Fixation to Consensus on Tree-related Graphs
We study a continuous time Markov process whose state space consists of an assignment of or to each vertex of a graph . The graphs that we treat are related to homogeneous trees of degree , such as finite or infinite stacks of such trees. The initial spin configuration is chosen from a Bernoulli product measure with density of spins. The system evolves according to an agreement inducing dynamics: each vertex, at rate 1, changes its spin value to agree with the majority of its neighbors. We study the long time behavior of this system and prove that, if is close enough to 1, the system reaches fixation to consensus. The geometric percolation-type arguments introduced here may be of independent interest.
In this work we study the long term behavior of continuous time Markov processes whose states assign either or (usually called a spin value) to each vertex in a graph . The graphs we consider are related to homogeneous trees of degree and include infinite stacks of homogeneous trees. These graphs will be specified in Section 2, where we will also discuss some earlier papers where such stacks of trees have been studied. The geometric and percolation theoretic methods we introduce to carry out our analysis (see especially Section 4 and Appendix B) are potentially of independent interest.
We denote by the value of the spin at vertex at time . Starting from a random initial configuration drawn from the independent Bernoulli product measure
the system then evolves in continuous time according to an agreement inducing dynamics: at rate 1, each vertex changes its value if it disagrees with more than half of its neighbors, and tosses a fair coin in the event of a tie. Our arguments and results easily extend to many other types of dynamics, as discussed in Remark 2.3 below; these include processes in discrete time, as in , and different rules for tie-breaking.
Our process corresponds to the zero-temperature limit of Glauber dynamics for a stochastic Ising model with ferromagnetic nearest neighbor interactions and no external magnetic field (see, e.g.,  or ). This process has been studied extensively in the physical and mathematical literature – primarily on graphs such as the hyper-lattice and the homogeneous tree of degree , . A physical motivation is the behavior of a magnetic system following the extreme case of a deep quench, i.e., when a system has reached equilibrium at infinite temperature and is instantaneously reduced to zero temperature. For references on this and related problems see, e.g.,  or . The main focus in the study of this model is the formation and evolution of boundaries delimiting same spin cluster domains: these domains shrink or grow or split or coalesce as their boundaries evolve. An interesting question is whether the system has a limiting configuration, or equivalently does every vertex eventually stop flipping? Whether
exists for almost every initial configuration, realization of the dynamics and for all in the underlying graph depends on and on the structure of the underlying graph . Nanda, Newman and Stein  investigated this question when and and found that in this case the limit does not exist, i.e., every vertex flips forever. Their work extended an old result of Arratia , who showed the same on for or . One important consequence of the methods of  is that does exist for almost every initial configuration, realization of the dynamics and every if the graph is such that every vertex has an odd number of neighbors, such as for example for odd.
Another question of interest is whether sufficient bias in the initial configuration leads the system to reach consensus in the limit. I.e., does there exist , such that for ,
We will refer to (3) as fixation to consensus (of ). Kanoria and Montanari  studied fixation to consensus on homogeneous trees of degree for a process with synchronous time dynamics. Their process has the same update rules as ours, except that all vertices update simultaneously and at integer times . For each , Kanoria and Montanari defined the consensus threshold to be the smallest bias in such that the dynamics converges to the all configuration, and proved upper and lower bounds for as a function of . Fixation to consensus was also investigated on for the asynchronous dynamics model. It was conjectured by Liggett  that fixation to consensus holds there for all . Fontes, Schonmann and Sidoravicius  proved consensus for all with strictly less than but very close to 1 and Morris  proved that as .
In  Howard investigated the asynchronous dynamics in detail on and showed how fixation takes place. On this tree graph, vertices fixate in spin chains (defined as doubly infinite paths of vertices of the same spin sign). Though no spin chains are present at time 0 when , Howard showed that for any , there are (almost surely) infinitely many distinct and spin chains at time . He also showed the existence of a phase transition in : there exists a critical such that if , spin chains do not form almost surely, whereas if they almost surely form in finite time. Our work is motivated by that of Howard, but for more general tree-related graphs.
2 Statements of Theorems
Let denote a doubly infinite stack of homogeneous trees of degree , i.e., the graph with vertex set and edge set specified below. The main focus of this paper is proving fixation to consensus on for the process started with an independent identically distributed initial configuration of parameter . Such infinite stacks of trees have been studied before in the context of Bernoulli percolation  and Ising models . More general nonamenable graphs have also been studied — see, e.g., . One motivation for these studies is that as the parameter of the model varies, the behavior is sometimes like that on a simple homogeneous tree and sometimes like that on a simple amenable graph like .
We express as
where , and think of this as a decomposition of the infinite stack into layers . Let the edge set of , , be such that any two vertices are connected by an edge if and only if:
for some , and the corresponding and are adjacent vertices in ; or
for some and ; or
for some and .
For a more detailed description of the Markov process than the one given in Section 1, we associate to each vertex a rate 1 Poisson process whose arrival times we think of as a sequence of clock rings at . We will denote the arrival times of these Poisson processes by and take the Poisson processes associated to different vertices to be mutually independent. We associate to the ’s independent Bernoulli random variables with values or , which will represent the fair coin tosses to be used in the event of a tie. Let be the probability measure for the realization of the dynamics (clock rings and tie-breaking coin tosses), and denote by the joint probability measure on the space of initial configurations and realizations of the dynamics; an element of will be denoted .
The main result of this paper is the following theorem, which shows fixation to consensus for nontrivial ; its proof is given in Section 4. Unlike Kanoria and Montanari , here we do not attempt to obtain good upper bounds on though we expect to approach with increasing degree . We restrict ourselves to proving fixation to for close enough to 1 with the standard majority update rule: when its clock rings, each vertex updates to agree with the majority of its neighbors or tosses a fair coin in the event of a tie. See Remark 2.3 below for other update rules to which our arguments and results apply.
Given , there exists such that for the process on fixates to consensus.
Homogeneous trees of degree .
Finite width stacks of homogeneous trees of degree with free or periodic boundary conditions. These are graphs, which we will denote by or , with vertex set and edge set or . and are defined similarly to the edge set of : two vertices are connected by an edge if and only if either condition i above holds; or
for and either condition ii or iii holds; or
and ; or
Any two vertices are connected by an edge if and only if either condition i holds; or
for and either condition ii or iii holds; or
and or ; or
and or .
Semi-infinite stacks of homogeneous trees of degree with free boundary conditions. These are graphs, which we will denoted by , with vertex set and edge set . Two vertices are connected by an edge if and only either condition i holds; or
for and either condition ii or iii holds; or
Fix and and let be one of the following graphs: , , or . There exists such that for the process on fixates to consensus.
Our results have natural extensions to other dynamics. Let be the maximum number of neighbors of a vertex in the graph where is any of the graphs of Theorem 2.2; for some , we can change (arbitrarily) the update rules for those vertices whose number of neighbors is strictly less than , and the conclusions of Theorem 2.1 or 2.2 remain valid with the same proof. For large , can be taken much larger than . For example, on the infinite stack of -trees, and for , one can take (as is readily seen from the proof of Theorem 2.1). A special case of this type of extension of our results is to modify the update rule in the event of a tie: e.g., instead of flipping a fair coin, flip a biased coin with any bias or do nothing. We can also change from two-valued spins to any fixed number of spin values, say . The initial configuration is given by the measure where and and the updating is done via a majority rule, e.g., by a rule that respects majority agreement of neighbors on, say, color 1. We can then think of color as the spin from before, and the other colors together representing the spin. If is close enough to 1, we again obtain fixation to consensus. All our results also apply to the synchronous dynamics of .
In order to prove Theorem 2.1 we will show that if we take close enough to 1, then already at time 0 there are stable structures of vertices, which are fixed for all time. We will choose these structures to be subsets (denoted ) of the layers in the decomposition of such that they are stable with respect to the dynamics. We will define a set as the union of for all , and show that for close enough to 1, the complement of is a union of almost surely finite components.
3.1 A Set of Fixed Vertices in
For fixed, let be the union of all subgraphs of that are isomorphic to with .
We point out that is stable for , since every has out of neighbors of spin and for . Not only is this set stable with respect to the dynamics on as in Theorem 2.1, but it’s also stable with respect to the dynamics on and the other graphs of Theorem 2.2. Let represent the union of across all levels , i.e.,
where for shorthand notation, .
If , as defined above is not stable with respect to the dynamics. In these cases the argument will be changed somewhat as discussed in Section 4.
3.2 Asymmetric Site Percolation on
The goal of this subsection is to state and prove a geometric probability estimate, Proposition 3.1, which concerns asymmetric site percolation on distributed according to the product measure with:
This equals the distribution of restricted to the layers , and therefore applies to these graphs as well. The statement and proof of Proposition 3.1 require a series of definitions. The first of these defines graphical subsets of , whereas the second concerns probabilistic events for subgraphs of that have a specific orientation. Later, in the proof of Theorem 2.1 which is given in Section 4, Proposition 3.1 will be applied to certain subsets of .
Certain rooted subtrees of Let in be two adjacent vertices, and denote by the connected component of in – see Figure 1.
Let be three adjacent vertices in , such that and are neighbors of . Denote by the connected component of in – see Figure 2.
Random -ary trees of spin with a certain
Let be a deterministic subtree of with at least two vertices,
and be a leaf of ; i.e., has a neighbor in and
neighbors in . is the event
that there is a subgraph of isomorphic to
and containing , such that –
see Figure 3.
Let be a deterministic subtree of with at least five vertices, and be a 2-point of (i.e., a vertex of with exactly two neighbors in ) that is also good (i.e., both its neighbors in are also 2-points of ). Let be the two neighbors of in and let be ’s other neighbor in . is the event that there is a subgraph of isomorphic to and containing and , such that ; here is the graph with vertex set and edge set – see Figure 4.
For distinct leaves of , the events are defined on disjoint subsets of , and are therefore independent; they are also identically distributed. The same is true for for disjoint pairs . The following is essentially the same as Definition 3.1, with the only difference being that here we define the graph on , whereas before we defined the same random graph on .
Let be the union of all subgraphs of that are isomorphic to with .
The next proposition estimates the probability that none of the vertices of a given set belong to any random -ary tree of spin (see Definition 3.3). This proposition is a main ingredient in the proof of Theorem 2.1.
For any such that for and any deterministic finite nonempty subset of ,
Let be the minimal spanning tree containing all the vertices of . We will call the vertices of the special vertices of . Note that all the leaves of are special vertices.
We first suppose ; the simpler case will be handled at the end of the proof. By the distinctness and disjointness results of Lemma C.1 from the Appendix, there exist constants depending only on , such that for each such tree , one or both of the following is valid:
there are at least leaves in , with the events mutually independent, and/or
there are at least edges having endpoints in with a good special 2-point, and the events mutually independent.
Let us first suppose that a) holds. We claim that, for any leaf of ,
The claim follows from a string of inclusions. First,
But if is a leaf of , then
Labeling of the leaves in a) as , we restrict the above intersection to the leaves of , so that
Since the events are mutually independent,
implying the claim.
Alternatively, suppose that b) holds. Now we claim that
where is a good special 2-point of and is one of ’s neighbors. This claim also follows from a string of inclusions. First,
If are adjacent and is a good special 2-point of , then
As with the proof of the previous claim, we label of the pairs of vertices given in b) as . Then
The second claim follows from the mutual independence of the events . The two claims imply (7) for by taking
If , suppose the only vertex in is 0, a distinguished vertex. Then
where are the neighbors of 0 and is defined as in Definition 3.3 with the tree containing only vertices 0 and . Then (7) follows in this case by taking and using Equation (43) and Lemma A.1. This completes the proof.
4 Main Results
We study the connected components of as a subgraph of , and show that if is close enough to 1 these connected components are finite almost surely. We will show that each of these finite connected components of vertices shrinks and is eliminated in finite time leading to fixation of all vertices to +1.
For any , is the connected component of in : is the set of vertices s.t. , i.e., there exists a path in with every .
Given , there exists such that for , is a union of almost surely finite connected components.
It suffices to show that is finite almost surely, where 0 is a distinguished vertex in . Since implies a.s., it suffices to show .
Let represent any site self-avoiding path in of length starting at 0, then by standard arguments
where by we mean that all the vertices of belong to .
To show the sum is finite we need to bound . Suppose the vertex set of is , where for each , is a nonempty subset of for some with the distinct. We now apply Proposition 3.1 to in each of the layers , which are isomorphic to . This shows that for any such that for ,
Since the are subsets of distinct levels of , the events are mutually independent. Therefore for ,
Equation (21) and the above bound on imply
where is the number of self-avoiding paths of length starting at 0. It is easy to see that
The proof is finished by choosing for . ∎
Proof of Theorem 2.1 for .
Taking as in Proposition 4.1, is a union of almost surely finite connected components:
where the ’s are nonempty, disjoint and almost surely finite with for some .
Fix any ; it suffices to show that is eliminated by the dynamics in finite time. By this we mean that there exists such that for any , and so the droplet fixates to . We proceed to show this.
For any set , let
so is stable with respect to the dynamics and for any , .
Since is finite it contains a longest path, . Since cannot be extended to a longer path, must have neighbors in . When ’s clock first rings, flips to and fixates for all later times. This argument can be extended to show is eliminated (i.e., the vertices are all flipped to ) by the dynamics in finite time as follows. Consider the set of vertices in which have not yet flipped to by some time , and take to infinity. Suppose this limiting set is nonempty. Since this set is finite, it contains a longest path . But now of ’s neighbors have spin as , implying that had no clock rings in for some finite . This event has zero probability of occurring, which contradicts the supposition of a nonempty limit set. ∎
The proof of Theorem 2.1 for and 4 is slightly different than for , since for (respectively, ) the ’s of Definition 3.1 are not stable with respect to the dynamics: each vertex has 2 (resp., 3) neighbors of spin , which is always less than a strict majority. The proof for or 4 requires a different decomposition of the space and definition of stable subsets. With this purpose in mind, we express as
where (see Equation (4) for a comparison). We call a vertex or its partner in , , doubly open if both and ; this occurs with probability . We proceed by defining a set of fixed vertices in in the spirit of Section 2.1.
For fixed, let be the union of all subgraphs of that are isomorphic to such that , is doubly open.
It is easy to see that is stable for or 4 with respect to the dynamics on . Let denote the union of across all levels , i.e.,
Proof of Theorem 2.1 for and .
We map one independent percolation model, on with parameter , to another one, on with parameter , by defining (resp., ) if is doubly open (resp., is not doubly open). Propositions 3.1 and 4.1 applied to imply that Proposition 4.1 with replaced by is valid for the percolation model. The rest of the proof proceeds as in the case for . ∎
Proof of Theorem 2.2.
The proof proceeds analogously to that of Theorem 2.1, except that the conclusion of Proposition 4.1, that almost surely has no infinite components (for close to ), is replaced by an analogous result for with an appropriately defined . We next specify a choice of for each of our graphs and leave further details (which are straightforward given the proof of Proposition 4.1) to the reader.
For with any , we simply label (see Definition 3.4). For , depends on like it did for - i.e., for , we take
and for or we take
For or with , we take
For or with or , the choice of depends on whether is even or odd since in the odd case the layers cannot be evenly paired. If is even, then we take
For odd (and ), we pair off the first layers and then use the final layers to define in which the use of doubly open sites for is replaced by triply open sites; then we take
Appendix A Galton-Watson Lemmas
The goal of this section is to show that the quantity
which appears at the end of the proof of Proposition 3.1, converges to as . Here is a leaf of a subtree of , is a pair of adjacent vertices of such that is a good 2-point (as in Definition 3.3), and are fixed constants.
Let be a subtree of such that contains and is a leaf of – see Figure 6.
Let be one of the neighbors of (other than 0) and be a subtree of containing and (but not ) such that is a good 2-point of – see Figure 7.
We consider the events with respect to and with respect to (see Definition 3.3) and estimate , by analyzing a related Galton-Watson process.
For any vertex , let denote the spin cluster of , that is, is the set of vertices in such that the path from to (including and ) includes only vertices , with .
where represents the graph distance(i.e., the minimum number of edges ) between and . if and only if , and in general is the number of vertices in at distance from that are in ’s spin cluster. is a Galton-Watson branching process with offspring distribution .
Let denote a tree with root , such that has coordination number and all the other vertices have coordination number . The following definition is close to that of (see Definition 3.3 and Figure 3), except that here the -ary tree in question is rooted.
Random rooted -ary trees of spin
Consider two vertices such that is a neighbor of
. Let denote the event that there exists a subgraph
of isomorphic to
which contains and is contained in , such that for all .
Consider three vertices such that and are neighbors of . Let be the event that there exists a subgraph of isomorphic to which contains and is contained in (see Figure 2), such that for all .
and are independent for by construction. The event is equivalent to the spin at being and the vertices being the roots of -ary trees of spin , so that
The proof is a consequence of Proposition 5.30 from  (about occurrence of -ary subtrees in Galton-Watson processes). ∎
The event is equivalent to , so that, by the independence of the events and ,
This result follows as in the proof of Lemma A.1. ∎
Appendix B Geometric Lemmas
Let be a finite tree with vertices and maximal coordination number . of ’s vertices are labeled special, such that all of ’s leaves are special vertices. We remark that in Section 4, we start with special vertices in and then is the minimal subtree of that contains all the special vertices.
Let be the number of leaves in , the number of -points (vertices with exactly two edges in ), , the number of -points (vertices with exactly edges in ); . Then
The proof can be found, for example, as part of Theorem 8.1 in . ∎
Recall that a good 2-point in is a 2-point both of whose neighbors are 2-points. A bad 2-point is a 2-point that is not a good 2-point – see Figure 8.
There exist constants , depending only on , such that either:
there are at least special good 2-points.
By Lemma B.1
and since ,