Finitely Dependent Insertion Processes

A $q$-coloring of $\mathbb Z$ is a random process assigning one of $q$ colors to each integer in such a way that consecutive integers receive distinct colors. A process is $k$-dependent if any two sets of integers separated by a distance greater than $k$ receive independent colorings. Holroyd and Liggett constructed the first stationary $k$-dependent $q$-colorings by introducing an insertion algorithm on the complete graph $K_q$. We extend their construction from complete graphs to weighted directed graphs. We show that complete multipartite analogues of $K_3$ and $K_4$ are the only graphs whose insertion process is finitely dependent and whose insertion algorithm is consistent. In particular, there are no other such graphs among all unweighted graphs and among all loopless complete weighted directed graphs. Similar results hold if the consistency condition is weakened to eventual consistency. Finally we show that the directed de Bruijn graphs of shifts of finite type do not yield $k$-dependent insertion processes, assuming eventual consistency.


Introduction
A proper q-coloring of Z is a sequence of colors (x i ) i∈Z with x i ∈ [q] := {1, . . . , q} such that x i = x i+1 for all i. A random q-coloring (X i ) i∈Z is stationary if (X i ) i∈Z and (X i+1 ) i∈Z are equal in law. A stationary q-coloring is k-dependent if (X i ) i<0 and (X i ) i≥k are independent, and finitely dependent if it is k-dependent for some k ≥ 0.
The simplest examples of stationary finitely dependent processes are the block factors. These are stochastic processes of the form {f (Y i , . . . , Y i+k )} i∈Z where f is deterministic and {Y i } i∈Z are an i.i.d. sequence. In the 1960s, Ibragimov and Linnik first suggested that there may exist non-block factor stationary finitely dependent processes [15,16]. Since then, examples of such processes have been constructed by several authors in the course of studying properties of finitely dependent processes [1,2,7,9,17]. Until recently, it has been believed that most 'natural' finitely dependent processes are block factors [6].
Yet block factors have subtle limitations. For example, these processes are never supported on proper colorings [3]. It turns out that finitely dependent processes do not have this limitation, although this fact is highly non-obvious and remains to be fully understood. This was discovered by Holroyd and Liggett in a recent breakthrough [12], in which they disproved a conjecture of Schramm [14] by showing that stationary finitely dependent colorings of the integers exist. These are perhaps the first natural non-block factor finitely dependent processes.
Specifically, Holroyd and Liggett constructed symmetric 3-and 4colorings with these properties. It is remarkable that, while their construction produces a q-coloring for each integer q ≥ 2, only when q ∈ {3, 4} is the coloring finitely dependent. Using a more complicated construction, these authors later obtained symmetric q-colorings for all q ≥ 4 [13].
As described in [12], the q-colorings therein have the following characterization. For each integer q ≥ 2, let Z = (Z 1 , . . . , Z n ) be a sequence of independent random variables each taking the values 1, 2, . . . , q with equal probability. Let σ be an independent uniformly random permutation of 1, . . . , n, which we interpret as meaning that the symbol Z σ(i) arrives at time i. Let E be the event that, for every time t = 1, . . . , n, the subsequence of Z formed by those symbols that arrived up to time t (ordered as in the original sequence Z) forms a proper coloring (i.e. no two consecutive elements in the subsequence are equal). The conditional law of Z given E equals the law of (X 1 , . . . , X n ), where X is the q-coloring constructed in [12].
It was observed by Holroyd (personal communication) that the proper coloring condition in the previous paragraph may be replaced by a graph adjacency condition. The case of a q-coloring corresponds to the complete graph with vertex set {1, . . . , q}, denoted K q . A general graph will encode which pairs of vertices may appear consecutively. See e.g. [5,Example 2.5] for more on this perspective. Since few stationary finitely dependent colorings are currently known, it is natural to pursue this generalization in the search for new finitely dependent processes.
Fix a finite graph G containing at least one edge. Let Z = (Z 1 , . . . , Z n ) be a sequence of independent uniformly random vertices of G. Let σ be an independent uniformly random permutation of 1, . . . , n, which we interpret as meaning that the vertex Z σ(i) arrives at time i. Let E be the event that, for every time t = 1, . . . , n, the subsequence of Z formed by those vertices that arrived up to time t (ordered as in the original sequence Z) forms a path in G. Finally, let (Y 1 , . . . , Y n ) denote random variables whose law equals the conditional law of Z given E.
To produce a stochastic process (X i ) i∈Z from these finite marginals, we must apply a limiting procedure. Let P n denote the probability mass function of (Y 1 , . . . , Y n ). If for all n the mass functions of (Y 1 , . . . , Y n−1 ) and (Y 2 , . . . , Y n ) equal P n−1 , then by the Kolmogorov Extension Theorem [10] there exists a unique stochastic process (X i ) i∈Z such that (X 1 , . . . , X n ) has mass function P n for all n ≥ 1. In this case we say that {P n } ∞ n=1 is consistent and that G satisfies property (C). We say the marginals are eventually consistent if the preceding condition holds for all sufficiently large n, in which case G satisfies property (EC). In this case, we construct a process (X i ) i∈Z in the same manner as before by taking a projective limit [18] over n sufficiently large, generalizing the previous construction when G has property (C). We call (X i ) i∈Z the insertion process associated to G.
Holroyd and Liggett showed that for all q ≥ 2, the complete graph K q has property (C) [12,Proposition 10]. Furthermore, they discovered that the insertion process associated to K 4 is 1-dependent and the insertion process associated to K 3 is 2-dependent. Moreover, these are the only values of q for which the process is finitely dependent [12,Proposition 13].
One may embellish these examples in our general setting. Replacing each of the 3 (resp. 4) colors with r copies of itself yields the complete multipartite graphs K r,r,r,r (resp. K r,r,r ). Both of these graphs are easily seen to have property (C) and have a 1-(resp. 2-)dependent insertion process. We establish the remarkable fact that these are essentially the only (EC) graphs with a finitely dependent insertion process. Theorem 1. Consider a finite graph G having property (C). Then the insertion process associated to G is k-dependent if and only if either: G = K r,r,r and k ≥ 2, or G = K r,r,r,r and k ≥ 1.
If instead we assume that G has property (EC), then the same characterization holds except G may in addition be a disjoint union of one of the above graphs with a collection of isolated vertices.
We deduce Theorem 1 from Theorem 2, which is a result about weighted graphs. In a weighted digraph, an edge from vertex i to vertex j has weight w(i, j). There is no assumption that w(i, j) = w(j, i), and a weight of 0 signifies that no edge is present. There is a natural extension of the insertion process to the setting of weighted digraphs, provided that the digraph satisfies a natural analogue of property (EC) 1 . In lieu of a precise description of these properties for weighted digraphs, 1 See Section 2; the only difference is that instead of conditioning on the sequence of vertices forming a path, a bias is applied depending on certain edge weights.
we present the following iterative sampling algorithm for the insertion procedure on weighted digraphs. Algorithm 1.
(i) Let P 1 assign uniform weight to all v ∈ V .
(ii) Given a random X ∈ V n with mass function P n , sample I ∈ {0, . . . , n} and V ∈ V with probability proportional to where the first or second factor is left out when i = 0 or i = n, respectively. (iii) Output the sequence (X 1 , . . . , X I , V, X I+1 , . . . , X n ). It has mass function P n+1 .
A weighted digraph is uniform of weight w if w(i, j) ∈ {0, w} for all vertices i and j, and if in addition w(i, j) = w(j, i) and w(i, i) = 0. We remark that uniform weight graphs underly a model of endpointweighted insertion introduced in [11, §4].
Theorem 2. Consider a finite uniform weight graph G having property (C). Then the insertion process associated to G is k-dependent if and only if either: G = K r,r,r and k ≥ 2, or G = K r,r,r,r and k ≥ 1.
If instead we assume that G has property (EC), then the same characterization holds except G may in addition be a disjoint union of one of the above graphs with a collection of isolated vertices.
Theorems 1 and 2 demonstrate that the requirement of satisfying property (EC) and having a finitely dependent associated insertion process is extremely restrictive on a graph. This trend continues beyond the setting of uniform weight graphs.
Theorem 3. Let G be a weighted digraph satisfying property (C) such that w(i, i) = 0 and w(i, j) > 0 for all distinct vertices i and j. Then the insertion process associated to G is k-dependent if and only if G is unweighted and either: G = K 3 and k ≥ 2; or G = K 4 and k ≥ 1.
As described previously, we are considering generalizations of a family of colorings constructed in [12] in the search for new finitely dependent processes. We have considered more general graph constraints in place of colorings. However, there is an even broader type of local constraint to consider, known as a shift of finite type [14,19]. A loopless shift of finite type is a set of the form The de Bruijn graph [8] associated to S has vertex set W and edge set Theorem 4. Consider a loopless shift of finite type with de Bruijn graph G. Suppose that G has property (EC). Then the insertion process associated to G extends to a process supported on the shift of finite type. Moreover this process is not finitely dependent.
When n = 2, shifts of finite type correspond to edge sets of directed graphs. In analogy with Theorems 1 to 3, one might expect K 3 and K 4 to appear in Theorem 4. The reason this is not the case is that Theorems 1 to 3 pertain to insertion processes on the vertex set, whereas Theorem 4 corresponds to insertion processes on the edge set.
Open questions. We have presented a general algorithm that produces a sequence of mass functions on paths of growing length. One must impose some condition on these mass functions in order to obtain a limiting process on Z. We have considered two such conditions: property (C), and property (EC), corresponding to consistency and eventual consistency of the mass functions, respectively.
But there are other ways to obtain a limiting process on bi-infinite paths. For example, one may use weak limits. Question 1. Does Theorem 1 continue to hold if we replace property (EC) with the assumption that the measures induced by the mass functions P n (constructed above) converge weakly?
In Theorem 3, we do not even know if property (C) can be weakened to property (EC), let alone to weak limits. Question 2. Does Theorem 3 continue to hold for strongly connected weighted digraphs if we replace property (C) with property (EC)?
While we have established several results in the general setting of weighted digraphs, there remains a relatively simpler context that remains unresolved. Question 3. Does Theorem 1 continue to hold for strongly connected unweighted digraphs?
Lastly, recall that in Theorem 3 we must assume strict positivity of the edge weights, which is an open condition (in the topological sense). It may come as a surprise that there does not appear to be a simple limiting argument that allows one to remove this assumption. One way to see this is that the conclusions of the theorem must change when one removes these positivity conditions, due to complete multipartite graphs with more than two vertices in each of the partite sets.
Question 4. If we remove the assumption that w(i, j) > 0 for all distinct vertices i and j but retain the hypotheses that property (C) holds and that the insertion process is k-dependent in Theorem 3, does it follow that G is unweighted and either G = K r,r,r and k ≥ 2 or G = K r,r,r,r and k ≥ 1?
Overview. Section 2 presents the main construction underlying the remainder of the paper, which consists of a sequence of mass functions on paths of growing length in a weighted digraph. Theorems 1 and 2 are proven in Section 3. Uniform weight graphs appear in Subsection 3.1, complete multipartite graphs in Subsection 3.2, and the proofs of Theorems 1 and 2 are in Subsection 3.3. Section 4 combines Theorem 2 with some additional arguments to deduce Theorem 3. Lastly, we deduce Theorem 4 in Section 5 from Lemma 8 in Subsection 2.2.

Weighted Insertion
In this section we introduce a generalization of the construction in [13, §2] to weighted digraphs.
Let V be a finite alphabet. A word (of length n) is a finite sequence x = (x 1 , x 2 , . . . , x n ) ∈ V n , which we sometimes abbreviate to x 1 x 2 · · · x n . The word of length 0 is denoted by ∅. Let S n be the symmetric group of all permutations of 1, . . . , n. Let x ∈ V n be a word and let σ ∈ S n be a permutation. We interpret σ as meaning that at time t = 1, . . . , n symbol x σ(t) arrives in (relative) position σ(t).
Let Min(σ) and Max(σ) denote the sets of running minima and maxima of σ, respectively. For example, Min(σ) is given by If σ(t) is neither a running minimum nor maximum, then at time t the symbol For example, when σ = id is the identity permutation of 1, . . . , n we have w(x; id) = w(x 1 , x 2 ) · · · w(x n−1 , x n ). We denote this quantity by w(x). When the length of x is at most 1, we have that w(x) = 1. Right: Multiply the edge weights to get w(x 1 · · · x 7 ; σ).
If we imagine building the word dynamically using σ, then when This is a generalization of the building number defined in [12], which is the special case consisting of the weighted digraph w with vertex set {1, . . . , q} and weight function w(i, j) Clearly B(x) > 0 if and only if there exists σ ∈ S n such that w(x; σ) > 0. In this case σ = id works. Word x has positive weight if either of the equivalent conditions B(x) > 0 or w(x) > 0 holds.
For a word x = x 1 x 2 · · · x n , we write Proof. Fix an index i. For any σ which satisfies i = σ(n), with the obvious modifications when i = 1, n. Summing over all σ ∈ S n yields the result.
2.1. Eventual Consistency. Let G be a weighted digraph with (finite) vertex set V . Say that G is recurrent if there exist arbitrarily long words of positive weight. We remark that this is equivalent to there existing positive-weight words of every length, since if w(x 1 x 2 · · · x n ) > 0 then also w(x 1 x 2 · · · x k ) > 0 for all 1 ≤ k ≤ n. Furthermore, it is easy to see that G is recurrent if and only if y∈V n B(y) > 0 for all n ≥ 0.
If G is a recurrent digraph, then for all n ≥ 0 we define a probability mass function P n on V n by setting It is a simple consequence of Lemma 5 that Algorithm 1 presented in the introduction does in fact sample according to the mass function P n in (3). Moreover, Algorithm 1 is well-defined provided that G is recurrent (otherwise, it terminates early if it reaches an n such that there are no positive-weight words of length n).
If G is recurrent and the mass functions {P n } ∞ n=0 satisfy we say that G satisfies property (C) and that the mass functions are consistent. Likewise if G is recurrent and (4) holds for all sufficiently large n, we say that G satisfies property (EC) and that the mass functions are eventually consistent. Clearly, equation (4) is equivalent to the existence of constants C n > 0 such that (5) Thus, a recurrent graph satisfies property (C) if and only if there exists C n > 0 such that (5) holds for all n ≥ 0. Similarly, a recurrent graph satisfies property (EC) if and only if for all sufficiently large n, there exists C n > 0 such that (5) holds. When the associated mass functions are eventually consistent, the Kolmogorov extension theorem [10] establishes that there exists a unique process (X i ) i∈Z with marginal distributions given by for all sufficiently large n. This is the insertion process associated to an (EC) weighted digraph. It follows by construction that the insertion process is stationary. Lemma 6. Suppose that G has property (EC). Then the associated insertion process is k-dependent if and only if for all n, m ∈ N sufficiently large, there are positive constants C n,m > 0 for which x ∈ V n and y ∈ V m .
Proof. By normalizing (6) we see it is equivalent to This is equivalent to the associated insertion process (X n ) n∈Z having the property that (X i ) i∈I and (X j ) j∈J are independent for any pair of sufficiently large intervals I and J separated by a distance of k or greater. Since restriction preserves independence, we may remove the adjective 'sufficiently large' from the previous sentence.
Holroyd and Liggett [12] proved that the (unweighted, undirected) complete graph K q has property (C) for all q ≥ 2. Furthermore, they showed that the associated insertion process is k-dependent if and only if q = 3 and k ≥ 2 or q = 4 and k ≥ 1. In Section 3 we show that apart from minor modifications, these are the only unweighted and undirected graphs with property (EC) for which the associated insertion process is finitely dependent.

2.2.
Necessary conditions for k-dependence. Consider a weighted digraph G satisfying property (EC). In this subsection, we present necessary conditions for the associated insertion process to be finitely dependent. We show that such digraphs must contain directed triangles (Lemma 8) and are not too far from being strongly connected (Lemma 9).
A directed triangle on (a, b) is a triple of vertices (a, b, c) ∈ V 3 that satisfy w(a, b) w(b, c) w(a, c) > 0. We consider weighted digraphs that lack directed triangles. There is a natural interpretation of this condition in relation with Algorithm 1: it is equivalent to requiring all insertions to occur at endpoints of the interval. Lemma 7. For a weighted digraph lacking directed triangles, any x ∈ V n satisfies B(x) = 2 n−1 w(x).
which implies the claim by induction on the length of the word.
Lemma 8. If a weighted digraph satisfies property (EC) and the associated insertion process is finitely dependent, then it contains a directed triangle.
Proof. Suppose there was a weighted digraph G lacking directed triangles that satisfies property (EC) and whose associated insertion process is k-dependent for some k ≥ 0. Let V denote its vertex set. Applying Lemma 6, it follows that for all sufficiently large m and n and for all vertices i, j ∈ V we have that Let A denote the adjacency matrix of G, which is the V × V matrix with A ij = w(i, j). Using Lemma 7 to express the left hand side of (7) in terms of the adjacency matrix we obtain that Combining (7) with (8)  In fact, all eigenvalues must vanish. To see why, consider Tr A = i∈V w(i, i). Since (i, i, i) is a directed triangle, our hypothesis that G lacks directed triangles implies that w(i, i) = 0 and thus Tr A = 0. Combined with the previous paragraph, this implies that all eigenvalues vanish and thus A is nilpotent.
From this it follows that for all but finitely many words x ∈ ≥0 V we must have w(x) = 0. This implies that G is not recurrent, contradicting the assumption that G satisfies property (EC).
A weighted digraph G is strongly connected if for every pair of vertices i, j ∈ V , there is a positive-weight word x ∈ ≥1 V that begins with i and ends with j. We denote the existence of such an x by writing i → j. The strongly connected components (SCCs) of a weighted digraph are the equivalence classes of the relation Note that i → i because singleton words are defined to have weight 1 (see the remark prior to Definition 1). Also, recall from our discussion following Definition 1 that a word has positive weight if and only if w(x) > 0, which occurs if and only if B(x) > 0.
Recall that a weighted digraph is a function w : V 2 → R ≥0 . A subset U ⊆ V induces a subdigraph by restriction of w to U 2 .
Say that an SCC is recurrent if the induced subdigraph of G is recurrent. By our remarks at the beginning of Subsection 2.1, an SCC with vertex set C is recurrent if and only if there are arbitrarily large n such that there exists a positive-weight word x ∈ C n .
Note that any SCC with at least two vertices is recurrent, for if a and b are any two such vertices, then each of the words (ab) n ∈ C 2n have positive weight. Also observe that for a directed acyclic graph, the SCCs are singletons, none of which is recurrent. Moreover, a singleton SCC is recurrent if and only if its vertex has a self-loop. In general, each directed cycle is contained in a recurrent SCC.
For the remainder of this section, we restrict attention to weighted digraphs satisfying property (EC) whose associated insertion process is finitely dependent. We will show in Lemma 9 that such digraphs are not too far from being strongly connected.
Lemma 9. Let G be a weighted digraph with property (EC) whose associated insertion process is finitely dependent.
(i) G has a unique recurrent SCC.
(ii) Vertices not in the recurrent SCC belong to singleton SCCs that have no directed path to or from the recurrent SCC. (iii) The subdigraph of G induced by the recurrent SCC also has property (EC). Moreover, its associated insertion process coincides with that of G. (iv) If moreover G has property (C), then it is strongly connected.
In particular, it follows from Lemma 9 that the family of finitely dependent insertion processes associated to weighted digraphs is unchanged if one restricts attention to strongly connected weighted digraphs.
Proof. Let (X i ) i∈Z denote the insertion process on G. Suppose without loss of generality that it is (k + 1)-dependent, for some k ≥ 0. In particular, we have the i.i.d. subsequence (X ik ) i∈Z .
Let C 1 , . . . , C denote the strongly connected components of G. The weighted digraph G induces the weighted digraph G SCC on the set of SCCs, denoted V SCC := {C 1 , . . . , C }, given by the weight Observe that C i → C j in G SCC if and only if for some u ∈ C i and v ∈ C j , there is a positive-weight word x composed of vertices of G that begins with u and ends with v. Thus if C i → C j and C j → C i , then C i ∪ C j is also an equivalence class, whereupon C i = C j .
Consider the function f : V → V SCC that assigns to each vertex its strongly connected component. Then the sequence f (X i ) i∈Z has the property that for all i < j, we have f (X i ) → f (X j ) in G SCC almost surely. But if f (X i ) occurs once in the sequence, it a.s. occurs infinitely many times (by passing to an i.i.d. subsequence). Thus there exists > j such that f (X ) = f (X i ). Consequently the sequence is a.s. constant. Furthermore, the unique value C rec that it takes must be a recurrent SCC, since the marginals of the process (X i ) i∈Z then yield arbitrarily long positive-weight words in ≥1 C rec . On the other hand, each vertex v ∈ V belonging to a recurrent SCC occurs infinitely often in the sequence (X i ) i∈Z . Indeed, we can fix a sufficiently long positive-weight word containing v and observe that it occurs with positive density in (X i ) i∈Z by finite dependence. Thus by the previous paragraph, it follows that C rec is the a unique recurrent SCC of G, establishing property (i).
Property (ii) follows by combining our previous observation that an SCC containing distinct vertices is recurrent with the following argument. If there was a directed path joining C rec to another strongly connected component denoted C , then there would be arbitrary long positive-weight words that contain vertices in C . But then C appears in the sequence f (X i ) i∈Z , by consideration of a sufficiently long marginal of the insertion process. This contradicts (9), proving (ii).
For all n > |V |, every word x ∈ V n contains some vertex at least twice. Thus by (ii), it follows that if such a word satisfies B(x) > 0, then necessarily x ∈ C n rec . Hence and it follows that all sufficiently long marginals of the insertion processes associated to G and the subdigraph induced by C rec coincide. This yields (iii). Finally, (iv) follows from the observation that B(x) = 1 for all words of length 1. Thus if G satisfies property (C), the random variable X 0 in the associated insertion process is uniformly random on the vertex set V . But by (9) we have that X 0 ∈ C rec almost surely, implying that V = C rec and therefore G is strongly connected.

Uniform Weight Graphs
We establish Theorem 2 in this section, and deduce Theorem 1 as a corollary. The plan of attack is as follows: (i) Subsection 3.1 defines the special class of weighted digraphs to which Theorem 2 applies, which we call uniform weight graphs.
We deduce necessary structural properties for a uniform weight graph to satisfy property (EC). (ii) Subsection 3.2 is devoted to complete multipartite graphs of uniform weight. It is shown that understanding finite dependence for such graphs reduces to the analysis of complete graphs, which has already been undertaken in [12]. (iii) Subsection 3.3 proves Theorem 2 by using a combinatorial argument to show that the structural properties in Subsection 3.1 imply that the graph is complete multipartite, then applying the results of Subsection 3.2.
3.1. Uniform weights. Uniform weight graphs are weighted digraphs that are undirected and of constant weight. Their associated insertion processes are sufficiently general to encompass the end-weighted insertion processes of [11, Section 4] as a special case. Definition 2. A uniform weight graph with weight w > 0 is a weighted digraph whose weight function satisfies that for all i, j ∈ V , (i) either w(i, j) = 0 or w(i, j) = w, and (ii) w(i, j) = w(j, i), and (iii) w(i, i) = 0. We regard uniform weight graphs as being undirected. As such, we refer to their strongly connected components simply as connected components (when applying Lemma 9, for instance). For uniform weight graphs, we say that vertices i and j are adjacent if and only if w(i, j) > 0, in which case w(i, j) = w.
The recurrence (5) simplifies to the following in the case of a positiveweight word in a uniform weight graph It is convenient to fix some notation regarding alternating words. For any positive integer n that may be even or odd, we define (ab) n/2 to be the unique alternating word in {a, b} n beginning with a. Lemma 10. If w(a, b) > 0 and w(b, v) > 0, then , w(a, v) = w and w = 1 2 n−1 (n + 1), w(a, v) = w and w = 1.
Proof. First suppose that w(a, v) = 0. By (10) we have that Applying this inductively yields the first case of (11). Next, suppose that w(a, v) = w and w = 1. Again by (10), Now the second case of (11) follows by induction, starting from the base case B(av) = 2w. The final case of (11) follows from a simplification of the previous calculation.
Lemma 11. If G is a non-empty connected uniform weight graph satisfying property (EC), then the following conditions hold.
Recall that B (ab) n/2 = (2w) n−1 . Thus by (5), it follows that for all sufficiently large n if w = 1. Likewise if w = 1, for all sufficiently large n we have that C n = 2(d − t) + (n + 1)t. In either case, both d and t are determined by the values of C n for n sufficiently large. Thus d and t take the same value, for all vertices b and all edges (a, b).
Consider the graph that appears in Figure 2.

First observe that we have the weaker inequality
Indeed, each σ ∈ S n +1 with w(yv; σ) > 0 is seen to satisfy w(xv; σ) > 0 as well. Since the graph has uniform weight, it therefore follows that w(xv; σ) ≥ w(yv; σ) and the claim follows upon summation.

3.2.
Complete multipartite graphs. In this section we relate property (EC) for complete multipartite graphs to property (EC) for complete graphs. This allows us to extend results of Holroyd and Liggett [12] to handle complete multipartite graphs. We use the following notation: K q denotes the complete graph on q vertices, K r,...,r denotes the complete multipartite graph with q parts each of size r, and w · K r,...,r denotes the uniform weight graph in which each edge of the corresponding complete multipartite graph has weight w > 0. In all cases, we take the vertex set to be [qr] := {1, . . . , qr} (with r = 1 in the case of K q ), and we take the edge set to be {(i, j) : i ≡ j mod q}. In the case of w · K r,...,r , the weight function is given by Note that the graph K r,...,r is a Turán graph [4].
Lemma 13. For any r ≥ 1 and k ≥ 0, the graph w · K r,...,r satisfies property (EC) and its associated insertion process is k-dependent if and only if the same holds for w · K q .
Proof. Consider the mapping f : from which it follows that B(x) = B f (x) . Thus the following are equivalent: By (5), it follows that w · K q satisfies property (EC) if and only if w · K r,...,r satisfies property (EC). Similarly, the following are equivalent: The result now follows by Lemma 6.
Lemma 14. Suppose that w = 1 and q ≥ 3. Then the graph w · K q does not satisfy property (EC).
Proof. Consider an integer n ≥ 3. Fix distinct vertices a, b, c ∈ K q . Keeping in mind our notation for alternating words, set x = (ba) n/2 ∈ {a, b} n , y = c (ab) (n−1)/2 ∈ {a, b, c} n (regardless of the parity of n). Consider the quantities We will prove by induction on n that Q n > R n for all w > 1 and Q n < R n for all w ∈ (0, 1). From (10) we deduce the recurrences B(xv) = wB( x 1 v)+w 2 B( x n v)+ wB(x) and B(yv) = wB( y 1 v)+w 2 B( y 2 v)+w 2 B( y n v)+wB(y), yielding that Q n = w + 1 2 Q n−1 + qw − w(w + 1) Subtracting the previous equations and substituting the recurrence B(y) = wB( y 1 ) + w 2 B( y 2 ) + wB( y n ) yields that 2B(y)(Q n − R n )/w equals When w > 1, we leave out the first term, using (w + 1) 2 ≥ 4w to obtain Since the right side vanishes for n = 3, it follows that Q n > R n by induction.
Next suppose that w < 1. This case requires a tighter bound. We begin by establishing that for all m ≥ 3, Indeed, we deduce the bound inductively from Now we plug the bound (15) into (14) to obtain a bound which is suitable for induction: As before, when n = 3 the right side vanishes. Thus it follows by induction that if 0 < w < 1, we have R n > Q n for all n ≥ 3. Combined with the previous case, we conclude that when w = 1 and q ≥ 3, the graph w · K q does not satisfy property (EC).
Combining several results allows us to determine which uniform weight complete multipartite graphs satisfy property (EC) and have a finitely dependent associated insertion process. Lemma 15. The graph w · K r,...,r for q ≥ 3 and r ≥ 1 has property (EC) and has a k-dependent associated insertion process if and only if w = 1 and either: q = 3 and k ≥ 2; or q = 4 and k ≥ 1.
Proof. By Lemma 13, it suffices to consider w · K q . By Lemma 14, w = 1. Applying Propositions 10 and 13 of [12], the insertion process on K q is k-dependent if and only if q = 3 (k ≥ 2) or q = 4 (k ≥ 1).

3.3.
Extension to uniform weight graphs. We now combine the results of Subsections 3.1 and 3.2 to extend the conclusion of Lemma 15 to all uniform weight graphs. We establish that the only uniform weight graphs satisfying property (EC) that have a k-dependent associated insertion process are K r,r,r and K r,r,r,r , and unions thereof with isolated vertices. Note that these graphs have weight w = 1, even though we allow w > 0 to be arbitrary a priori. These graphs are closely related to the graphs K 3 and K 4 corresponding to the colorings discovered by Holroyd and Liggett; in fact, the graphs K r,r,r and K r,r,r,r are obtained from K 3 (resp. K 4 ) by replacing each vertex with r copies of itself. Similarly, the insertion process on K r,r,r (resp. K r,r,r,r ) is obtained from the corresponding process on K 3 (resp. K 4 ) by replacing each instance of vertex i with an i.i.d. choice of one of its r copies in K r,r,r (resp. K r,r,r,r ).
The following graph-theoretic lemma allows us to reduce the general uniform weight case to that of the complete multipartite graphs treated in Subsection 3.2. We will use Lemma 16 to show that any uniform weight graph either contains a kite, or is complete multipartite. Lemma 16. Let G be a kite-free connected loopless graph containing a triangle abc. Then every vertex d ∈ V is adjacent to at least two of {a, b, c}. Proof. By Lemma 12, no vertex d ∈ V is adjacent to exactly one of abc. Hence it suffices to show that every vertex d is adjacent to abc.
Suppose to the contrary that some d ∈ V is non-adjacent to abc (Figure 3). Choose a minimal path joining d to abc. We show that a kite is present near the intersection of the path with abc, from which we obtain the desired contradiction. Let d denote the path vertex adjacent to abc. There are two cases to consider: either d is adjacent to a single vertex of abc (left half of Figure 4), or it is adjacent to more than one vertex (right half of Figure 4).
If d is adjacent to a single vertex, then (abc; d ) is a kite. Now suppose that d is adjacent to more than one vertex. Without loss of generality, suppose that d is adjacent to a and c. Let d denote a neighbor of d on the path from d to d . By minimality of the path, d is non-adjacent to abc. Consequently (d ac; d ) is a kite.
We are now in a position to prove Theorem 2.
Proof of Theorem 2. Suppose that G is a uniform weight graph with property (EC) and suppose that the insertion process associated to G is k-dependent. We will deduce that either G = K r,r,r,r and k ≥ 2, or G = K r,r,r and k ≥ 1, or G is a disjoint union of one of these graphs with a collection of isolated vertices.
Since uniform weight graphs are undirected, Lemma 9 takes on a simpler form in the present context. Indeed, it implies that both the (EC) property and the insertion process are unchanged by deletion of isolated vertices, and moreover it implies that the resulting graph is connected. Hence it suffices to consider the connected case.
We show in this case that G is complete multipartite. Consider the relation This relation is reflexive and symmetric by definition of a uniform weight graph. Once we establish transitivity, it will follow that the graph G is complete multipartite with partite sets given by the equivalence classes of this relation. Suppose to the contrary that transitivity did not hold. Then there would exist vertices a, b, d such that w(a, b) > 0 yet w(a, d) = w(b, d) = 0. By Lemma 8 and Lemma 11, there are t ≥ 1 triangles on every edge. In particular, we may complete the edge (a, b) into a triangle abc. By Lemma 12, G lacks kites, and by assumption it is connected. Moreover by definition of uniform weight, G is loopless.
Thus we have verified the conditions of Lemma 16. Consequently, the vertex d is adjacent to at least two of a, b, c. This contradicts the hypothesis that w(a, d) = w(a, b) = 0. Thus the relation (16) is transitive, and we deduce that it is an equivalence relation.
Decompose the vertex set into equivalence classes of (16). Then G is complete multipartite, with partite sets are given by the equivalence classes of (16). By Lemma 11, G is a regular graph and thus the parts have equal sizes. Thus G = w · K r,...,r , for some w > 0 and r ≥ 1.
Applying Lemma 8 again, we see that G contains a triangle and therefore q ≥ 3. The result now follows when G has property (EC) by Lemma 15. Finally, observe that if G also satisfies property (C), then there can be no isolated vertices by Lemma 9(iv).

Complete weighted digraphs with property (C)
The results in this section apply to loopless complete weighted digraphs satisfying property (C). That is, the digraphs under consideration satisfy w(i, i) = 0 and w(i, j) > 0 for all distinct vertices i and j, as well as property (C). We will establish Theorem 3, which states that the only such graphs for which the associated insertion process is k-dependent are the (unweighted) graphs K 3 (for k ≥ 1) and K 4 (for k ≥ 2). We will establish this result by reducing it to the case of a uniform weight graph and applying results from Section 3.
For distinct indices i, j ∈ V , we define the quantity where we use the convention 0 0 := 1. Lemma 17. Fix a loopless complete weighted digraph satisfying property (C) and fix an integer n ≥ 1. Then the value of T n (i, j) is constant over all vertices i and j with w(i, j) > 0. Figure 5. Let x be as in the proof of Lemma 17. For σ ∈ S n+1 , let be as in (19). Consider the nearest neighbor graph on {1, . . . , n + 1}, except we modify vertex n + 1 to have degree . This graph is drawn above with vertex i labeled with the i th symbol of the word xv. Then w(xv; σ) = e=(i,j) w (xv) i , (xv) j , where the product is taken over edges of the graph above.
Proof. Let x ∈ {i, j} n be the unique alternating word ending in i.
Recall from Definition 1 that Since w(i, i) = w(j, j) = 0 by assumption, for all permutations with w(xv; σ) > 0 we have that where is given by the formula This is straightforward to verify from the definitions and Figure 5. By (19), we see that ranges over {1, . . . , n} as σ ranges over S n+1 . Thus substituting (18) into (17) implies that there are integers d 1 , . . . , d n > 0 whose values depend only on n such that Summing over all vertices v ∈ V , we obtain v∈V B(xv) = w(x) n =1 d T (i, j).
By (5) we have that v∈V B(xv) = C n B(x). Furthermore B(x) = 2 n−1 w(x), by Lemma 7 applied to the subgraph induced by {i, j}. As w(x) > 0, we have that n =1 d T (i, j) = 2 n−1 C n .
Since d 1 , . . . , d n > 0, we may explicitly solve this system of equations to obtain that . . .
In particular, it follows that T n (i, j) is independent of the pair (i, j).
In light of Lemma 17, we write T n in place of T n (i, j) from now on. Lemma 18. For all pairs of distinct indices (i, j) and (i , j ), we have w(i, j) = w(i , j ).
Proof. Let z 1 > z 2 > · · · denote the set of distinct positive values attained by w(i, v)w(j, v) as v ranges over V . Let V denote the vertices which contribute to z , given by V = {v ∈ V : w(i, v)w(j, v) = z }, and let a = v∈V w(i, v). Rewriting the expression for T 2n+1 yields T 2n+1 = a z n .
Next, observe that z 1 = inf n→∞ (T 2n+1 ) 1/n and a 1 = inf n→∞ T 2n+1 z n 1 . Hence the parameters a 1 and z 1 can be reconstructed given the sequence {T n }. Applying the same procedure to T 2n+1 − a 1 z n 1 allows us to iteratively reconstruct all of the parameters. Thus a and z are uniquely determined by the {T n }, so they are independent of the choice (i, j). Finally, observe that a = v =j w(i, v). Therefore which shows that for i = j, the value of w(i, j) is constant.
Combining Lemma 18 with the results of Section 3 allows us to conclude Theorem 3.
Proof of Theorem 3. Suppose that G is a loopless weighted digraph such that w(i, j) > 0 for all distinct vertices i and j. Moreover, suppose that G has property (C) and that the associated insertion process is k-dependent process. We show that either: G = K 3 and k ≥ 2; or G = K 4 and k ≥ 1.
By Lemma 18, the graph G has uniform weight, and by Lemma 9 it is strongly connected. Applying Theorem 2, we conclude that either G = K r,r,r and k ≥ 2, or G = K r,r,r,r and k ≥ 1. Since w(i, j) > 0 for all i = j, it follows that r = 1. Thus G is a complete graph, and applying [12,Proposition 13] we deduce that G is either K 3 (for k ≥ 2) or K 4 (for k ≥ 1).
Conversely, by the main result of [12] the graphs K 3 and K 4 are k-dependent for k ≥ 2 and k ≥ 1 respectively.

Shifts of finite type
We turn to the proof of Theorem 4, which states that if the de Bruijn graph of a loopless shift of finite type satisfies property (EC), then the associated insertion process on the shift of finite type is not finitely dependent.
Proof of Theorem 4. Let G denote the de Bruijn graph of the shift of finite type and let {Y } ∈Z denote the insertion process associated to G. We write Y = (x , . . . , x +n−1 ) ∈ [q] n .
Since {Y } is almost surely a path in G, the overlapping elements in adjacent tuples Y and Y +1 almost surely coincide. We extend the insertion process from G to the shift of finite type by considering the random sequence {x } ∈Z .
Since the edge (a, c) is present in G, we must have (x 2 , x 3 , . . . , x n ) = (x 3 , . . . , x n+1 ). Thus x 2 = x 3 = · · · = x n+1 , so b is a constant sequence. This contradicts our assumption that the vertex set of G lacks elements of the form (i, . . . , i). Therefore the associated insertion process is not k-dependent.