 1

2(tan β − cot β) < w < 1

2 tan β, ℜ E is increasing, so its maximum in this range will occur
at the endpoint w = 1

2 tan β. At that maximum,

ℜ E = 1

2

�π

2 + tan−1
�1

2 tan β
�
− 2β
�
tan β − ln

�

1 + 1

4 tan2 β
(A.10)

Treating this quantity as a function of β and using Mathematica [14] to determine its zeros, we
find that it is negative for .9474 < β < π

2, an interval that includes the full range of interest.
Hence ℜ E(w) < 0 for any β in the range π

3 < β < π

2, and the contribution of the contour C−
1 is
again exponentially suppressed.
The contour C−
2 is basically a reflection, and gives the identical suppression. Let

x = 1 − iv,
0 ≤ v ≤ 1

2 tan β
(A.11)

Then

h(x) = −(1 − iv)
�
ln
√

1 + v2 − i tan−1 v
�
− iv
�πi

2 + ln v
�

= π

2 v + v tan−1 v − ln
√

1 + v2 + imaginary part
(A.12)

and
ℜ E =
�π

2 + tan−1 v − 2β
�
v − ln
√

1 + v2
(A.13)

8


which exactly matches (A.7). This match is not accidental; it follows from the fact that

ℜh(1 − iv) = ℜh(iv) = ℜh(−iv)

as long as we stay on the same branch of the logarithm.

Error estimates

The integral (4.13) is based on a quadratic approximation to E(x). In this case, we can also
get control over the errors. Let x = 1

2(1 − u). It is then easy to check that for n ≥ 2,

dnh
dun = −1

2
(n − 2)!
(1 − u)n−1 − (−1)n

2
(n − 2)!
(1 + u)n−1
(A.14)

Now expand E(x) around x0. Since u0 = i tan β is imaginary, the two terms in (A.14) evaluated
at x0 are complex conjugates; the odd derivatives are imaginary, while the even derivatives are
real. The Taylor expansion for E(x) around x0, with x − x0 real, is then

ℜE(x) = ℜE(x0) −

∞
�

n=1

22n

2n(2n − 1)[cos2n−1β][cos(2n − 1)β] (x − x0)2n
(A.15)

where (4.10) has been used to evaluate ℜ(1 − u0)−(2n−1). Hence
��ℜ(E(x) − E(x0) + 2 cos2 β(x − x0)2)
��

≤

∞
�

n=2

22n

2n(2n − 1)| cos2n−1β|| cos(2n − 1)β| (x − x0)2n

≤

∞
�

n=2

22n

2n(2n − 1)

�1

2

�4n−1
=

∞
�

n=2

2−2n

n(2n − 1)
(A.16)

using the facts that | cos β| ≤ 1

2 and |x − x0| ≤ 1

2. The sum evaluates to

3
2 ln 3

2 + 1

2 ln 1

2 − 1

4 ≈ 0.0116

We can thus state, for instance, that on the line 0 < ℜx < 1, ℑx = − i

2 tan β with tan β > 0—that
is, the line through the saddle point x0—the exponent E(x) is negative as long as

| cos β| < 2 · 3−3/2e−1/4 ≈ 0.4942
(A.17)

which in turn yields (4.15). We do not know whether this is a sharp limit.

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