Random walks on groups and monoids with a Markovian harmonic measure

We consider a transient nearest neighbor random walk on a group G with ﬂnite set of generators §. The pair ( G; §) is assumed to admit a natural notion of normal form words where only the last letter is modiﬂed by multiplication by a generator. The basic examples are the free products of a ﬂnitely generated free group and a ﬂnite family of ﬂnite groups, with natural generators. We prove that the harmonic measure is Markovian of a particular type. The transition matrix is entirely determined by the initial distribution which is itself the unique solution of a ﬂnite set of polynomial equations of degree two. This enables to e–ciently compute the drift, the entropy, the probability of ever hitting an element, and the minimal positive harmonic functions of the walk. The results extend to monoids.


Introduction
Consider a Markov chain whose state space is the Cayley graph of a group (G, * ) with a finite set of generators Σ.If the Markov chain moves from a node g to a node g * h with a probability µ(h) depending only on h, we say that it is a randow walk.If the support of µ is included in Σ, we say that we have a nearest neighbor random walk (with respect to Σ).
A remarkable and well known result is that the harmonic measure of a transient nearest neighbor random walk on a finitely generated free group is Markovian [6,19,25].This can be viewed as a consequence of the multiplicative structure of the Green kernel.If one tries to extend the setting by assuming that the probability µ defining the random walk is not of nearest neighbor type but only of finite support, then the result fails to be true, see [17].In other words, having a Markovian harmonic measure is a property which depends not only on the group but also on the choosen set of generators.
Consider now a nearest neighbor random walk on a free product of finitely many finite groups.It is easily seen that the Green kernel still has a multiplicative structure.The Markovian nature of the harmonic measure is therefore natural.What we prove is that it is a Markovian measure with a very specific structure that we call Markovian multiplicative.In particular, the Markovian measure is entirely determined by its initial distribution r, where r is the unique solution of a finite set of polynomial equations that we call the Traffic Equations.Understanding this point enables us to define precisely the setting in which the result holds: nearest neighbor random walks on a class of pairs (G, Σ) that we coin as 0-automatic pairs.
A pair (G, Σ) formed by a group (group law * , unit element 1 G ) and a finite set of generators is 0-automatic if the set of words L(G, Σ) = {u 1 • • • u k | ∀i, u i * u i+1 ∈ Σ ∪ 1 G } is a crosssection of G.Such pairs were first considered by Stallings [27] under another name.Consider a group G = F(S) G 1 • • • G k which is a free product of a finitely generated free group and a finite family of finite groups, also called plain group.Consider the natural (but not necessarily minimal) set of generators Σ = S S −1 i G i \{1 G i }.Then the pair (G, Σ) is 0-automatic.Now consider an arbitrary 0-automatic pair (G, Σ).Then G is isomorphic to a plain group.On the other hand, Σ may be strictly larger than a natural set of generators of the group seen as a free product.(See Example 3.5.)And what is relevant in our context is the pair group -generators rather that the group itself, since the generators form the support of the measure defining the nearest neighbor random walk.The larger is the set of generators, the more general is the model.
With the above definition, we can describe more precisely the main result.Let (G, Σ) be a 0-automatic pair and let (X n ) n be a realization of a transient nearest neighbor random walk on G.The random walk can be viewed as evolving on L(G, Σ).Let L ∞ be the set of rightinfinite words whose finite prefixes belong to L(G, Σ).Then X ∞ = lim n X n is a well-defined r.v.living on L ∞ .The law µ ∞ of X ∞ is called the harmonic measure of the random walk.Intuitively, µ ∞ gives the directions in which the random walk escapes to infinity.We prove that the harmonic measure is a Markovian measure.Furthermore, let r and P be respectively the initial distribution and the transition matrix of the Markov chain defining µ ∞ .Then, P is entirely determined by r, and r is itself characterized as the unique solution to the finite set of Traffic Equations.
Mutatis mutandis, the result extends to random walks on 0-automatic pairs formed by a monoid and a finite set of generators.
Apart from the structural aspect, the above mentioned result also provides a simple and direct approach for computational purposes.The drift and the entropy of the random walk exist as a direct consequence of Kingman's Subadditive Ergodic Theorem.But subadditive limits are in general hard to compute explicitly, and drift and entropy are no exceptions to the rule.Here, using the shape of the harmonic measure, we can express the drift and the entropy in a 'semiexplicit' way, as a simple function of r.In comparison, the 'classical' approach of [25] consists in expressing the drift as a functional of the first-passage generating series of the random walk.This provides also a potential method for explicit computations, but a less direct one (and which does not work for the entropy).Using the Traffic Equations yields more chances to solve the model completely and to get closed form formulas.Examples of drift computations are given in §6.In the companion paper [22], we perform a more precise and systematic study for free products of cyclic groups.
Let us clarify a couple of points.
• Following Dynkin & Malyutov [6], random walks on plain groups have been extensively studied in the 1980ies.General formulas for the harmonic measure are available, see in particular [30,Section 6].For a detailed account and bibliographic references, see the monograph of Woess [31], and in particular the Chapters II-9, III-17, III-19, and IV-26.There are two main differences between the previous results and the work presented here.First, the previous results are usually stated in a more general framework, and are therefore less simple and less precise.Second, they are centered around the probabilities of hitting elements instead of the quantity r.We come back to this point at the end of §4.3.
• All the random walks considered here belong to the general setting of random walks on regular languages studied in [18].In [18], local limit theorems are proved.Also, our random walks can be viewed as random walks on a tree with finitely many cone types in the sense of [24], but they are specific since they arise from a group, and since they have one-step moves at distance 1 and 2. In [24], it is proved that the harmonic measure is Markovian for random walks on a tree with finitely many cone types, but for one-step moves at distance 1. (See also [28].)Our method of proof is different from the one in [24], and the nature of the result is different as well (Markovian multiplicative versus Markovian).We discuss this point in §5.3.
• Some important aspects of the theory of random walks on discrete infinite groups are barely touched upon here, for instance, boundary theory, or the asymptotics of P {X n = 1 G }.The interested reader should consult [14,29,31] and the references there.Of a different nature, but somehow related, are the 'homesick random walks' on Cayley graphs of groups, see [20].
• There exists a completely orthogonal approach to the study of random walks on monoids.The random walk is viewed as evolving in the egg-box diagram of the monoid.In general, any box of the diagram may contain monoid elements of arbitrary lengths (with respect to a given finite set of generators).For instance if the monoid is a group, then the egg-box diagram is trivial.See [13] for details.
The paper is organized as follows.Zero-automatic pairs are defined in §3.The main results for groups are then proved in §4, and the extension to monoids is done in §5.
The results presented here, as well as the ones from the companion paper [22], were announced without proofs in the Proceedings of the International Colloquium of Mathematics and Computer Science held in Vienna in Septembre 2004 [23].
Notations.Let N be the set of non-negative integers and let N * = N\{0}.We also set R * + = R + \{0}.We denote the support of a random variable by supp.If µ is a measure on a group (G, * ), then µ * n is the n-fold convolution product of µ, that is the image of the product measure µ ⊗n by the product map The symbol is used for the disjoint union of sets.Given a finite set Σ, a vector x ∈ R Σ , and S ⊂ Σ, set x(S) = u∈S x(u).

Finitely Generated Groups and Random Walks
Given a set Σ, the free monoid it generates is denoted by Σ * .As usual, Σ is called the alphabet, the elements of Σ and Σ * are called respectively letters and words, and the subsets of Σ * are called languages.The empty word is denoted by 1 and we set Σ + = Σ * \{1}.The concatenation product of two elements u and v of Σ * is written uv.The length (number of letters) of a word u is denoted by |u| Σ .The longest prefix of two words u and v is denoted by u ∧ v.
Consider a finitely generated group (G, * ) with unit element 1 G .The inverse of the element u is denoted u −1 .Let Σ ⊂ G be a finite set of generators of G. Throughout the paper, we always assume that 1 G ∈ Σ and that u ∈ Σ =⇒ u −1 ∈ Σ. Denote by π : Σ * → G the monoid homomorphism which associates to a word The length with respect to Σ of a group element g is: A representative u of g is a The Cayley graph X(G, Σ) of a group G with respect to a set of generators Σ is the directed graph with set of nodes G and with an arc from g to h if g −1 * h ∈ Σ.As usual, an arc from g to h is represented graphically by g −→ h.It is often convenient to view X(G, Σ) as a labelled graph with set of labels Σ (with For examples of Cayley graphs, see Figures 1 or 2. For simplicity, we draw a single undirected edge g −−g * a instead of the two directed arcs g g * a. Observe that |g| Σ is the geodesic distance (in number of arcs) from 1 G to g in the Cayley graph.
Consider a relation R ⊂ Σ * × Σ * , and let ∼ R be the least congruence on Given a set S, denote by F(S) the free group generated (as a group) by S. A monoid presentation of Consider a relation R ⊂ (S S −1 ) * × (S S −1 ) * .Let G be isomorphic to the quotient group (F(S)/ ∼ R ).We say that We write, respectively, when necessary if we are dealing with a monoid or a group presentation.
Given two groups G 1 and G 2 , let G 1 G 2 be the free product of G 1 and G 2 .Roughly, the elements of G 1 G 2 are the finite alternate sequences of elements of and the group law is the concatenation with simplification.More rigorously, set where Following [12], define a plain group as the free product of a finitely generated free group and a finite family of finite groups.
is a set of generators for G that we call natural generators.(Even for a fixed decomposition (F(S), G 1 , . . ., G k ) of G, the set of natural generators is not unique as soon as #S ≥ 2, because of the flexibility in choosing the free generators of the free group.) Random walks.Let µ be a probability distribution over Σ.Consider the Markov chain on the state space G with one-step transition probabilities given by: ∀g ∈ G, ∀a ∈ Σ, P g,g * a = µ(a).This Markov chain is called the (right) random walk (associated with) (G, µ).It is a nearest neighbor random walk: one-step moves occur between nearest neighbors in the Cayley graph X(G, Σ).Let (x n ) n be a sequence of i.i.d.r.v.'s distributed according to µ. Set The sequence (X n ) n is a realization of the random walk (G, µ).The law of X n is µ * n .
Drift, entropy.The first natural question concerning the random walk (G, µ) is whether it is recurrent or transient.Assuming transience, the next natural question is the speed at which the random walk escapes all the finite sets.
Subadditive Ergodic Theorem [16] yields the following (first noticed by Guivarc'h [11]).There for all 1 ≤ p < ∞.We call γ the drift of the random walk.Intuitively, γ is the rate of escape to infinity of the walk.
Another quantity of interest is the entropy.The entropy of a probability measure µ with finite support S is defined by The entropy of the random walk (G, µ), introduced by Avez [2], is a.s. and in L p , for all 1 ≤ p < ∞.The existence of the limits as well as their equality follow again from Kingman's Subadditive Ergodic Theorem, see [2,4].

Zero-Automatic Pairs
Let (G, * ) be a group with finite set of generators Σ ⊂ G.A language L of Σ * is a cross-section of G (over the alphabet Σ) if the restriction of π to L defines a bijection, that is if every element of G has a unique representative in L. A word of L is then called a normal form word.The map φ : G → L which associates to a group element its unique representative in L is called the normal form map.
Definition 3.1.Let G be a group with finite set of generators Σ. Define the language L(G, Σ) ⊂ Σ * of locally reduced words by: We say that the pair Such pairs were introduced and studied by Stallings in [27] under the name unique factorization groups, a terminology that we did not adopt for several reasons.It is convenient to introduce the sets: Then we can reformulate (7) as: Here are some consequences of Definition 3.1.First, let φ : and the analog for φ(a * g) also holds.Second, the language L(G, Σ) is regular and recognized by the following automaton: Set of Third, the Cayley graph X(G, Σ) has a tree-like structure.In particular, if G is infinite then X(G, Σ) has uniform node-connectivity 1, that is, the removal of any node disconnects the graph.Fourth, all the group elements have a unique geodesic representative with respect to Σ, and the set of these geodesic representatives is precisely L(G, Σ).Fifth, the set of simple circuits going through the node 1 G in X(G, Σ) is finite.In fact, this last property is equivalent to the property that L(G, Σ) be a cross-section, see [12].
Let us now describe the pairs which are 0-automatic.The following is easy to prove.Proposition 3.2.Consider a plain group G and a set of natural generators Σ (see §2).Then the pair (G, Σ) is 0-automatic.(8), can be explicited: Conversely, next Proposition can be proved using results from Stallings [27]: The proof of Stallings is constructive and provides more precise information.Let ϕ be any isomorphism from G to the plain group G. Then there exists a set S of natural generators of G such that S ⊂ ϕ(Σ).The inclusion may be strict, but for u ∈ ϕ(Σ), we must have |u| S = 1, 2, or 3.This is illustrated in Example 3.5.
Plain groups are hyperbolic in the sense of Gromov [10] and automatic in the sense of Epstein & al [7].Besides, (G, Σ) is 0-automatic iff (Σ, L(G, Σ)) is an automatic pair (in the sense of [7]) satisfying the 0-fellow traveller property.This is our justification for the chosen terminology.Example 3.4.Consider the group G = Z Z/2Z Z/4Z.Let a, b, and c be the respective generators of Z, Z/2Z, and Z/4Z.Set S = {a, a −1 , b, c, c −1 }.We have represented the Cayley graph of G with respect to S in Figure 1.Observe that (G, S) is not 0-automatic.For instance, c * c has two representatives in L(G, S): cc and c −1 c −1 ; hence L(G, S) is not a cross-section.Now, let us switch from the minimal set of generators S to the natural (but non-minimal) set of generators Σ = {a, a −1 , b, c, c 2 , c 3 = c −1 }.According to Proposition 3.2, (G, Σ) is 0-automatic.This shows that 0-automaticity is not intrinsic: it depends on the set of generators.Zero-automatic pairs are studied in detail in [3,21].In [21], we compile several characterizations of 0-automatic pairs, some known and some new.The following natural question is also answered in [3,21]: Given a plain group G and a finite set of generators Σ, is the pair (G, Σ) 0-automatic?
Another simple result proved in [21], and which will be used several times below, is the following.The graph of successors Next(G, Σ) of a 0-automatic pair (G, Σ) is the directed graph with: nodes: Σ, arcs: Lemma 3.6.Let (G, Σ) be a 0-automatic pair with G infinite and not isomorphic to Z.The graph of successors Next(G, Σ) is strongly connected.

Random Walks on Zero-Automatic Pairs
Consider a 0-automatic pair (G, Σ).Assume that G is infinite and not isomorphic to Z or Z/2Z Z/2Z.Let µ be a probability measure on Σ which generates the whole group, that is, This random walk is always transient.Indeed, in a non-amenable group, any random walk supported by the whole group is transient and has a strictly positive drift (see [11] and [ by A word belongs to L ∞ iff all its finite prefixes belong to L(G, Σ).The set L ∞ is the end boundary, as well as the hyperbolic boundary, of G. Using the symbolic dynamic terminology, it is a one-sided subshift of finite type.
Equip Σ N with the Borel σalgebra associated with the product topology.This induces a σ-algebra on L ∞ .Given a measure ν ∞ on L ∞ and a ∈ Σ, define the measure aν ∞ by: f Proposition 4.1.Let (X n ) n be a realization of the random walk (G, µ) and set Y n = φ(X n ).
There exists a r.v.Y ∞ valued in L ∞ such that a.s.
meaning that the length of the common prefix between Y n and Y ∞ goes to infinity a.s.Let µ ∞ be the distribution of Y ∞ .The measure µ ∞ is µ-stationary and is the only µ-stationary probability on L ∞ .
We call µ ∞ the harmonic measure of (G, µ).Intuitively, the harmonic measure gives the direction in which (X n ) n goes to infinity.
Proposition 4.2.The drift and the entropy of the random walk are given by: where dx −1 µ ∞ /dµ ∞ is the Radon-Nikodym derivative of x −1 µ ∞ with respect to µ ∞ .
In words, γ is the expected change of length of an infinite normal form distributed according to µ ∞ , when left-multiplied by an element distributed according to µ.The entropy h is the mutual information of Y 1 and Y ∞ , or equivalently, the relative entropy of the law of (Y 1 , Y ∞ ) with respect to the product law of Y 1 and Y ∞ .In the context of the free group, Propositions 4.1 and 4.2 are proved for instance in [19, Theorem 1.12, Theorem 4.10, Corollary 4.5].Concerning the entropy, see also [5,15].The proofs adapt easily to the present setting.In adapting the proof of [19, Theorem 1.12], one has to argue that L ∞ is the unique closed G-invariant and non-empty subset included in L ∞ .But this follows from Lemma 3.6.Several of the key arguments for Propositions 4.1 and 4.2 go back to Furstenberg [8,9], see [19] for precise references.

Markovian multiplicative measure
It is the transition matrix of a Markov Chain on the state space Σ.According to Lemma 3.6, this Markov chain is irreducible.Let (U n ) n be a realization of the Markov chain with transition matrix P and starting from U 1 such that We call ν ∞ the Markovian multiplicative probability measure associated with r.
The measure ν ∞ is in general non-stationary with respect to the translation shift τ : Σ N → Σ N , (x n ) n → (x n+1 ) n .Indeed, the distribution of the first marginal is r which is different in general from the stationary distribution of P .x(u)

Markovian harmonic measure
The Traffic Equations are closely related to the harmonic measure of (G, µ).
Lemma 4.4.If the harmonic measure µ ∞ is the Markovian multiplicative measure associated with r ∈ B, then r is a solution to the Traffic Equations (17).Conversely, if the Traffic Equations admit a solution r ∈ B, then the harmonic measure µ ∞ is the Markovian multiplicative measure associated with r.
Proof.Assume that the harmonic measure µ ∞ is the Markovian multiplicative measure associated with some r ∈ B. According to Proposition 4.1, µ ∞ is µ-stationary.So, applying (12), we get, for all a ∈ Σ, Simplifying using ( 16), we obtain Therefore, r is a solution to the Traffic Equations.
Conversely, let r ∈ B be a solution to the Traffic Equations.Let ν ∞ be the Markovian multiplicative measure associated with r.According to Proposition 4.1, the measure ν ∞ is the harmonic measure if and only if it is µ-stationary, i.e. iff it satisfies: for all Set q(a) = r(a)/r(Next(a)) for all a.The previous equation holds if and only if: This is equivalent to: By definition, a ∈ Next(y) iff φ(y * a) = ya.Using that x * y ∈ Σ and applying (9), we deduce that φ(x * (y * a)) = (x * y)a.In particular a ∈ Next(x * y).So we have Next(y) ⊂ Next(x * y).Now using that y = x −1 * (x * y), we conclude easily that (20) holds.
It follows that r(Next(u 1 )) = r(Next(y)).So Equation ( 19) is equivalent to: This last equation holds since r is a solution to the Traffic Equations.We conclude that ν ∞ is indeed the harmonic measure.
Since the harmonic measure is uniquely defined, a corollary of the above is that the Traffic Equations ( 17) have at most one solution x ∈ B. We are now ready to state the main result.
Theorem 4.5.Let G be an infinite group with finite set of generators Σ, such that (G, Σ) is 0-automatic.Assume that G is not isomorphic to Z or Z/2Z Z/2Z.Let µ be a probability measure on Σ such that n∈N * supp µ * n = G.Then the Traffic Equations ( 17) have a unique solution r ∈ B. The harmonic measure of the random walk is the Markovian multiplicative measure associated with r.
When specializing Theorem 4.5 to the case G = F(Σ), the free group generated by Σ, we recover the classical results of [6,19,25].
Corollary 4.6.Under the assumptions of Theorem 4.5, the drift is given by and the entropy is given by where ∀a ∈ Σ, q(a) = r(a)/r(Next(a)).
In particular, if the probabilities µ(a), a ∈ Σ, are algebraic numbers, then the drift and the entropy are also algebraic numbers.
We are going to prove Theorem 4.5 using a fixed point argument.
Proof of Theorem 4.5.Define the application Φ : ) Hence, a solution to the Traffic Equations is a vector r ∈ B such that r = Φ(r).
Let us prove that Φ( B) ⊂ B. Consider X ∈ B. We consider separately the contributions from the terms A, B, and C in a∈Σ Φ(X)(a).We have Collecting the three contributions, we obtain a∈Σ Φ(X)(a) = a∈Σ µ(a) = 1.We have proved that Φ( B) ⊂ B.
The set B is a convex compact subset of R Σ , which suggests to look for a fixed point argument.However, Φ cannot in general be extended continuously on B. Indeed, set Φ(X) a = A(X) + B(X) + C(X) with the same decomposition as in (23).Clearly the maps A(X) and B(X) can be extended continuously on B. On the other hand, C(x) can be extended iff v∈Next(u) X(v) = 0 for all u ∈ Prev(a).
For x ∈ B\ B, let Φ(x) ⊂ B be the set of possible limits of Φ(x n ), x n ∈ B, x n → x.We have extended Φ to a correspondence Φ : B B. Clearly this correspondence has a closed graph and nonempty convex values.Therefore, we are in the domain of application of the Kakutani-Fan-Glicksberg Theorem, see [1,Chapter 16].The correspondence has at least one fixed point: ∃x ∈ B such that x ∈ Φ(x).Now let us prove that a fixed point x satisfies x ∈ B. Using the shape of the Traffic Equations, we get that: The graph G with nodes Σ and arcs a → b, b ∈ Prev(a), is the graph obtained from the graph Next(G, Σ) by reverting the direction of the arcs.Hence G is strongly connected according to Lemma 3.6.We deduce that x must necessarily belong to B.
Finally, according to Lemma 4.4, there is at most one fixed point in B, by uniqueness of the harmonic measure.Therefore there is exactly one r in B such that Φ(r) = r and the Markovian multiplicative measure associated with r is the harmonic measure.

Probability of ever hitting a group element
Asymptotic properties of random walks on plain groups have been extensively studied via the Green kernel, see [31,.The purpose of this section is to clarify the link between this classical approach and the results in §4.2.
For all g ∈ G, let q(g) be the probability of ever hitting g: Clearly, 0 , by the strong Markov property, we have q(g) = q(u 1 )q(u 2 ) Therefore, all we need to compute are the quantities q(a), a ) a∈Σ has to be a solution of the set of equations: ∀a ∈ Σ, Let us justify it.The first two terms on the right-hand side are more or less obvious.Now assume that the random walk starts with an initial step of type c ∈ Σ\Σ a .Given the tree-like structure of the Cayley graph, it has to go back to 1 G before possibly reaching a.Now, the probability of ever hitting 1 G starting from c is equal to the probability of ever hitting c −1 starting from 1 G .This accounts for the third right-hand term in (25).
The Equations ( 25) are close enough to the Traffic Equations (17).Let (r(a)) a∈Σ be the unique solution in B of the Traffic Equations (Theorem 4.5).We have: , Hence q(a) = r(a)/r(Next(a)), a ∈ Σ, is a solution to (25).To conclude that q = q, it is enough to show that the Equations ( 25) have a unique solution.
Lemma 4.7.The Equations ( 25) have a unique solution in (0, 1) Σ , which is In particular, we recover the classical fact that the Martin boundary and minimal Martin boundary of the random walk coincide with the end boundary L ∞ .(This holds in much more general contexts, see [31, and the references therein.)The pair (L ∞ , µ ∞ ) is the Poisson boundary.
Comparison with the literature.The importance of the Equations ( 25) in q is well-known.
In the seminal paper of Dynkin & Malyutov [6], these equations are explicitely solved in the free group case, and the harmonic functions are then derived as above.In [25], the authors prove that µ ∞ is Markovian for the free group as follows: they use the expression for q obtained in [6], they define r as in (27), and then they prove that the measure defined by ν ∞ (u is the harmonic measure.The approach of [30] is similar.For trees with finitely many cone types, the proof in [24] that the harmonic measure is Markovian is also centered around the analog of the Equations (25).The series version of ( 25) is still the main ingredient in getting central or local limit theorems.See for instance [25,18,24] for nearest neighbor random walks, and [17] for finite range random walks on free groups.
Here the proof that µ ∞ is Markovian multiplicative is different and based on the Traffic Equations (17) instead of the Equations ( 25).This is a more direct path and the only way to proceed in the general case since we cannot retrieve a solution to the Traffic Equations from a solution to (25).This point becomes even clearer in the context of monoids, see §5.

From Groups to Monoids and Beyond
At the cost of some added technicalities in the definitions, Theorem 4.5 holds when replacing groups by monoids.The main result is Theorem 5.3.The notion of zero-automaticity for monoids is discussed in more details in [21].

Zero-automaticity for monoids
Let (M, * ) be a finitely generated monoid with unit element 1 M .Let Σ ⊂ M be a finite set of generators.We always assume that 1 M ∈ Σ.Let π : Σ * → M be the corresponding monoid homomorphism.
Mutatis mutandis, the following notions and notations are defined as in §2: a representative word u ∈ π −1 (m), a cross-section of M , a normal form word, the normal form map, the length |•| Σ , a geodesic, the Cayley graph X(M, Σ).Define the language L(M, Σ) ⊂ Σ * of locally reduced words by: Define the sets Next(.),Prev(.) as in (8).
We now want a notion of 0-automaticity for monoids.The condition that L(M, Σ) is a crosssection of M is not sufficient to get an analog of Theorem 4.5.We propose the following definition:

Random walks
Let M be a monoid with finite set of generators Σ.Let µ be a probability measure on Σ such that n∈N * supp µ * n = M .The (right) random walk (associated with) (M, µ) is defined as in the group case.The drift is defined as in (5) and the entropy as in (6).¿From now on, assume that the pair (M, Σ) is 0-automatic.We make three additional assumptions.First, that the graph of successors Next(M, Σ) is strongly connected.Second, that M is infinite.Third, that the random walk (M, µ) is transient.Define L ∞ as in (11).Define a µ-stationary probability measure as in (12).The analog of Proposition 4.1 holds.Indeed, the proof of Proposition 4.1 given in [19,Theorem 1.12] translates to the new setting.In particular, to get the uniqueness of the µ-stationary measure, one uses the assumption that Next(M, Σ) is strongly connected.So the harmonic measure µ ∞ of the random walk is well-defined.The formulas for the drift and the entropy are now: The Traffic Equations are defined by: ∀a ∈ Σ, The proof of Lemma 4.4 carries over.The only modification is that the equality [Next(x * y) = Next(y)] which was proved in the group case (see (20)), is now true by definition (see (31)).
All the ingredients are in place to state the extension of Theorem 4.5 for monoids.The proof of Theorem 4.5 can be used, mutatis mutandis, to prove Theorem 5.3.Define q(a), the probability of ever hitting a, as in (24).The results of §4.3 do not extend to the monoid case.Indeed there is no direct link anymore between the Traffic Equations and the equations defining q.This is best understood on an example.There is an additional term in the first equation with respect to the second.

Beyond monoids
To delimit the precise role of the group or monoid assumption in Theorems 4.
Let µ be a probability measure on Σ, let (x n ) n∈N be a sequence of i.i.d.r.v.'s with law µ.Define the Σ * -valued Markov chain (Y n ) n by Y n = Ψ(x 0 • • • x n−1 ).This is a generalization of the previous group and monoid random walks, with f replacing the group or monoid law.
Using (34), we get and, with a similar argument, for all a ∈ Σ, = r(a) .
This completes the proof.
As opposed to the group or monoid case, the measure µ ∞ is Markovian but not necessarily Markovian multiplicative.Hence the crucial link, via Lemma 4.4, with the Traffic Equations is cut.It is not possible anymore to characterize µ ∞ using the Traffic Equations.To illustrate this last point, consider the following example.This last equation shows that µ ∞ is not Markovian multiplicative.
Trees with finitely many cone types.In [24], the authors prove that a transient Nearest Neighbor (NN) random walk on a Tree with Finitely Many Cone Types (TFMCT) has a Markovian 'harmonic measure'.Let us compare more closely the result from [24] and Theorems 4.5 and 5.3.Let (M, Σ) be a 0-automatic pair, and let φ : M → L(M, Σ) be the normal form map. Define the directed and labelled tree T (M, Σ) with set of nodes M and arcs: Let T x (M, Σ), x ∈ M, be the cone rooted at x, that is, the subtree of T (M, Σ) with nodes {u | φ(u) ∈ φ(x)Σ * }.Then T x (M, Σ) and T y (M, Σ) are isomorphic as soon as φ(x) and φ(y) end up with the same letter.In particular, T (M, Σ) is a TFMCT.Now consider the random walk (M, µ) where µ is a probability on Σ.There are three possible types of one-step moves, see (30): the lines 1 and 3 in (30) correspond to one-step moves between On the other hand, the values (12 + 3 √ 2)/28 and (6 + √ 3)/12 cannot be retrieved using such elementary arguments.Here it is necessary to solve the Traffic Equations.

Figure 1 :
Figure 1: The Cayley graph of the group Z Z/2Z Z/4Z.
This last assumption is not restrictive.If M is in fact a group, see the discussion at the beginning of §4.If M is not a group, then (M, µ) is transient.Indeed, there must exist b ∈ Σ with no inverse.Consider c ∈ Σ such that b * d = c for some d ∈ Σ.Clearly, c has no inverse either.Let (Y n ) n be a realization of the random walk on L(M, Σ).In view of (30), if Y n equals an element ending with the letter b, then we have ∀m ≥ n, |Y m | Σ ≥ |Y n | Σ .The transience follows easily.
with obvious notations for the entropy.Once again, this is proved by slightly modifying the arguments in[19, Theorem 4.10, Corollary 4.5].

Theorem 5 . 3 .
Let M be an infinite monoid with finite set of generators Σ, such that (M, Σ) is 0-automatic.Assume that the graph of successors Next(M, Σ) is strongly connected.Let µ be a probability measure on Σ. Assume that n∈N * supp µ * n = M and that the random walk (M, µ) is transient.Then the Traffic Equations have a unique solution r ∈ B. The harmonic measure of the random walk is the Markovian multiplicative measure associated with r.The drift is given by: γ = a∈Σ µ(a) − b|a * b=1 M r(b) + r(Next(a)) , and the entropy is given by: log p(a * b) p(b) r(b) + log[p(a)]r(Next(a)) , where ∀a ∈ Σ, p(a) = r(a)/r(Next(a)).

Figure 4 :
Figure 4: The simple random walk on Z 2 Z 2 Z 2 B.