Cluster size in bond percolation on the Platonic solids
aa r X i v : . [ m a t h . P R ] D ec Cluster size in bond percolationon the Platonic solids
Nicolas Lanchier ∗ and Axel La Salle Abstract
The main objective of this paper is to study the size of a typical cluster of bondpercolation on each of the five Platonic solids: the tetrahedron, the cube, the octahedron,the dodecahedron and the icosahedron. Looking at the clusters from a dynamical point ofview, i.e., comparing the clusters with birth processes, we first prove that the first and secondmoments of the cluster size are bounded by their counterparts in a certain branching process,which results in explicit upper bounds that are accurate when the density of open edges issmall. Using that vertices surrounded by closed edges cannot be reached by an open path, wealso derive upper bounds that, on the contrary, are accurate when the density of open edges islarge. These upper bounds hold in fact for all regular graphs. Specializing in the five Platonicsolids, the exact value of (or lower bounds for) the first and second moments are obtainedfrom the inclusion-exclusion principle and a computer program. The goal of our program isnot to simulate the stochastic process but to compute exactly sums of integers that are toolarge to be computed by hand so these results are analytical, not numerical.
1. Introduction
Bond percolation on a simple undirected graph is a collection of independent Bernoulli randomvariables with the same success probability p indexed by the set of edges, with the edges associatedto a success being referred to as open edges, and the ones associated to a failure being referred toas closed edges. The open cluster containing a vertex x is the random subset of vertices that areconnected to vertex x by a path of open edges. This stochastic model was introduced in [5] to studythe random spread of a fluid through a medium.Bond percolation is traditionally studied on infinite graphs such as the d -dimensional integerlattice in which case the quantity of interest is the percolation probability, the probability that thecluster of open edges containing the origin is infinite. For bond percolation on integer lattices, itfollows from Kolmogorov’s zero-one law that the existence of an infinite cluster of open edges is anevent that has probability either zero or one. This, together with a basic coupling argument, impliesthat there is a phase transition at a critical value p c for the density of open edges from a subcriticalphase where all the open clusters are almost surely finite to a supercritical phase where there isat least one infinite cluster of open edges. It is known that the cluster size decays exponentially inthe subcritical phase [12] and that there is a unique infinite cluster of open edges called the infinitepercolation cluster in the supercritical phase [1, 2]. Using planar duality, coupling arguments andthe uniqueness of the infinite percolation cluster, it can also be proved that the critical value intwo dimensions is equal to one-half [10]. We refer the interested reader to [7] for additional resultsabout bond percolation on integer lattices, and to [11, chapter 13] for a brief overview.Bond percolation has also been studied on fairly general finite connected graphs [4]. Impor-tant particular cases are the complete graph, in which case the set of open edges form the very ∗ Nicolas Lanchier was partially supported by NSF grant CNS-2000792.AMS 2000 subject classifications: Primary 60K35Keywords and phrases: Bond percolation, Platonic solids, branching processes, inclusion-exclusion identity.1
Nicolas Lanchier and Axel La Salle
Tetrahedron (4, 6, 4) Cube (8, 12, 6) Octahedron (6, 12, 8) Dodecahedron (20, 30, 12) Icosahedron (12, 30, 20)
Figure 1 . Picture of the five Platonic solids. The numbers between parentheses refer to the number of vertices, thenumber of edges, and the number of faces, respectively. Note that the tetrahedron is dual to itself, the cube and theoctahedron are dual to each other, and the dodecahedron and icosahedron are dual to each other. popular Erd˝os-R´enyi random graph [6], as well as the hypercube [3]. Due to the finiteness of theunderlying graph, all the open clusters are finite so whether there exists an infinite percolationcluster or not becomes irrelevant. Such processes, however, still exhibit a phase transition in thesense that, in the limit as the number of vertices goes to infinite, there is a giant component ofopen edges (an open cluster whose size scales like the size of the graph) if and only if p is exceedsa certain critical value. In particular, most of the works about bond percolation on finite graphs isconcerned with asymptotics in the large graph limit.In contrast, the objective of this paper is to study (the first and second moments of) the sizedistribution of a typical cluster of bond percolation on each of the five Platonic solids: the tetrahe-dron, the cube, the octahedron, the dodecahedron and the icosahedron. The motivation originatesfrom our previous works [8, 9] that introduce a mathematical framework based on Poisson pro-cesses, random graphs equipped with a cost topology and bond percolation to model the aggregateloss resulting from cyber risks. Insurance premiums are based on the mean and variance of theaggregate loss which, in turn, can be easily expressed using the first and second moments of thesize of the percolation clusters. Estimates for the size of the clusters are given in [8] for the processon finite random trees and in [9] for the process on path, ring and star graphs. Even though ourpresent work does not have any applications in the field of cyber insurance (because the Platonicsolids are not realistic models of insurable networks), studying the size of percolation clusters onthe Platonic solids is a very natural question in probability theory.
2. Main results
Having a simple undirected graph G = ( V , E ), let x = Uniform ( V ) and ζ ( e ) = Bernoulli ( p ) , e ∈ E be a vertex chosen uniformly at random and a collection of Bernoulli random variables with thesame success probability p on the set of edges. The edges with ζ ( e ) = 1 are said to be open whilethe edges with ζ ( e ) = 0 are said to be closed, and we let C x = { y ∈ V : there is a path of open edges connecting x and y } be the percolation cluster containing x . The main objective of this paper is to study the first andsecond moments of S = card ( C x ) = the size of this percolation cluster when the graph G consists of luster size in bond percolation on the Platonic solids TH 2TH 3TH 1 TH 1TH 2TH 3TH 1TH 2TH 4 Sim.TH 1TH 2TH 4TH 1TH 2TH 4 Sim.TH 1TH 2TH 4
Figure 2 . First moment on the left and second moment on the right of the size distribution of bond percolationclusters on the tetrahedron (top), cube (middle) and octahedron (bottom) as functions of the probability p . The thicksolid lines show the exact expressions in TH 3 and 4, the thick dashed lines show the second moment obtained fromthe average of one hundred thousand independent realizations of the process for various values of p , and the othercurves show the upper bounds in TH 1 and TH 2 for the appropriate values of D and N . each of the five Platonic solids depicted in Figure 1. Our first result gives upper bounds for the firstand second moments of the cluster size that apply to all finite regular graphs and are not restrictedto the Platonic solids. The idea is to think of the cluster C x as a dynamical object described by a Nicolas Lanchier and Axel La Salle
Sim.TH 1TH 2TH 5 Sim.TH 1TH 2Sim.TH 1TH 2TH 5 Sim.TH 1TH 2
Figure 3 . First moment on the left and second moment on the right of the size distribution of bond percolationclusters on the dodecahedron (top) and icosahedron (bottom) as functions of the probability p . The thick dashedlines show the second moment obtained from the average of one hundred thousand independent realizations of theprocess for various values of p while the other curves show the upper bounds in TH 1 and TH 2 for the appropriatevalues of the degree D and the number of vertices N , and the lower bounds in TH 5. birth process starting with one particle at x and in which particles give birth with probability p onto vacant adjacent vertices. The size of the cluster is equal to the ultimate number of particlesin the birth process which, in turn, is dominated stochastically by the number of individuals up togeneration card ( V ) − Theorem 1 – For every D -regular graph with N vertices, E ( S ) ≤ Dp (cid:18) − ν R − ν (cid:19) E ( S ) ≤ (cid:18) Dp (cid:18) − ν R − ν (cid:19)(cid:19) + Dp (1 − p )(1 − ν ) (cid:18) (1 − ν R )(1 + ν R +1 )1 − ν − Rν R (cid:19) where ν = ( D − p and R = N − . Taking D and N in the theorem to be the degree and the number of vertices in each of the Platonicsolids, we get the solid curves in Figures 2 and 3. Note that these upper bounds are only accurate luster size in bond percolation on the Platonic solids for p small. To have upper bounds that are accurate for p large, we simply use that a vertex y = x cannot be in the percolation cluster C x when all the edges incident to x are closed. This gives thefollowing result that again applies to all finite regular graphs. Theorem 2 – For every D -regular graph with N vertices, E ( S ) ≤ N − ( N − − p ) D E ( S ) ≤ N − ( N − N − − p ) D + ( N − N − − p ) D − . Taking D and N in the theorem to be the degree and the number of vertices in each of the Platonicsolids, we get the dashed curves in Figures 2 and 3.Our last results are specific to the five Platonic solids and we denote by S f the size of a percolationcluster on the solid with f faces. To explain these results, we first observe that the mean clustersize can be easily expressed using the probability that each vertex belongs to the open cluster C x which, in turn, is equal to the probability that at least one of the self-avoiding paths connecting x to this vertex is open. In particular, identifying all the self-avoiding paths connecting x to any othervertex and using the inclusion-exclusion identity give an exact expression for the first moment. Thesame holds for the second moment looking instead at all the pairs of paths connecting x to twoother vertices. This approach also shows that the first and second moments of the cluster size arepolynomials in p with integer coefficients and degree (at most) the total number of edges so, tostate our next results and shorten the notation, we let P k = ( p , p , p , . . . , p k ) T for all k ∈ N . The main difficulties following this strategy is to identify all the self-avoiding paths and computethe probability that any sub-collection of paths are simultaneously open. Recall that, when dealingwith n events, the inclusion-exclusion identity consists of a sum of 2 n − − − ,
023 for the second moment . In particular, we compute the first moment by hand whereas for the second moment we rely on acomputer program that returns the exact value of the (seven) coefficients.
Theorem 3 (tetrahedron) – For all p ∈ (0 , , E ( S ) = (1 , , , , − , , − · P and E ( S ) = (1 , , , , − , , − · P . The cube and the octahedron both have twelve edges. There are respectively •
15, 16, 18 self-avoiding paths connecting two vertices at distance 1, 2, 3 on the cube, •
26, 28 self-avoiding paths connecting two vertices at distance 1, 2 on the octahedron.In particular, the first moment of the cluster size cannot be computed by hand for the cube andthe octahedron because the number of terms in the inclusion-exclusion identity ranges from tensof thousands to hundreds of millions. Identifying all these paths and using the same computerprogram as before, we get the following theorem.
Nicolas Lanchier and Axel La Salle
Theorem 4 (cube and octahedron) – For all p ∈ (0 , , E ( S ) = (1 , , , , , , − , − , , , − , , − · P E ( S ) = (1 , , , , − , − , , , − , , − , , − · P . The dodecahedron and the icosahedron both have thirty edges. For these two solids, even writingdown all the self-avoiding paths connecting two vertices is beyond human capability so we onlyfocus on the paths of length at most five for the dodecahedron and of length at most three for theicosahedron. Using that two vertices are in the same open cluster if (but not only if) at least oneof the paths is open, together with the inclusion-exclusion identity and our computer program, weget the following lower bounds for the mean cluster size.
Theorem 5 (dodecahedron and icosahedron) – For all p ∈ (0 , , E ( S ) ≥ (1 , , , , , , − , − , − , , − , , − , , − , , , − , − , , , , , , − , , , , , , · P E ( S ) ≥ (1 , , , , − , − , , , − , − , − , , − , , − , − , , , − , , − , , , , , , , , , , · P . The first and second moments in Theorem 3 and the first moments in Theorem 4 are representedby the thick solid curves in Figure 2. These curves fit perfectly with numerical solutions obtainedfrom one hundred thousands independent realizations of the percolation process. The lower boundsfor the first moments in Theorem 5 are represented by the dotted curves in Figure 3.
3. Proof of Theorem 1 (branching processes)
This section is devoted to the proof of Theorem 1. Though our focus is on the Platonic solids, werecall that the theorem applies to every finite D -regular graph G = ( V , E ). The basic idea of theproof is to use a coupling argument to compare the size of the percolation cluster starting at agiven vertex with the number of individuals in a certain branching process. Birth process.
Having a vertex x ∈ V and a realization of bond percolation with parameter p on the graph, we consider the following discrete-time birth process ( ξ n ). The state at time n is aspatial configuration of particles on the vertices: ξ n ⊂ V where ξ n = set of vertices occupied by a particle at time n. The process starts at generation 0 with a particle at x , i.e., ξ = { x } . • For each vertex y adjacent to vertex x , the particle at x gives birth to a particle sent tovertex y if and only if edge ( x, y ) is open.The children of the particle at x are called the particles of generation 1. Assume that the processhas been defined up to generation n >
0, and let Y n = card ( ξ n \ ξ n − )be the number of particles of that generation. Label arbitrarily 1 , , . . . , Y n the particles of genera-tion n and let x n, , x n, , . . . , x n,Y n be their locations so that ξ n \ ξ n − = { x n, , x n, , . . . , x n,Y n } . Then, generation n + 1 is defined as follows: luster size in bond percolation on the Platonic solids Bond percolation Generation 0 Generation 1Generation 4Generation 3Generation 2
Figure 4 . Example of a construction of the birth process from a realization of bond percolation (top left picture)on the dodecahedron. The thick lines represent the open edges, the black dots represent the vertices occupied by aparticle at each generation, an the arrows represent the birth events, from parent to children. • For each vertex y adjacent to x n, , the first particle of generation n gives birth to a particlesent to y if and only if y is empty and edge ( x n, , y ) is open. • For each vertex y adjacent to x n, , the second particle of generation n gives birth to a particlesent to y if and only if y is empty and edge ( x n, , y ) is open. • · · · • For each vertex y adjacent to x n,Y n , the Y n th particle of generation n gives birth to a particlesent to y if and only if y is empty and edge ( x n,Y n , y ) is open.Note that two particles i and j with i < j might share a common neighbor y in which case achild of particle i sent to y prevents particle j from giving birth onto y . For a construction of thebirth process from a realization of bond percolation on the dodecahedron, we refer to Figure 4. Theprocess is designed so that particles ultimately occupy the open cluster starting at x . In particular,the total number of particles equals the cluster size, as proved in the next lemma. Lemma 6 – The cluster size is given by S = card ( C x ) = card ( ξ N − ) = Y + Y + · · · + Y N − where N = card ( V ) . Proof.
To begin with, we observe that • Because particles can only give birth to another particle sent to an empty vertex, each vertexis ultimately occupied by at most one particle.
Nicolas Lanchier and Axel La Salle • The open cluster containing x can be written as C x = { y ∈ V : there is a self-avoiding path ofopen edges connecting vertex x and vertex y } . • The set of vertices occupied by a particle of generation n is ξ n \ ξ n − = { y ∈ C x : the shortest self-avoiding path ofopen edges connecting x and y has length n } . These three properties imply that all the vertices in the open cluster C x are ultimately occupiedby exactly one particle whereas the vertices outside the cluster remain empty therefore S = card ( C x ) = card ( ξ ) + card (cid:18) ∞ [ n =1 ( ξ n \ ξ n − ) (cid:19) = card ( ξ ) + ∞ X n =1 card ( ξ n \ ξ n − ) = ∞ X n =0 Y n . (1)In addition, because the graph has N vertices, the shortest self-avoiding path on this graph musthave at most N − ξ n = ξ n − and Y n = card ( ξ n \ ξ n − ) = 0 for all n > N. (2)Combining (1) and (2) gives the result. (cid:3) Coupling with a branching process.
The next step is to compare the number of particles in thebirth process with the number of individuals in a branching process ( X n ). The process coincideswith the birth process when the graph is a tree and is defined by X = 1 and X n +1 = X n, + X n, + · · · + X n,X n for all n ≥ X n,i representing the offspring distribution (number of offspring ofindividual i at time n ) are independent and have probability mass function X , = Binomial ( D, p ) and X n,i = Binomial ( D − , p ) for all n, i ≥ . This branching process can be visualized as the number of particles in the birth process abovemodified so that births onto already occupied vertices are allowed. In particular, the branchingprocess dominates stochastically the birth process.
Lemma 7 – For all n ≥ , we have the stochastic domination Y n (cid:22) X n . Proof.
As for the branching process, for all n ≥ i ≤ Y n , we let Y n,i = i th particle of generation n in the birth process . Because the edges are independently open with the same probability p and there are exactly D edges starting from each vertex, the number of offspring of the first particle is Y = Y , = Binomial ( D, p ) . (3)For each subsequent particle, say the particle located at z , we distinguish two types of edges startingfrom z just before the particle gives birth. luster size in bond percolation on the Platonic solids • There are m edges ( z, y ) that are connected to an occupied vertex y . Because parent andoffspring are located on adjacent vertices, we must have m ≥ • There are D − m edges ( z, y ) that are connected to an empty vertex y . These edges have notbeen used yet in the construction of the birth process, i.e., there has been no previous attemptto give birth through these edges, therefore each of these edges is open with probability p independently of the past of the process.From the previous two properties, we deduce that, for all n > i ≤ Y n , P ( Y n,i ≥ k ) = E ( P ( Y n,i ≥ k | Y , , Y , , . . . , Y n,i − )) ≤ P (Binomial ( D − , p ) ≥ k ) = P ( X n,i ≥ k ) . (4)The stochastic domination follows from (3) and (4). (cid:3) Number of individuals.
It directly follows from Lemmas 6 and 7 that E ( S k ) = E (( Y + Y + · · · + Y N − ) k ) ≤ E (( X + X + · · · + X N − ) k ) (5)for all k >
0. In view of (5), the last step to complete the proof of Theorem 1 is to show thatthe upper bounds in the theorem are in fact the first and second moments of the total number ofindividuals up to generation R = N − E ( ¯ X R ) and E ( ¯ X R ) where ¯ X R = X + X + · · · + X R . The rest of this section is devoted to computing these moments.
Lemma 8 – Let ν = ( D − p . Then, E ( ¯ X R ) = 1 + Dp (cid:18) − ν R − ν (cid:19) for all R > . Proof.
For i = 1 , , . . . , X , let¯ Z i = number of descendants of the i th offspring of the first individualup to generation R , including the offspring . Then ¯ X R = 1 + ¯ Z + · · · + ¯ Z X and the ¯ Z i are independent of X so E ( ¯ X R ) = E ( E ( ¯ X R | X )) = E ( E (1 + ¯ Z + · · · + ¯ Z X | X ))= E (1 + X E ( ¯ Z i )) = 1 + E ( X ) E ( ¯ Z i ) = 1 + DpE ( ¯ Z i ) . Because ¯ Z i is the number of individuals up to generation R − D − , p ), we deduce from [8, Theorem 2] that E ( ¯ X R ) = 1 + Dp (cid:18) − ( µp ) R − µp (cid:19) = 1 + Dp (cid:18) − ν R − ν (cid:19) where ν = µp = ( D − p. This completes the proof. (cid:3)
Using the same decomposition as in the previous lemma, we now compute the second momentof the number of individuals up to generation R = N − Nicolas Lanchier and Axel La Salle
Lemma 9 – Let ν = ( D − p . Then, for all R > , E ( ¯ X R ) = (cid:18) Dp (cid:18) − ν R − ν (cid:19)(cid:19) + Dp (1 − p )(1 − ν ) (cid:18) (1 − ν R )(1 + ν R +1 )1 − ν − Rν R (cid:19) . Proof.
Using again ¯ X R = 1 + ¯ Z + · · · + ¯ Z X and independence, we get E ( ¯ X R ) = E ( E ((1 + ¯ Z + · · · + ¯ Z X ) | X ))= E ( E (1 + 2( ¯ Z + · · · + ¯ Z X ) + ( ¯ Z + · · · + ¯ Z X ) | X ))= E (1 + 2 X E ( ¯ Z i ) + X E ( ¯ Z i ) + X ( X − E ( Z i )) )= 1 + 2 E ( X ) E ( ¯ Z i ) + E ( X ) E ( ¯ Z i ) + E ( X ( X − E ( Z i )) . (6)In addition, using that X = Binomial ( D, p ), we get E ( X ( X − X ) + ( E ( X )) − E ( X )= Dp (1 − p ) + D p − Dp = D ( D − p . (7)Combining (6) and (7) gives E ( ¯ X R ) = 1 + 2 DpE ( ¯ Z i ) + DpE ( ¯ Z i ) + D ( D − p ( E ( ¯ Z i )) = 1 + 2 DpE ( ¯ Z i ) + Dp (Var( ¯ Z i ) + ( E ( ¯ Z i )) ) + D ( D − p ( E ( ¯ Z i )) = 1 + 2 DpE ( ¯ Z i ) + Dp ( Dp + 1 − p )( E ( ¯ Z i )) + Dp Var( ¯ Z i )= (1 + DpE ( ¯ Z i )) + Dp (1 − p )( E ( ¯ Z i )) + Dp Var( ¯ Z i ) . Then, applying [8, Theorem 2] with µ = D − σ = 0, we get E ( ¯ X R ) = (cid:18) Dp (cid:18) − ν R − ν (cid:19)(cid:19) + Dp (1 − p ) (cid:18) − ν R − ν (cid:19) + Dp ν (1 − p )(1 − ν ) (cid:18) − ν R − − ν − (2 R − ν R − (cid:19) . Observing also that Dp (1 − p ) (cid:18) − ν R − ν (cid:19) + Dp ν (1 − p )(1 − ν ) (cid:18) − ν R − − ν − (2 R − ν R − (cid:19) = Dp (1 − p )(1 − ν ) (cid:18) (1 − ν )(1 − ν R ) + ν (1 − ν R − )1 − ν − (2 R − ν R (cid:19) = Dp (1 − p )(1 − ν ) (cid:18) − ν R + 2 ν R +1 − ν R +1 − ν − (2 R − ν R (cid:19) = Dp (1 − p )(1 − ν ) (cid:18) − ν R + 2 ν R +1 − ν R +1 + (1 − ν ) ν R − ν − Rν R (cid:19) = Dp (1 − p )(1 − ν ) (cid:18) (1 − ν R )(1 + ν R +1 )1 − ν − Rν R (cid:19) completes the proof. (cid:3) Theorem 1 directly follows from (5), and from Lemmas 8 and 9. luster size in bond percolation on the Platonic solids
4. Proof of Theorem 2
Theorem 2 relies on the following simple observation: vertex y = x cannot be in the percolationcluster starting at x when all the edges incident to y are closed. In contrast with the comparisonwith branching processes, this result leads to a good approximation of the moments of the sizedistribution when the probability p approaches one. To prove the theorem, note that E ( S k ) = E (cid:18) X y ∈ V { y ∈ C x } (cid:19) k = X y ,...,y k ∈ V E ( { y ∈ C x } · · · { y k ∈ C x } )= X y ,...,y k ∈ V P ( x ↔ y , . . . , x ↔ y k ) (8)for all integers k . To estimate the last sum, we let B y be the event that all the edges incident to y are closed. Using that there are exactly D edges incident to each vertex, and that there is at mostone edge connecting any two different vertices, say y = z , we get P ( B y ) = (1 − p ) D P ( B y ∪ B z ) = P ( B y ) + P ( B z ) − P ( B y ∩ B z ) ≥ − p ) D − (1 − p ) D − . (9)In addition, we have the inclusion of events B y ⊂ { x y } for all y = x. (10)Combining (9) and (10), we get P ( x y or x z ) ≥ ( (1 − p ) D when card { x, y, z } = 22(1 − p ) D − (1 − p ) D − when card { x, y, z } = 3 . (11)Using (8) with k = 1 and (11), we deduce that E ( S ) = 1 + X y = x P ( x ↔ y ) = 1 + X y = x (1 − P ( x y )) ≤ X y = x (1 − (1 − p ) D ) = 1 + ( N − − (1 − p ) D ) = N − ( N − − p ) D . Similarly, applying (8) with k = 2, observing thatcard { ( y, z ) ∈ V : card { x, y, z } = 2 } = 3( N − { ( y, z ) ∈ V : card { x, y, z } = 3 } = ( N − N − , and using (11), we deduce that E ( S ) ≤ N − − (1 − p ) D ) + ( N − N − − − p ) D + (1 − p ) D − )= N − N − − p ) D − ( N − N − − p ) D − (1 − p ) D − )= N − ( N − N − − p ) D + ( N − N − − p ) D − . This completes the proof of Theorem 2. Nicolas Lanchier and Axel La Salle
5. Proof of Theorems 3–5 (inclusion-exclusion identity)
Theorems 3–5 follow from an application of the inclusion-exclusion identity. To begin with, weprove a result (see (15) below) that holds not only for all five Platonic solids but also a larger classof finite regular graphs. Fix a vertex x ∈ V , let r be the radius of the graph, and defineΛ s = { y ∈ V : d ( x, y ) = s } and N s = card (Λ s ) for s = 0 , , . . . , r. At least for the Platonic solids, N s does not depend on the choice of x . Fixing y s ∈ Λ s for all s = 0 , , . . . , r, and applying (8) with k = 1, we get E ( S ) = X y ∈ V P ( x ↔ y ) = r X s =0 X y ∈ Λ s P ( x ↔ y ) = r X s =0 N s P ( x ↔ y s ) . (12)To compute the probabilities p s = P ( x ↔ y s ), we label the edges 0 , , . . . , n −
1, think of eachself-avoiding path π as the collection of its edges, and let π ( y s ) , . . . , π K s ( y s ) = all the self-avoiding paths x → y s A i = the event that π i ( y s ) is an open path for i = 1 , , . . . , K s . Because the edges are independently open with the same probability p , P ( A i ∩ · · · ∩ A i j ) = P ( π i ( y s ) , . . . , π i j ( y s ) are open paths)= P ( e is open for all e ∈ π i ( y s ) ∪ · · · ∪ π i j ( y s ))= p card ( π i ( y s ) ∪ ··· ∪ π ij ( y s )) for all 0 < i < · · · < i j ≤ K s . Here card refers to the number of edges in the subgraph thatconsists of the union of the self-avoiding paths. Using that x ↔ y s if and only if at least one of thepaths connecting the two vertices is open, and the inclusion-exclusion identity, we deduce that P ( x ↔ y s ) = P (cid:18) K s [ j =1 A j (cid:19) = K s X j =1 ( − j +1 X
For the tetrahedron, all the vertices are distance one apart and thereare exactly five self-avoiding paths connecting any two vertices (see first table in Figure 5). Callingthese paths π , . . . , π in the order they are listed in the table, and writingcard ( π i ∪ π i ∪ · · · ∪ π i j ) = | π i ,i ,...,i j | for short, one can easily check that | π | = 1 | π , | = 3 | π , , | = 5 | π , , , | = 6 | π , , , , | = 6 | π | = 2 | π , | = 3 | π , , | = 5 | π , , , | = 6 | π | = 2 | π , | = 4 | π , , | = 5 | π , , , | = 6 | π | = 3 | π , | = 4 | π , , | = 5 | π , , , | = 6 | π | = 3 | π , | = 4 | π , , | = 5 | π , , , | = 5 | π , | = 4 | π , , | = 6 | π , | = 4 | π , , | = 5 | π , | = 4 | π , , | = 5 | π , | = 4 | π , , | = 5 | π , | = 5 | π , , | = 5This, together with (14), implies that, for all x = y , P ( x ↔ y ) = ( p + 2 p + 2 p ) − (2 p + 7 p + p ) + (9 p + p ) − ( p + 4 p ) + p = p + 2 p − p + 7 p − p = (0 , , , , − , , − · P . (16)Using also (15) and that N = 3 for the tetrahedron, we conclude that E ( S ) = 1 + 3 (0 , , , , − , , − · P = (1 , , , , − , , − · P which proves the first part of Theorem 3.To compute the second moment, we observe that any three distinct vertices of the tetrahedronalways form a triangle (regardless of the choice of the vertices) and, for all x ∈ V ,card { ( y, z ) ∈ V : card { x, y, z } = 2 } = 3 × { ( y, z ) ∈ V : card { x, y, z } = 3 } = 3 × . Using also (8) with k = 2, we get E ( S ) = P ( x ↔ x ) + 9 P ( x ↔ y ) + 6 P ( x ↔ y, x ↔ z ) (17) Nicolas Lanchier and Axel La Salle
01, 32, 51, 4, 52, 3, 4123 105, 69, 110, 4, 71, 4, 62, 5, 73, 4, 117, 8, 90, 1, 5, 70, 2, 4, 60, 3, 7, 90, 4, 8, 111, 2, 4, 71234 3, 4, 7, 81, 3, 5, 111, 3, 6, 92, 5, 8, 112, 6, 8, 90, 1, 5, 8, 110, 1, 6, 8, 90, 2, 3, 5, 110, 2, 3, 6, 91, 2, 3, 7, 91, 2, 4, 8, 111, 3, 5, 7, 82, 3, 4, 6, 854 1, 32, 85, 96, 110, 1, 80, 2, 31, 4, 92, 7, 113, 4, 55, 10, 116, 7, 86, 9, 100, 1, 7, 110, 2, 4, 9234 0, 1, 7, 9, 100, 2, 4, 10, 110, 3, 5, 7, 102, 3, 4, 7, 101, 4, 7, 8, 100, 4, 5, 7, 110, 4, 6, 7, 90, 4, 6, 8, 105 0, 3, 6, 70, 4, 5, 81, 4, 10, 112, 7, 9, 103, 4, 6, 105, 7, 8, 10470, 4, 83, 6, 110, 1, 2, 6, 110, 1, 5, 8, 90, 1, 5, 10, 110, 4, 9, 10, 111, 2, 3, 4, 82, 3, 5, 8, 92, 3, 5, 10, 113, 6, 8, 9, 100, 1, 2, 6, 8, 9, 100, 2, 4, 5, 6, 9, 111, 2, 3, 4, 9, 10, 111, 3, 4, 5, 6, 8, 107135 23 3, 43, 54, 50, 1, 20, 1, 40, 1, 50, 2, 30, 2, 41, 2, 31, 2, 58, 910, 110, 1, 5, 70, 4, 7, 91, 4, 5, 82, 3, 5, 72, 5, 6, 113, 6, 7, 100, 1, 2, 6, 7, 100, 1, 3, 5, 6, 110, 2, 3, 4, 5, 80, 3, 4, 6, 8, 100, 3, 4, 6, 9, 111, 2, 3, 4, 7, 91, 2, 4, 6, 8, 101, 2, 4, 6, 9, 11246 0, 1, 71, 4, 82, 3, 72, 6, 115, 8, 95, 10, 110, 1, 3, 6, 110, 2, 3, 4, 80, 4, 5, 7, 91, 4, 9, 10, 112, 6, 8, 9, 103, 5, 6, 7, 100, 1, 3, 6, 8, 9, 100, 2, 3, 4, 9, 10, 110, 2, 4, 6, 7, 9, 100, 3, 4, 5, 6, 8, 100, 3, 4, 5, 6, 9, 111, 3, 4, 6, 7, 9, 1035735 402 167 4523 0 111 8 9100 12 3 4567 8 91011
Figure 5 . The three pictures on the left show planar representations of the tetrahedron, the cube and the octahedron,along with an arbitrary labeling of their edges. The tables on the right give the list of the self-avoiding paths connectingthe two vertices (or pairs of self-avoiding paths connecting the three vertices) represented by the black, dark grey,light grey and/or white dots in the pictures. Each path is represented by the collection of its edges using the labelsshown in the pictures. The numbers in the first column of each table indicate the length of the paths. where vertices x, y, z are arbitrary but all three distinct. In addition, letting γ , γ , . . . , γ K be thepairs of self-avoiding paths connecting all three vertices, and using the same argument as beforebased on the inclusion-exclusion identity, we get P ( x ↔ y, x ↔ z ) = X B ⊂ [ K ]: B = ∅ ( − card ( B )+1 p card (cid:0) S i ∈ B γ i (cid:1) (18) luster size in bond percolation on the Platonic solids which can be viewed as the analog of (14). For the tetrahedron, there are K = 10 such paths (seethe second table in Figure 5). As previously, computingcard (cid:18) [ i ∈ B γ i (cid:19) for every B ⊂ [10] = { , , . . . , } is straightforward in the sense that it does not require any logical thinking. However, having tenself-avoiding paths, the sum in (18) is now over2 − ,
023 termsand is therefore unrealistic to compute by hand. Also, to compute (18), we designed a computerprogram that goes through all the possible subsets B ⊂ [10] and returns six (= number of edgesof the tetrahedron) coefficients a , a , . . . , a . These seven coefficients are initially set to zero andincrease or decrease by one according to the following algorithm:replace a j → a j + 1 each time card (cid:16) S i ∈ B γ i (cid:17) = j and card ( B ) is oddreplace a j → a j − (cid:16) S i ∈ B γ i (cid:17) = j and card ( B ) is even . (19)In other words, because the tetrahedron contains six edges, the right-hand side of (18) is a poly-nomial with degree at most six, and the algorithm returns the value of the seven coefficients ofthis polynomial. We point out that the values we obtain are exact because the computer is usedto add a large number of integers rather than to simulate the percolation process. Therefore, theexpression of the second moment in the theorem is indeed exact even though we rely on the use ofa computer. The input of the program is the ten self-avoiding paths represented by the subsets ofedges in the second table of Figure 5, and the output of the program is a = 0 , a = 0 , a = 3 , a = 5 , a = − , a = 15 , a = − . This, together with (16) and (17), implies that E ( S ) = 1 + 9 (0 , , , , − , , − · P + 6 ( a , a , a , a , a , a , a ) · P = 1 + 9 (0 , , , , − , , − · P + 6 (0 , , , , − , , − · P = (1 , , , , − , , − · P . This completes the proof of Theorem 3. (cid:3)
Proof of Theorem 4.
The idea is again to compute the sum (15) explicitly by first collectingthe self-avoiding paths connecting two vertices and then using the computer program mentionedabove to obtain the exact value of the coefficients of the polynomial.
Cube.
For the cube, there are respectively fifteen, sixteen and eighteen self-avoiding paths con-necting any two vertices at distance one, two, and three from each other, as shown in Figure 5.Because the cube has twelve edges, the sum consists of a polynomial with degree 12. The first fourcolumns in the first table of Figure 6 show the coefficients computed by our program from the listof all the self-avoiding paths. The first column simply means that, with probability one, a vertex Nicolas Lanchier and Axel La Salle − − − − −
11 0002425 − − − − −
14 000006 − − − − E ( S ) s = 0 N s = 1 N s = 3 N s = 3 N s = 1 s = 1 s = 2 s = 3 13612912 − − − − −
11 00 − − − − − − − − − s = 1 s = 2 s = 0 N s = 1 N s = 1 N s = 4 E ( S )141220 − − − − − Figure 6 . Coefficients returned by algorithm (19) for the cube (left) and the octahedron (right) using the self-avoidingpaths listed in Figure 5. The last column of each table is equal to the linear combination of the other columns withweight given by the value of the N s in the second row, which corresponds to the coefficients of the polynomial in p equal to the first moment of the cluster size. is in the open cluster starting from itself while the second column means that a vertex of the cubeat distance one of vertex x is in the open cluster starting at x with probability(0 , , , , − , , − , − , , , − , , − · P = p + 2 p − p + 8 p − p − p + 67 p − p + 55 p − p . The second row in the first table of Figure 6 shows the value of N s for the cube. The last columnis simply the linear combination of the first four columns where column s has weight N s . By (15),this is the expected value of the cluster size so the proof for the cube is complete. Octahedron.
Because the radius of the octahedron is two, two distinct vertices can only be atdistance one or two apart. There are respectively twenty-six and twenty-eight self-avoiding pathsconnecting any two vertices at distance one and two from each other (see Figure 5). Note that thesum in (14) for two vertices of the octahedron at distance two apart now contains2 − , ,
455 termsso the use of a computer is absolutely necessary to compute this sum explicitly. The sum againconsists of a polynomial with degree 12, the common number of edges in the cube and the octahe-dron, and the program gives the coefficients reported in the second table of Figure 6. The rest ofthe proof is exactly the same as for the cube. (cid:3)
Proof of Theorems 5.
For the dodecahedron and the icosahedron, not only the sum (15)cannot be computed by hand, but also the number of self-avoiding paths connecting two verticesis beyond human capability. However, we can find lower bounds for the mean cluster size by onlytaking into account a subset of paths. More precisely, given x = y , and letting • π , π , . . . , π J be the self-avoiding paths of length ≤ c connecting x and y , • π J +1 , π J +2 , . . . , π K be the self-avoiding paths of length > c connecting x and y , luster size in bond percolation on the Platonic solids (2)(2)(2)(2)(2) (2)(2)(2)(2) (2)(2)(2) (2)(2)(2) (2)(2)(2) (2) s = 4 s = 5 s = 1 s = 2 s = 2 s = 3 s = 3 s = 1 s = 2 s = 3 s = 3 Figure 7 . Picture of the self-avoiding paths with length at most five connecting two vertices of the dodecahedron atrespectively distance 1, 2, 3, 4, and 5, of each other, and picture of the self-avoiding paths with length at most threeconnecting two vertices of the icosahedron at respectively distance 1, 2, and 3, of each other. The label (2) next tosome pictures means that the mirror image of the path is another path connecting the same two vertices.8
Nicolas Lanchier and Axel La Salle N s = 3 N s = 1000000000000000000 0000000000000000 s = 0 s = 1 s = 2 s = 3 s = 4 s = 5 N s = 1 N s = 3 N s = 6 N s = 6 E ( S ) ≥ s = 0 N s = 1 N s = 1 s = 1 s = 2 s = 32 − − N s = 5 N s = 5 E ( S ) ≥ − − − − −
122 10 − − − − − − − − − − − − − − − − − −
11 12 − −
11 111 − −
22 122 − − − − −
31 24 − − − − −
52 6 − − − − − − − − − − − − − − Figure 8 . Coefficients returned by algorithm (19) for the dodecahedron (left) and the icosahedron (right) using theself-avoiding paths represented in Figure 7. The last column of each table is equal to the linear combination of theother columns with weight given by the value of the N s in the second row. Because we only look at a subset of theself-avoiding paths connecting two vertices the last column now gives the coefficients of a polynomial in p that issmaller than the first moment of the cluster size. we deduce from (14) that P ( x ↔ y ) = X B ⊂ [ K ]: B = ∅ ( − card ( B )+1 p card (cid:0) S i ∈ B π i (cid:1) ≥ X B ⊂ [ J ]: B = ∅ ( − card ( B )+1 p card (cid:0) S i ∈ B π i (cid:1) . (20)The inequality follows from an inclusion of events: if at least one of the first J paths is open thenat least one of the K paths is open. For both the dodecahedron and the icosahedron, we choosethe cutoff c to be the radius of the graph, meaning that we only consider self-avoiding paths withlength at most five for the dodecahedron and self-avoiding paths with length at most three forthe icosahedron. These paths are drawn in Figure 7. Because both graphs have thirty edges, theright-hand side of (20) is a polynomial with degree at most 30. Fixing a labeling of the edges forboth graphs to turn the self-avoiding paths into subsets of { , , . . . , } , and using these subsets asinputs, our program returns the values shown in the first table of Figure 8 for the dodecahedron andthe values shown in the second table for the icosahedron. As previously, multiplying each columnby the appropriate N s listed in the first row of each table gives the coefficients of the polynomialon the right-hand side of (20), which completes the proof of Theorem 5. These polynomials have luster size in bond percolation on the Platonic solids degree less than 30 because we only take into account the shortest self-avoiding paths. (cid:3) In conclusion, using the inclusion-exclusion identity and independence, we proved that comput-ing the expected value of the size of an open cluster reduces to finding all the self-avoiding pathsconnecting two vertices at distance 1 , , . . . , r apart. Whenever finding all these paths is possiblelike for the tetrahedron, the cube and the octahedron, our program returns the exact value of thecoefficients of the polynomial representing the expected value. When finding all the paths is notpossible like for the dodecahedron and the icosahedron, one can still obtain lower bounds by onlylooking at a subset of self-avoiding paths. References [1] M. Aizenman, H. Kesten, and C. M. Newman. Uniqueness of the infinite cluster and continuityof connectivity functions for short and long range percolation.
Comm. Math. Phys. , 111(4):505–531, 1987.[2] M. Aizenman, H. Kesten, and C. M. Newman. Uniqueness of the infinite cluster and relatedresults in percolation. In
Percolation theory and ergodic theory of infinite particle systems(Minneapolis, Minn., 1984–1985) , volume 8 of
IMA Vol. Math. Appl. , pages 13–20. Springer,New York, 1987.[3] M. Ajtai, J. Koml´os, and E. Szemer´edi. Largest random component of a k -cube. Combinatorica ,2(1):1–7, 1982.[4] N. Alon, I. Benjamini, and A. Stacey. Percolation on finite graphs and isoperimetric inequali-ties.
Ann. Probab. , 32(3A):1727–1745, 2004.[5] S. R. Broadbent and J. M. Hammersley. Percolation processes. I. Crystals and mazes.
Proc.Cambridge Philos. Soc. , 53:629–641, 1957.[6] P. Erd˝os and A. R´enyi. On random graphs. I.
Publ. Math. Debrecen , 6:290–297, 1959.[7] G. Grimmett.
Percolation , volume 321 of
Grundlehren der Mathematischen Wissenschaften .Springer-Verlag, Berlin, second edition, 1999.[8] P. Jevti´c and N. Lanchier. Dynamic structural percolation model of loss distribution for cyberrisk of small and medium-sized enterprises for tree-based LAN topology.
Insurance Math.Econom. , 91:209–223, 2020.[9] P. Jevti´c, N. Lanchier, and A. La Salle. First and second moments of the size distribution ofbond percolation clusters on rings, paths and stars.
Statist. Probab. Lett. , 161:108714, 6, 2020.[10] H. Kesten. The critical probability of bond percolation on the square lattice equals . Comm.Math. Phys. , 74(1):41–59, 1980.[11] N. Lanchier.
Stochastic modeling . Universitext. Springer, Cham, 2017.[12] M. V. Menshikov. Coincidence of critical points in percolation problems.
Dokl. Akad. NaukSSSR , 288(6):1308–1311, 1986., 288(6):1308–1311, 1986.