Path length distribution in two-dimensional causal sets
PPath Length Distributionin Two-Dimensional Causal Sets
M. Aghili ∗ , L. Bombelli † , B.B. Pilgrim ‡ May 18, 2018
Abstract
We study the distribution of maximal-chain lengths between two elements of a causal setand its relationship with the embeddability of the causal set in a region of flat spacetime.We start with causal sets obtained from uniformly distributed points in Minkowski space.After some general considerations we focus on the 2-dimensional case and derive a recursionrelation for the expected number of maximal chains n k as a function of their length k andthe total number of points N between the maximal and minimal elements. By studying thesetheoretical distributions as well as ones generated from simulated sprinklings in Minkowskispace we identify two features, the most probable path length or peak of the distribution k and its width ∆, which can be used both to provide a measure of the embeddability ofthe causal set as a uniform distribution of points in Minkowski space and to determine itsdimensionality, if the causal set is manifoldlike in that sense. We end with a few simpleexamples of n k distributions for non-manifoldlike causal sets. Introduction
In the causal set approach to quantum gravity, the Lorentzian manifold used in general relativ-ity and other theories of gravity to represent the spacetime geometry is simply the large-scaleview of a locally finite partially ordered set, a causal set [1]. Part of the motivation for thisapproach is the observation [2, 3] that if one chooses a sufficiently dense set of uniformly dis-tributed random points in a spacetime manifold, one can recover the spacetime geometry onscales larger than the one determined by the point density simply by using the causal orderingof the points. In this view then, spacetime is a purely combinatorial structure, a collection C ofevents with a relation p ≺ q which is reflexive, transitive, symmetric, and makes C locally finitein the sense that for any p , q ∈ C the interval or Alexandrov set I ( p, q ) := { r | p ≺ r ≺ q } has ∗ [email protected] † [email protected] ‡ [email protected] a r X i v : . [ g r- q c ] M a y nly a finite number of elements; for the purposes of this paper, without loss of generality fromnow on we will assume that C is actually of finite cardinality and corresponds to a spacetimeregion of finite volume.Given a Lorentzian manifold without closed causal curves, any choice of a finite number N of points in it, with the partial order induced by causality, will produce a causal set; if thepoints are randomly distributed with uniform density, the causal set samples the continuumgeometry uniformly and can be seen as a discretization of it at the volume scale given by theinverse density provided there are no length scales in the manifold of the order or smaller thanthe average distance between the points. On the other hand, given a large number N , thevast majority of the causal sets that can be constructed out of N elements cannot arise asdiscretizations of Lorentzian manifolds. One of the most general questions causal set theorymust address then is what makes it possible, in this approach, for the causal sets arising outof the dynamics to be such that they are seen as Lorentzian manifolds at large scales. Somepreliminary steps towards answering this question consist in showing that if a large causalset is manifoldlike then the manifold is approximately unique in an appropriate sense [3, 4],establishing criteria for manifoldlikeness of causal sets, possibly involving their embeddabilityin Lorentzian manifolds [5, 6, 7, 8, 9], and identifying procedures for using the structure ofa manifoldlike causal set to determine properties of the corresponding manifold, such as itsdimensionality, topology or curvature [10, 11, 12, 13].In this paper, we use the distribution of maximal-chain lengths between pairs of points in acausal set to find a criterion for manifoldlikeness. As shown in Section 3, the path distributionhas an approximately Gaussian shape, with a maximum and a width that can be used as twoparameters characterizing the structure of that interval in the causal set. The Path Length Distribution
In this section we will derive an expression for the distribution of maximal-chain lengths ina causal set sprinkled uniformly at random in an Alexandrov set I ( p, q ) of 2-dimensionalMinkowski spacetime, and show the results of some numerical simulations. This distributionprovides a good opportunity for statistical analysis of properties of the causal set. We willrefer to an element of such a sprinkling that is contained in some smaller region within I ( p, q )simply as a point in that region, and we will call a region empty if it does not contain any ofthe randomly distributed points. A maximal chain of length k between points x and x k +1 isdefined by k + 1 related points x ≺ x ≺ ... ≺ x k ≺ x k +1 such that for each i the Alexandrovset I i,i +1 = I ( x i , x i +1 ) is empty; in the following, a maximal chain will be called a path .The number of k -paths between two points p = x and q = x k +1 in a causal set uniformlyembedded in Minkowski space is a random variable, whose mean value can be evaluated ana-lytically by picking k − x i , calculating the probability thatone causal set point is found in each infinitesimal neighborhood d x i and each interval I i,i +1 isempty, and integrating over all the x i . hen N points are sprinkled uniformly in a volume V , the probability of finding exactly k of them within a region of volume V inside it is given by the binomial distribution, P k = (cid:18) Nk (cid:19) (cid:18) VV (cid:19) k (cid:18) − VV (cid:19) N − k . (1)In the small V /V limit, as in the case of an infinitesimal d x i , this probability can be ap-proximated by a Poisson distribution of density ρ = N/V . Thus, for each of the d d x i theprobability that it contains exactly one point can be written as ρ d d x i e − ρ d d x i ≈ ρ d d x i , andall of those probabilities can be considered to be independent as long as the number of linksin the chain is much smaller than the total number of points, k (cid:28) N . The intervals I i,i +1 however may not be small, and for those we will use the binomial distribution. In particular,the probability that a region of volume V is empty is given by P = (1 − V /V ) N , and for theunion I , ∪ · · · ∪ I k,k +1 of all Alexandrov sets between pairs of points that probability can bewritten as P = (cid:32) − (cid:80) ki =1 V i,i +1 V (cid:33) N . (2)Putting these together we then get, following the same approach as in Ref. [10], P ( x , . . . , x k ) d d x · · · d d x k = ρ k − d d x · · · d d x k (cid:32) − (cid:80) ki V i,i +1 V (cid:33) N + higher-order terms . (3)We now identify the probability in Eq. 3 with the mean number of paths through thoselocations, which integrated over all x i gives the mean number of k -paths between p and q , (cid:104) n k (cid:105) = ρ k − (cid:90) I d d x · · · (cid:90) I k − d d x k (cid:32) − (cid:80) ki V i,i +1 V (cid:33) N , (4)where I i = I ( x i , q ) is the Alexandrov set between x i and the maximal element q in the manifold;for simplicity, from now on we will drop the angle brackets, (cid:104) n k (cid:105) (cid:55)→ n k . Using the binomialexpansion we can write n k = ρ k − N (cid:88) n =0 (cid:18) Nn (cid:19) (cid:18) − V (cid:19) n × n (cid:88) i =0 (cid:18) ni (cid:19) · · · i k − (cid:88) i k − =0 (cid:18) i k − i k − (cid:19) (cid:90) I d d x · · · (cid:90) I k − d d x k ( V ) n − i · · · ( V k,k +1 ) i k − . (5)In two dimensions the volume of an Alexandrov set can be easily calculated using the nullcoordinates u = ( t + x ) / √ , v = ( t − x ) / √ , (6)in terms of which V i,i − = ( u i − u i − )( v i − v i − ) . (7) ith these expressions for the volumes, in the d = 2 case the integrals in Eq. 5 give n k = N k − N (cid:88) i =0 (cid:18) Ni (cid:19) ( − i Γ( i + 1)Γ( i + k ) f i,k (8)where we used N = ρV and f i,k = i (cid:88) i =0 Γ(1 + i − i ) × i (cid:88) i =0 Γ(1 + i − i ) · · · i k − (cid:88) i k − =0 Γ(1 + i k − − i k − ) i k − (cid:88) i k =0 Γ(1 + i k − − i k ) Γ( i k + 1) (cid:124) (cid:123)(cid:122) (cid:125) f ik − , (cid:124) (cid:123)(cid:122) (cid:125) f ik − , (cid:124) (cid:123)(cid:122) (cid:125) f i ,k − . (9)As suggested by the underbraces, this definition implies the recursion relation f i,k = i (cid:88) j =0 Γ(1 + i − j ) f j,k − , (10)with f i, := Γ( i + 1) . Using Eqs. 8–10, n k may now be calculated for any N and k . Results of Simulations
We wish to compare the results of the analytical distribution with actual manifoldlike causalsets obtained from numerical simulations of random sets of points sprinkled with uniformdensity in the Alexandrov set defined by two timelike related points in Minkowski space. FromFig. 1 it’s easy to see that the theory matches well with the average of the simulations, thoughit is worth pointing out that as the large error bars suggest, individual sprinklings can deviatesignificantly from the theory. This problem can be somewhat though not completely mitigatedby considering only the peak and width of the distribution rather than its entirety. The shapeof these distributions is nearly Gaussian, allowing us to characterize each curve with just thesetwo numbers. As one can see in Fig. 2 the relative errors in both the peak position andthe width decrease with N , implying that the larger errors in Fig. 1 are primarily due tofluctuations in the total number of paths rather than the shape of the curve considered as aprobability distribution; however, there is still enough error in the peak and width to causeus some concern. As a result, any application based on this distribution should account forstatistical fluctuations in evaluating a single causal set. For instance, if one wishes to use thisdistribution for its stated purpose of determining manifoldlikeness of causal sets, the failure ofa particular causal set to exactly match either the full analytical distribution or its peak and By width we mean full width at half maximum. .5 5.0 7.5 10.0 12.5 15.0 17.5050100150200250300350400 path dist. for 300 sprinklings ( N = 50) TheorySimulation 0 5 10 15 20020004000600080001000012000 path dist. for 100 sprinklings ( N = 100) TheorySimulation
Figure 1: A comparison of the average of many sprinklings of 50 and 100 points to their correspond-ing analytical distributions, on the left and right respectively. Due to numerical error, the theorycurve starts at 9.
10 15 20 25 30 35Peak position45678910 W i d t h Figure 2: A plot of width vs peak position for a variety of sizes of sprinkled causal sets.The color indicates the number N of elements in the causal set. width should not be taken as a sign that the causal set is not manifoldlike; rather, a fairlylarge range around the analytical distribution should be used, and causal sets which fall intothis range should be considered candidates for manifoldlike causal sets. We will discuss thisfurther in the next section. Manifoldlike Causal Sets
One of the motivations for this work was to explore the possibility of using the mean pathlength between two causal set elements p and q for a known value for the volume of I ( p, q )as a dimension estimator, similar to the use of the longest path length in Ref. [10], with thepossible computational advantage that sampling the set of paths between p and q and using Figure 3: Left: An artificially produced causal set whose purpose is to mimic both the heightand width of our path length distribution. Right: A similarly produced artificial causal set whicheliminates some redundancies to reduce the number of points while maintaining the path lengthdistribution; however, it still has far more points than its manifoldlike cohorts. an average length to estimate the mean may be easier than finding the longest path. Fromsimulations whose results are shown in this paper, as well as simulations in higher-dimensionalMinkowski space, it appears that the average length of a sample of a few paths is indeed avalid dimension estimator, though it is unclear whether it is computationally better than thelongest path method. One benefit of our approach and similar ones using the distribution ofpath lengths, however, is that it provides a criterion of manifoldikeness for causal sets.It is clear even from simple examples that quantities like the longest or the mean pathlength may be good dimension estimators only for causal sets known to be manifoldlike, anddo not by themselves distinguish those causal sets from non-manifoldlike ones. For example,the union of m chains of length k with minimal points and maximal points identified is acausal set that can always be embedded in 2D Minkowski space, but adjusting the values of m and k one can obtain a relationship between the total number of elements N = m ( k −
1) + 2and the longest or mean path length k that reproduces that of any Minkowski dimensionality.Similarly, a causal set could be constructed as the union of separate paths of various lengthsall sharing the same minimal and maximal element, and with no other overlap, as in the leftside of Fig. 3, with the number of chains of each length adjusted in a way that exactly matchesthe mean and width of the typical manifoldlike distribution. However, while the constructionwould yield the right value of n k for any length k , the total number of points in the causal set, N = (cid:80) k max k =2 n k ( k −
1) + 2 , would be quite different as the manifoldlike distribution would havemany paths sharing points and this contrived example does not. We could make the exampleslightly more realistic by forcing the paths to share all points not linked to the maximumpoint as in the right side of Fig. 3. This would limit the number of points significantly, witha total of N = (cid:80) k max k =2 n k + k max , where k max is the length of the longest path in the causalset. However, for N (cid:29) = 121 points. The path distribution for this set is alsosharply peaked, at k = 20 in this case, and not at all similar to that of a causal set from arandom distribution of spacetime points. Right: A 20-element causal set illustrating the Kleitman-Rothschild limit, which has a sharp peak for path length k = 2. use the average values of these 100-point sprinklings, the first method requires around 4 × points while the second one requires around 4 × points.What we propose as a first manifoldlikeness criterion based on the distribution of pathlengths n k is simply that any N -element causal set for which the mean value k and thewidth ∆ of that distribution are not consistent with the corresponding theoretical valueswithin statistical fluctuations cannot be manifoldlike. Based on the few examples we just saw,finding nonmanifoldlike causal sets that satisfy this condition is not trivial. Nevertheless, thiscondition is most likely not a sufficient one for manifoldlikeness. To further explore whichcausal sets meet or do not meet our criterion we will now provide two other types of examplesof nonmanifoldlike causal sets.One type includes causal sets that are not manifoldlike but are interesting for other reasons,and fail our criterion. The first example is a causal set that has one maximal and one minimalelement, with all other elements located between them and unrelated to each other (i.e., onelarge antichain with added minimal and maximal elements); the path length distribution isa Kronecker delta n k = δ k, , with a sharp peak at length 2 and zero for other lengths. Aregular “diamond lattice” (shown in the left side of Fig. 4) also has a path distribution sharplypeaked at some length k ≈ √ N , with no paths of other lengths. More generally, mostrandomly chosen causal sets of N elements with N (cid:29) N/ N/ n k = δ k, . The causal sets in these examples are all nonmanifoldlike,as one would not obtain them from uniform distributions of points in a Lorentzian manifold.The other type of examples includes causal sets that are still not manifoldlike, but arelikely to meet our criterion for manifoldlikeness. Fig. 5 shows the effect of adding one extrapoint to a causal set obtained from a 250-point random sprinkling in an Alexandrov set I ( p, q ) N u m b e r o f p a t h s Distribution difference ( N = 250) Figure 5: Left: Causal set obtained from sprinkling in 2D Minkowski space and one added pointwith “non-local” links. Right: Difference between the path length distributions. of 2D Minkowski space. The added point was to the future of a point approximately in themiddle of the sprinkled causal set, and linked directly to the maximal element q ; the left sideof the figure shows the resulting augmented causal set. Because the added point gives riseto additional paths which are shorter than the ones that go through the original, sprinkledcausal set, the new path length distribution will exhibit an additional small bump with a peaklength shorter than the overall k . The right side of the figure shows the difference betweenthe new path length distribution and the one without the additional point. This difference isvery small compared to the overall distribution, but the feature it shows may be identifiableas characteristic of this particular type of nonmanifoldlike causal set.We plan to continue studying the relationship between our criterion for manifoldlikeness anddifferent ways in which a causal set may fail to be manifoldlike, to establish which additionalcriteria are needed to exclude nonmanifoldlike causal sets that are not physically interesting,and possibly formulate a more quantitative way to take into account statistical fluctuations inthe path length distribution that would allow us to include causal sets which, strictly speaking,are not faithfully embeddable in a Lorentzian manifold, but may be physically interesting andwe may want to call manifoldlike (see, e.g., [5]).The results provided here will help us set up a procedure to establish whether a causalset is close to a spacetime manifold and address, at least in 2 dimensions, one of the mostfundamental questions in causal set theory. 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