Asymptotic behavior of the Eden model with positively homogeneous edge weights

Asymptotic behavior of the Eden model with positively homogeneous edge weights

Sébastien Bubeck111sebubeck@microsoft.com  and Ewain Gwynne222ewain@mit.edu Microsoft Research and Massachusetts Institute of Technology
Abstract

Let , , and let be locally Lipschitz and positively homogeneous of degree (e.g.  could be the th power of a norm on ). We study a generalization of the Eden model on wherein the next edge added to the cluster is chosen from the set of all edges incident to the current cluster with probability proportional to the value of at the midpoint of this edge, rather than uniformly. This model is equivalent to a variant of first passage percolation where the edge passage times are independent exponential random variables with parameters given by the value of at the midpoint of the edge.

We prove that the -weighted Eden model clusters have an a.s. deterministic limit shape if , which is an explicit functional of and the limit shape of the standard Eden model, and estimate the rate of convergence to this limit shape. We also prove that if , then there is a norm on (depending on ) such that if we set , then the -weighted Eden model clusters are a.s. contained in a Euclidean cone with opening angle for all time. We further show that there does not exist a norm on for which this latter statement holds for all ; and that there is no choice of function for which the above statement holds with .

Our basic approach is to compare the local behavior of the -weighted first passage percolation to that of unweighted first passage percolation with iid exponential edge weights (which is equivalent to the unweighted Eden model).

We include a list of open problems and several computer simulations.

1 Introduction

1.1 Overview

Let and equip with its standard cubic lattice structure. The Eden model is a simple statistical physics model introduced in [Ede61], defined as follows. Let be sampled uniformly from the set of edges of incident to 0, and set . Inductively, if , , and has been defined, let be sampled uniformly from the set of edges of incident to and set .

The Eden model is equivalent to first passage percolation with iid exponentially distributed edge passage times, which was first introduced in [HW65] (this is a consequence of the “memoryless” property of the exponential distribution). Under this representation, the Eden model has been studied extensively, but many aspects of this model are still poorly understood. For example, it is known that the clusters have a deterministic limiting shape in a rather strong sense (see [Ric73, CD81, Kes93] as well as Sections 1.4 and 3.1 below), but little is known about this limit shape besides that it is compact, convex, and satisfies the same symmetries as . We refer the reader to the survey articles [Kes86, Kes87, How04, Bla10, GK12, AHD15] and the references therein for more information on first passage percolation.

In this article, we will consider the following natural variant of the Eden model. Let be a weight function from the edge set of to the positive real numbers. The weighted Eden model with edge weights is the growing family of edge sets which is defined in the same manner as the Eden clusters above, except that each edge is sampled from the set of edges incident to with probability proportional to instead of uniformly. Like the standard Eden model, this model can also be expressed in terms of a variant of first passage percolation where the passage time of each edge is an independent exponential random variable with parameter (in fact, we will mostly focus our attention on this latter model, which seems to be easier to analyze), see Section 1.3.

We will primarily be interested in the following special case of the above model. Fix . Let be a strictly positive Lipschitz function on the boundary of the Euclidean unit ball . Let

(1.1)

so that is strictly positive, locally Lipschitz, and homogeneous of degree . We call such a function an -weight function. A particular example of an -weight function is the th power of some norm on , which corresponds to for . The -weighted Eden model is the weighted Eden model where the weight of each edge of is given by

(1.2)

where is the midpoint of . In the case where and , the -weighted Eden model is a slight variant of the Pólya urn model, so the -weighted Eden model can be viewed as higher-dimensional generalization of the Pólya urn model. The -weighted Eden model in the case where was first introduced as an open problem in [Bub15].

Weighted versions of the eden model have been studied elsewhere in the literature. Diffusion limited aggregation (DLA) on a -ary tree is equivalent to a weighted variant of the Eden model on the tree with edge weights which are an exponential, rather than polynomial, function of the distance to the root vertex. This model is studied in [AS88, BPP97]. In the computer science literature, the authors of [FKOV14] propose a weighted version of the Eden model on a general graph, which they call “adaptive diffusion”, as a protocol for spreading a message in a network while obscuring its source.

As we shall see, the asymptotic behavior of -weighted FPP in general dimension and for general choice of depends crucially on the homogeneity degree of . In particular, we will prove the following.

  • If , the -weighted FPP clusters (for any choice of weight function ) have a deterministic compact limit shape which is an explicit functional of and the standard Eden model limit shape . We also provide a rate of convergence estimate in the spirit of [Kes93, Ale97].

  • If , there exists a norm on depending on (which we can take to be an explicit functional of and ) such that with , the -weighted FPP clusters are a.s. contained in a certain Euclidean cone with opening angle at all times.

  • For any choice of the Lipschitz function in (1.1), there is a constant (again, depending explicitly on and ) such that if then a.s. the -weighted FPP clusters with weight function eventually hit all but finitely many edges in .

See Section 1.5 below for precise statements. We also include several open problems related to the weighted Eden model, see Section 6.

The main idea of our proofs is to compare the local behavior of -weighted FPP to the local behavior of standard FPP. This allows us to show that passage times in -weighted FPP are well-approximated by a deterministic metric , which is defined precisely in Section 1.4 and depends on and the standard FPP limit shape .

Remark 1.1.

In the open problem statement [Bub15], it is conjectured that for , the -weighted FPP clusters a.s. have a deterministic limit shape if and are a.s. contained in a Euclidean cone of opening angle at all times if . Our results confirm this conjecture in the case . In the case , our results show that this conjecture is false for sufficiently close to 1, but is true if we replace with a norm which is allowed to depend on . It is still an open problem to determine whether it holds for large enough that the -weighted FPP clusters with are a.s. contained in a Euclidean cone of opening angle for all times.

Remark 1.2.

We include several simulations of -weighted FPP clusters, which are scattered throughout Section 1. All of these simulations are produced using Matlab and are run for iterations. Particles are color-coded based on the time at which they are added to the cluster. In order to reduce the file size of the images, we re-sampled a subset of the particles in the clusters. This re-sampling does not significantly change the images, except that some of the images include small white dots corresponding to points which are contained in the cluster, but which were removed during the re-sampling.

Acknowledgments We thank Ronen Eldan, Shirshendu Ganguly, Christopher Hoffman, Yuval Peres, and David Wilson for helpful discussions. We thank two anonymous referees for helpful comments on an earlier version of this paper. This work was carried out while the second author was an intern with the Microsoft Research theory group in Redmond, WA.

1.2 Basic notations

Before stating our main results we record some (mostly standard) notations which we will use throughout this paper.

1.2.1 Intervals and asymptotics

For , we define the discrete intervals and .

If and are two quantities, we write (resp. ) if there is a constant (independent of the parameters of interest) such that (resp. ). We write if and .

If and are two quantities which depend on a parameter , we write (resp. ) if (resp. remains bounded) as (or as , depending on context). We write if for each .

Unless otherwise stated, all implicit constants in , and and and errors involved in the proof of a result are required to satisfy the same dependencies as described in the statement of said result.

1.2.2 Graphs

For a graph , we write for the set of vertices of and for the set of edges of .

For a graph and a subset of we write for the set of edges of not contained in which are incident to an edge of . For a subset of , we write for the set of vertices which are incident to vertices of not contained in .

Let be a graph and let . A path of length in is a sequence such that the edges can be oriented in such a way that the initial endpoint of coincides with the terminal endpoint of for each . We say that is simple if does not visit any vertex of more than once. We write for the length of .

1.2.3 Metrics

We will have occasion to consider several different metrics on and . We use the following notation to distinguish these metrics.

Let be a metric on . For and , we write for the closed ball of radius centered at in the metric . For a set , we write for the -diameter of . If is a norm on , we write for the metric induced by . We often abbreviate .

We write for the Euclidean norm on and for its unit ball.

1.3 Weighted first passage percolation model

In most of this paper we will consider the following weighted variant of first passage percolation instead of the weighted Eden model described above. The two models are shown to be equivalent in Lemma 2.3 below. We first define the model in the greatest possible generality, then describe the special case which is our primary interest.

Definition 1.3.

Let be a connected, countable graph in which all vertices have finite degree. Let be a marked vertex of . Let be a deterministic function which assigns a positive weight to each . The first passage percolation (FPP) clusters on started from with weights is the random increasing sequence of subgraphs of defined as follows.

  • For each edge , let be an exponential random variable with parameter . We take the ’s to be independent.

  • For a path in , let . For vertices , we write

  • For , let be the graph defined as follows. The set of vertices is the set of with . The set of edges is the set of such that for some path in with incident to and .

For we write for the -algebra generated by and for . We also let

be the first (possibly infinite) time at which the cluster is infinite.

Note that ordinary first passage percolation with exponential passage times corresponds to the special case when for each in Definition 1.3.

We are primarily interested in the following special case of the model of Definition 1.3, which is a continuous-time parametrization of the -weighted FPP model described in Section 1.1 (see Lemma 2.3 below). Fix . Let for (with the standard cubic lattice structure) and let . Let be a Lipschitz function and let be as in (1.1) and as in (1.2). Let , , and be as in Definition 1.3 with this choice of parameters. We call the above model -weighted FPP. We also introduce the notation

(1.3)

We note that it is easy to see (by considering a path from 0 to along a coordinate axis) that for our model a.s. whenever . It will follow from Theorem 1.7 (resp. the proof of Theorem 1.9) below that a.s.  whenever (resp. ).

1.4 Standard FPP limiting shape and weighted metric

Our main method for studying the model described in Section 1.3 is to compare it to standard FPP, i.e. the case where , which is equivalent to the unweighted Eden model. In this case, it is shown in [Ric73, CD81] that there exists a compact convex set which is symmetric about 0 such that the random sets converge a.s. as to in the following sense. For , let

(1.4)

be the “fattening” of , so that contains no isolated points and . Then for each ,

(1.5)

Not much is known rigorously about the limit shape besides that it is compact, convex, and has the same symmetries as . It is expected that is not the Euclidean unit ball, but even this is not known except in dimension  [CEG11]. See, e.g., [FSS85, BH91, ED14] for numerical studies of Eden clusters.

Let be the norm whose closed unit ball is , i.e.

(1.6)

We will have occasion to compare to the Euclidean unit ball. For this purpose we use the following notation.

Definition 1.4.

Let

(1.7)

Also let

(1.8)

be the set of points on furthest from 0.

In the remainder of this subsection, we will define a metric on which will turn out to be a good approximation for passage times in our weighted FPP model.

Definition 1.5.

A piecewise linear path in is a continuous map for some for which there exists a subdivision of such that is affine for each . We say that is parametrized by -length if the following is true. For , let be the largest with . Then

i.e. is the sum of the -lengths of the linear segments of traced up to time . In this case we write .

If is a piecewise linear path parametrized by -length, we define the -length of by

(1.9)

with the -weight function from (1.1). If is not necessarily parametrized by -length, we define the -length of to be the -length of the path obtained by parametrizing by -length. We define a metric on by

(1.10)

where the infimum is over all piecewise linear paths connecting and .

As we shall see in Section 3 below, is a good approximation for the passage time in the -weighted FPP process . The following lemma is immediate from the -homogeneity of and the definition (1.10) of .

Lemma 1.6.

Let and . Then

(1.11)

1.5 Main results

Throughout this section, we assume that we are in the special case of Definition 1.3 described in Section 1.3, so in particular , is an -weight function as in (1.1), and are the -weighted FPP clusters.

Let be the metric from Section 1.4. If , then it is easy to see by integration that is finite for each and that extends to a metric on all of . In particular, the -balls for are well-defined. Let . We note that Lemma 1.6 implies that

(1.12)

The set is the limiting shape of the -weighted FPP clusters for , in the following sense.

Theorem 1.7.

Let and

(1.13)

For , let be as in (1.4) (for a general choice of ). Then for ,

where here denotes a quantity which decays faster than any negative power of as (recall Section 1.2.1).

Theorem 1.7 gives in some sense a complete qualitative characterization of the asymptotic behavior of the -weighted FPP clusters when . However, we expect that the exponent in (1.13) is not optimal (in fact, we expect the theorem to be true at least for any ; c.f. Remark 3.1 below). Moreover, we cannot give a more explicit description of the limit shape than the one above. Indeed, we cannot even characterize the functions for which the set is convex. See Figures 1 and 2 for simulations of -weighted FPP clusters with , some of which appear to have a non-convex limit shape.

Figure 1: Left panel: A simulation of an -weighted FPP cluster with , where here is the norm (which restricts to the graph distance on ). The clusters appear to be converging to a deterministic limit shape (which we know is a.s. the case by Theorem 1.7), but it is not clear from the simulation whether this limit shape is convex. Right panel: A simulation of an -weighted FPP cluster with , where here is the norm. The clusters appear to be converging to a deterministic limit shape which is a slight rounding of the -unit ball.
Figure 2: Left panel: A simulation of an -weighted FPP cluster with , where here is the norm whose closed unit ball is the rectangle . Right panel: A simulation of an -weighted FPP cluster with the weight function given by the third power of the ratio of the norm to the Euclidean norm (so ). We note that in both figures, the limit shape appears to be non-convex.

In the case , matters are more complicated. The qualitative asymptotic behavior of the -weighted FPP clusters depends crucially on the function , rather than just the value of . In the case when , simulations like the ones in Figure 3 suggest that the -weighted FPP clusters for many choices of tend to grow in a single direction, rather than being ball-like like in the case when . We recall that . Our next theorem tells us that for each , there exists a norm on (depending on ) such that if is the th power of this norm, then is a.s. contained in a cone of opening angle .

Theorem 1.8.

For each and each (Definition 1.4), there exists a norm on and a such that the following is true. Let

be the Euclidean cone based at 0 with opening angle centered at the ray from 0 through . Also let and let the -weighted FPP process. Then a.s. either

(1.14)

We will actually prove a more quantitative version of Theorem 1.8 (see Theorem 5.2 below). This result says that the statement of Theorem 1.8 holds for all -weight functions satisfying certain conditions, which are satisfied for the -th powers of a certain class of norms on . The unit ball of a typical norm in this class is a “cylinder” of the form where is a compact convex subset of the hyperplane through the origin perpendicular to and is a large fixed parameter which tends to as . See Figure 4 for an illustration.

Figure 3: Left panel: A simulation of an -weighted FPP cluster with , where here is the norm whose closed unit ball is the rectangle . This norm is similar to the norm appearing in Theorem 1.8, although in Theorem 1.8 the rectangle may be rotated by some (non-explicit) angle which depends on the standard FPP limit shape . Right panel: A simulation of an -weighted FPP cluster with . The figure suggests that the clusters will be contained in a Euclidean cone with opening angle for all times , but we do not prove that this is the case for this particular choice of .
Figure 4: An illustration of a the unit ball (light blue) of a typical norm satisfying the conclusion of Theorem 1.8 when slightly bigger than 1 and . Also shown is the Eden model limit shape (pink) and the smallest Euclidean ball which contains it, namely (dashed boundary). The boundary of the cone is shown as a pair of dashed lines. As approaches 1, the opening angle of this cone approaches . However, we do not prove that the opening angle of approaches 0 as .

It is an open problem to give for each a reasonably (though perhaps not fully) general characterization of the choices of for which the conclusion of Theorem 1.8 holds. We expect that a rigorous proof of such a characterization may require additional knowledge about the standard FPP limit shape .

Theorem 1.8 focuses on the behavior of the FPP clusters up to time , which is a.s. finite for . It is natural to ask about the behavior of the clusters for . Straightforward tail estimates for sums of exponential random variables (see, e.g. [Jan14, Theorem 5.1, item (i)]) show that if , then it is a.s. the case that for each , the set contains all but finitely many vertices of . Hence there is no interesting macroscopic behavior after time .

One may wonder to what extent the norm and the cone in Theorem 1.8 can taken to be uniform in . It turns out that the condition on needed for (1.14) to hold a.s. differs from the condition needed for this result to hold with positive probability. In particular, our more quantitative statement Theorem 5.2 implies the following.

  • For any , we can choose and such that whenever and , the condition (1.14) holds a.s.

  • For any , we can choose and such that whenever , we have that (1.14) holds with positive probability.

We note that Theorem 1.9 below tells us that cannot be chosen uniformly for all .

Our next theorem tells us that there is no choice of the function of (1.1) for which the conclusion of Theorem 1.8 holds for every choice of . In fact, we will show that if is sufficiently close to 1 (depending on ), then a.s. contains all but finitely many vertices of . To quantify how close to 1 we need to be, we introduce some notation. For and , let be the set of piecewise linear paths (Definition 1.5) connecting and which can be decomposed into linear segments whose endpoints are all contained in and which each have Euclidean length at most . Let

(1.15)

be half the -circumference of . Since on , it is easy to see that depends only on , not on , and that for any choice of . Furthermore, if we take for some norm on , then depends on but is uniformly positive for in any bounded subset of .

Theorem 1.9.

Let be the Lipschitz function in (1.1). Let be as in (1.7), as in (1.3), and as in (1.15). Suppose

For , let

(1.16)

There is a constant , depending only on and , such that

(1.17)

In particular, a.s.  is a finite set.

Remark 1.10.

In the case when , it will be clear from the proof of Theorem 1.9 that a.s. , so .

See Figure 5 for simulations of -weighted FPP clusters in the setting of Theorem 1.9.

Figure 5: Left panel: A simulation of an -weighted FPP cluster with , where here is the norm. The figure illustrates the conclusion of Theorem 1.9, namely that the clusters will a.s. cover all but finitely many points of before reaching . However, these clusters need not grow in a symmetric manner Right panel: A simulation of an -weighted FPP cluster with (so ). The clusters do not appear to be converging toward a deterministic limit shape, but it is conceivable that they converge toward a random limit shape or that they converge toward a deterministic limit shape at a very slow rate.

1.6 Outline

The remainder of this paper is structured as follows. In Section 2, we prove some basic properties of the weighted FPP model of Definition 1.3 at a greater level of generality than what we will consider in the remainder of the paper. In Section 3, we prove several lemmas which allow us to approximate -weighted FPP passage times via the deterministic metric of (1.10). In Section 4, we use these estimates to prove Theorems 1.7 and Theorem 1.9. In Section 5, we prove Theorem 1.8. In Section 6, we list some open problems related to the model studied in this paper.

2 General results for weighted FPP

Throughout this section we assume we are in the setting of Definition 1.3 for a general choice of graph , starting vertex , and weights . We recall in particular the FPP clusters and the FPP filtration .

In this section we will point out some basic properties of the model of Definition 1.3. In later sections we will only need the case where , , and is as in (1.2), but it is no more difficult to treat the general case. In Section 2.1, we state the strong Markov property of our model (which follows from the fact that the passage times have an exponential distribution) and deduce some basic consequences. In Section 2.2, we will prove a lemma which allows us to compare weighted FPP to standard FPP (equivalently, the unweighted Eden model). In Section 2.3, we will prove a weak form of one-endedness for weighted FPP clusters in the case where the graph is infinite and the passage time to , , is a.s. finite.

2.1 Markov property and applications

The following lemma gives a Markov property for weighted FPP clusters, and is the reason why we consider exponential passage times.

Lemma 2.1 (Strong Markov property).

Let be a stopping time for the FPP filtration . The conditional law of the passage times of the explored edges, given is described as follows.

  • For , the conditional law of is the same as its marginal law.

  • For , the conditional law of is that of an exponential random variable of parameter plus , where is the minimum of over all paths in joining to an endpoint of .

  • The random variables are conditionally independent given .

Proof.

The case where is deterministic follows from the memoryless property of exponential random variables. From this, we immediately obtain the case where takes on only countably many possible values. The case of a general stopping time is proven by approximating by a sequence of stopping times which take on only countably many possible values. ∎

Lemma 2.1 motivates the following definition.

Definition 2.2.

For and an edge , let

(2.1)

where is as in Lemma 2.1. For a path in , let

(2.2)

By Lemma 2.1, if is a stopping time for the filtration , then the conditional law given of is that of a collection of independent exponential random variables where each has parameter . Furthermore, if is a path in with only one edge lying in , then

(2.3)

Lemma 2.1 easily implies the following, which gives the equivalence of the model of Definition 1.3 and the weighted Eden model described in Section 1.1.

Lemma 2.3.

Assume we are in the setting of Definition 1.3 with for each . Let and for , let be the smallest for which . Let . Then the law of the sequence of random sets is described as follows. Let . Let be chosen uniformly from the set of edges of incident to and let . Inductively, if and has been defined, let be sampled from the uniform measure on the set of edges adjacent to weighted by . Let .

We next record another application of the random variables of Definition 2.2, namely a monotonicity statement for realizations of the cluster when is a stopping time for .

Lemma 2.4.

Let be a stopping time for . Let be a subset of chosen in a manner which is measurable with respect to . Let be the smallest for which .

  1. is conditionally independent from given .

  2. Let and be two possible realizations of such that and the realizations of corresponding to and are the same. Then the conditional law of given stochastically dominates the conditional law of given .

Proof.

First we prove assertion 1. Let be the set of simple paths for which the following is true.

  1. connects to a vertex in .

  2. contains exactly one edge in .

  3. Let be the time for which . There is no path in whose last edge shares an endpoint with and which satisfies .

For , we write

Then a.s. .

Define the random variables for and the passage times and as in Definition 2.2. Note that condition 3 in the definition of implies that . Hence for ,

By (2.3), we obtain . Therefore,

(2.4)

is a deterministic functional of the set and the random variables for . By Lemma 2.1, the conditional law of this latter collection of random variables given depends only on , so this collection of random variables is conditionally independent from given . We thus obtain assertion 1.

Now suppose we are in the setting of assertion 2. Let be the set of simple paths whose first edge belongs to , none of whose other edges belong to , and whose last edge is incident to a vertex in . In the notation introduced at the beginning of the proof, is the set of paths for on the event . Define similarly but with in place of . For , let be the largest with and let . Then .

Let be a collection of independent exponential random variables, each with parameter . For a path in , let . By (2.4), the conditional law of given (resp. ) is the same as the law of

Since is a surjective map from to , we obtain the desired stochastic domination. ∎

2.2 Comparison to standard exponential FPP

In this subsection, we will record some observations which allow us to compare the model of Section 2.2 to standard FPP on (i.e. with all of the edge weights equal to 1). For this purpose we first define a collection of iid exponential random variables which are related to the weighted FPP passage times .

Definition 2.5.

For and an edge , let , with as in Definition 2.2. Also let be a collection of random variables whose conditional law given is that of a family of iid exponential random variables with parameter 1, independent from the random variables for . For a path in , let

(2.5)

For , also let be the FPP clusters started from corresponding to the collection of random variables , i.e.  if and only if there is a path in joining to with and is the set of endpoints of edges in .

We also define an FPP geodesic from to to be a path in such that is incident to , , and is minimal among all such paths. If we do not specify the point , we assume is the root vertex of . It is easy to see that there a.s. exists at most one FPP geodesic from to .

Lemma 2.6.

Let be a stopping time for the FPP filtration and define the random variables for and the clusters for as in Definition 2.5. Then the conditional law of given is that of a collection of iid exponential random variables, each of which has parameter . If and then the following holds.

  1. Suppose . Let be the FPP geodesic from to . Let be the last vertex in crossed by and let . Then .

  2. Suppose there exists a simple path in started from such that , , and with as in (2.5),

    Then .

Proof.

From the strong Markov property (Lemma 2.1), Definition 2.5, and the scaling property of exponential random variables, it is clear that the conditional law of given is as claimed.

Now suppose the hypotheses of assertion 1 are satisfied. Let be the (a.s. unique) integer for which . Then