A non-symmetric Kesten criterion and ratio limit theorem for random walks on amenable groups

We consider random walks on countable groups. A celebrated result of Kesten says that the spectral radius of a symmetric walk (whose support generates the group as a semigroup) is equal to one if and only if the group is amenable. We give an analogue of this result for finitely supported walks which are not symmetric. We also conclude a ratio limit theorem for amenable groups.


Introduction
Let G be a finitely generated countable group.Let µ be a probability measure on G, i.e. a function µ : G → R + such that g∈G µ(g) = 1.Let S µ := {g ∈ G : µ(g) > 0}, the support of µ.We say that µ is non-degenerate if S µ generates G as a semigroup.(We do not require S µ to be finite.) Let | • | be a word metric on G associated to some finite generating set.(We do not assume any connection between this set and S µ .)We say that µ has finite first moment if g∈G |g|µ(g) < ∞ and that µ has finite exponential moment of order c > 0 if g∈G e c|g| µ(g) < ∞.
We will be interested in the spectral radius of this random walk, defined by λ(G, µ) := lim sup n→∞ (µ * n (e)) 1/n .Clearly, 0 ≤ λ(G, µ) ≤ 1.A celebrated theorem of Kesten (which does not even require G to be finitely generated) says that if µ is symmetric then λ(G, µ) = 1 if and only if G is amenable [8].(We recall that G is amenable if and only if it admits a Banach mean, i.e. a linear functional M : ℓ ∞ (G) → R such that M (1) = 1, inf g∈G f (g) ≤ M (f ) ≤ sup g∈G f (g), and M (f g ) = M (f ), where f g (x) = f (gx).See the papers of Følner [5] and Day [2] for further discussion.)The aim of this note is to generalise Kesten's criterion to the non-symmetric case.
To state our generalisation, we need to consider the abelianisation of G. Since G is finitely generated, this has a finite rank k ≥ 0. Let G ab = G/[G, G] denote the abelianisation of G and let G ab T denote the torsion subgroup of G ab .Now set G = G ab /G ab T ∼ = Z k , for some k ≥ 0, (the torsion-free part of the abelianisation) and let π : G → G be the natural projection homomorphism.Write μ = π * (µ), i.e. μ(m) = g∈G π(g)=m µ(g).
The special case where µ has finite support originally appeared in Dougall-Sharp [4], where it is written in the language of subshifts of finite type and Gibbs measures.
A probability measure µ (with finite first moment) is said to be centred if for each homomorphism χ : G → R, we have g∈G χ(g)µ(g) = 0.
Any such homomorphism factors through G so it is easy to see that µ is centred if and only if either k = 0 or In particular, µ is centred if and only if μ is centred.
If, in addition, µ has a finite exponential moment of some order then we have the following result.
Corollary 1.2.Let G be a finitely group and let µ be a non-degenerate probability measure on G. Provided µ has a finite exponential moment of some order, we have λ(G, µ) = 1 if and only if G is amenable and µ is centred.
Theorem 1.1 allows us to prove a ratio limit theorem for amenable groups with an explicit limit.To avoid any parity issues, it is convenient to restrict to aperiodic walks.We say that (G, µ) is aperiodic if there exists n 0 ≥ 1 such that µ * n (e) > 0 for all n ≥ n 0 .Theorem 1.3 (Ratio limit theorem).Suppose that G is a finitely generated amenable and that µ is a non-degenerate probability measure on G. Assume in addition that (G, µ) is aperiodic.Then, for each g ∈ G, where ξ ∈ R k is the unique value for which λ(G, µ) = ϕ µ (ξ).
Remark 1.4.One should compare this with a theorem of Avez [1] that says that if G is amenable and µ is symmetric, non-degenerate and aperiodic then lim n→∞ µ * n (g)/µ * n (e) = 1, for all g ∈ G.
Remark 1.5.It should be noted that there is no a priori mechanism to guarantee that the ratios do indeed have a limit.However, notice that if one has the ratio limits for all g ∈ G, for some ξ then G is necessarily amenable.We give the short demonstration.From the hypothesis we have for any s ∈ G, µ * n (s −1 ) µ * n (e) = e ⟨ξ,π(s)⟩ .
we then have In particular ϕ µ (ξ) < ∞.We proceed with the proof assuming that ξ = 0 and deduce the general case after.Now using that µ * n (e) we see that (1.1) with ϕ µ (0) = 1 implies that lim sup n→∞ (µ * n (e)) 1/n = 1.This contradicts Day's [2] generalisation of Kesten's criterion to the random walk operator spectral radius -a consequence of which is that, for any non-degenerate probability, we have that if G is non-amenable then lim sup n→∞ (µ * n (e)) 1/n < 1.
For the general case ξ ̸ = 0, we have already shown that ϕ µ (ξ) < ∞, and so μ(g) = e ⟨ξ,π(g)⟩ ϕ µ (ξ) µ(g) is a well-defined probability measure on G with ratio limits equal to one, and we again conclude that G is amenable.
Let us now outline the contents of the rest of the paper.In Section 2, we recall results of Stone on random walks on Z k that are essential to the formulation of our results, and the rather general results of Gerl.In Section 3, we give a proof of Corollary 1.2 assuming Theorem 1.1.We prove Theorem 1.1 in Sections 4 and 5. Theorem 1.3 is proved in Section 6, as a consequence of equidistribution results for countable state shifts.

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We now state a ratio limit theorem due to Stone.
Then, for each m ∈ Z k , Ratio limit theorems are intimately related to the existence of harmonic functions.Given a random walk (G, µ), we define the random walk operator P µ : This may be written as a convolution (Some authors define f to be λ-harmonic if µ * f = λf .)If we write h ξ (m) = e −⟨ξ,m⟩ for the limit in Proposition 2.3 then we see that the function ȟξ (m ) is the smallest value for which there is a λ-harmonic function.
One may ask about ratio limit theorems on more general groups than Z k .Following earlier work by Avez [1] and Gerl [6], a rather general ratio limit theorem was proved by Gerl [7], where it is obtained as a corollary of the following limit theorem.A detailed account and discussion may be found in the recent note by Woess [18].Proposition 2.4 (Gerl's fundamental theorem [7]).Suppose that µ is a nondegenerate probability measure on G such that (G, µ) is aperiodic.Then we have Gerl used this to prove the following conditional ratio limit theorem.
Proposition 2.5 (Gerl's ratio limit theorem [7]).Suppose that µ is a non-degenerate probability measure on G such that (G, µ) is aperiodic.Suppose there is a set for some subsequence for all g ∈ G.
In particular, if we have uniqueness of a λ(G, µ)-harmonic function for (G, µ) then we know the ratio limit theorem holds.The advantage of our Theorem 1.3, for amenable groups, is that we don't consider arbitrary harmonic functions instead we directly work with functions coming from the abelianisation.

Proof of Corollary 1.2
In this section we prove Corollary 1.2, assuming Theorem 1.1.We note that ϕ µ = ϕ μ, so we can use Lemma 2.1.
Since μ is non-degenerate, ϕ µ (v) is strictly convex on the set where it is finite.The hypothesis of a finite exponential moment implies that ϕ µ (v) is finite and differentiable in some neighbourhood of 0 ∈ R k .Therefore, ϕ µ (v) has its unique minimum at v = 0 if and only if ∇ϕ µ (0) = 0. Suppose that λ(G, μ) = 1; then, by Lemma 2.1, ϕ µ has its minimum at 0 and so i.e. µ is centred.On the other hand, if λ(G, μ) < 1 then, again by Lemma 2.1, the unique minimum of ϕ µ is not at 0 and so In this section we show that if λ(G, µ) = λ(G, μ) then G is amenable.(In Kesten's original theorem, this was the harder direction but in our case it is the easier implication.)Noting that ϕ µ = ϕ μ, recall from Section 2 that there is a unique We define a new probability measure µ ξ on G (analogous to the measure ω ξ on Z k in the proof of Corollary 2.2) by Proof of Theorem 1.1 ( ⇐= ).Assume that G is non-amenable.By Theorem 1 of Day [2] (see also Theorem 5.4 of Stadlbauer [13]), we see that the probability measure µ ξ satisfies lim sup Unpicking the definitions, µ * n ξ (e) = ϕ µ (ξ) −n µ * n (e).Hence lim sup n→∞ (µ * n (e)) 1/n < ϕ µ (ξ).□

Proof of Theorem 1.1 ( =⇒ )
In this section we will show the harder implication that if G is amenable then λ(G, μ) ≤ λ(G, µ), and hence that λ(G, μ) = λ(G, µ).We remark that the proof given here is significantly easier and more direct than the one we gave in [4].Following that proof would introduce a family of measures, indexed by g ∈ G, on the space S N µ × G, each describing the paths that visit S N µ × {g}.These measures (which are also analysed in [14]) are not required here.
Let us begin by emphasising the following: though we know that μ has a λ(G, μ)harmonic function it plays no role in this part of proof !The first element of the proof is the following proposition.Subsequently, the rest of the section will be devoted to showing that show its hypothesis is satisfied with λ = λ(G, µ).
Proposition 5.1.If there is a homomorphism h : G → R >0 , the multiplicative group of positive real numbers, such that, for all n ∈ N, Proof.Suppose such a homomorphism h exists.Since h is positive so we can throw away the terms where −π(g) ̸ = 0 and obtain Hence, for any δ < 1, This says that λ(G, μ) ≤ λδ −1 .Since we can take δ arbitrarily close to 1 we are done.□ We view λ(G, µ) as a convergence parameter for the series This formulation is reminiscent of the Poincaré series used in the construction of Patterson-Sullivan measures on the limit sets of Kleinian groups and more general limit sets and, indeed, we employ ideas from this theory The most relevant reference here is Roblin [9], which covers the basic tools of Patterson-Sullivan theory and the amenability "trick" we will use in the proof of Proposition 5.6.The series ζ(t) does not necessarily diverge at t = λ(G, µ) but one can modify the series, in a controlled way, to guarantee divergence at this critical parameter.The following appears as Lemma 3.2 in Denker and Urbanski [3] (see also [14]).
n=1 be the sequence given by Lemma 5.2 for the series with a n = µ * n (e).We prefer to use c n = 1/b n , a decreasing sequence.Note that we have lim n→∞ c n−r /c n = 1 for all r ∈ N. We will work with a modified series ζ e c (t) defined by and also, for each g ∈ G, the series Proof.We begin by observing that, for every g, h ∈ G, we have and we may choose k ≥ 1 for which µ * k (gh −1 ) > 0. This gives us the inequality Since the numbers c n are positive and, for a fixed k, lim n→∞ c n /c n+k = 1, we have inf n c n /c n+k > 0. Hence .
Since g, h are arbitrary the lemma follows.□ By the previous lemma and a standard diagonal argument, there exists a sequence t n → λ(G, µ) for which the following limits are well-defined for all g ∈ G.A crucial observation is the following.
Lemma 5.4.For any r ∈ N, we have Proof.Fix r ∈ N and let ϵ > 0. Since lim n→∞ c n−r /c n = 1, we can choose n 0 such that 1 − ϵ ≤ c n−r /c n ≤ 1 + ϵ, for all n > n 0 .We will also use that Setting using that the terms in the series are non-negative.This gives and, since ϵ is arbitrary, For the lower bound, we have This gives λ r H(g) ≥ s∈G µ * r (s)H(s −1 g).

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Lemma 5.4 gives us the following corollary.
Corollary 5.5.For any fixed γ ∈ G, we have Proof.Given γ ∈ G, we can find x 1 , . . ., x k ∈ S µ , for some k ≥ 1, such that and so sup g∈G H(γg)/H(g) is finite.Now we put γg on the right hand side and choose y 1 , . . ., y ℓ ∈ S µ such that and so inf g∈G H(γg)/H(g) is positive.□ We are now ready to use the amenability of G.We use the existence of a Banach mean on ℓ ∞ (G) to "average over g" in the last lemma.

Proposition 5.6. There is a homomorphism
Proof.Since any homomorphism h : G → R >0 factors through G, it suffices to show that there is a homomorphism h : G → R >0 such that, for all n ∈ N, we have (5.1) Let M be a Banach mean on ℓ ∞ (G).Jensen's inequality says that if φ is convex then (This is more familiar when the linear functional is integration with respect to a probability measure.One can check that we only need monotonicity, finite additivity, and the unit normalisation M (1) = 1.)We apply this to the function g → (H(γg)/H(g)) in ℓ ∞ (G).(Note we have used Corollary 5.5 to know that g → log(H(γg)/H(g)) is in ℓ ∞ (G).)Thus we obtain We will show that h satisfies (5.1), recalling that M is only finitely additive.Let {g k } ∞ k=1 be an enumeration of G and, for N ≥ 1, let G N = {g 1 , . . ., g N }.Lemma 5.4 gives us that, for any n ≥ 1 and any N ≥ 1, we have Taking the supremum over N gives (5.1).It remains to show that h is a homomorphism.Notice that using translation invariance of M .The conclusion follows.□ Combining Proposition 5.1 and Proposition 5.6 shows that if G is amenable then λ(G, μ) ≤ λ(G, µ).

Equidistribution and proof of the ratio limit theorem
In this section we use Theorem 1.1 to prove a ratio limit theorem for amenable groups, Theorem 1.3.Our arguments will also give a new proof of Proposition 2.4 in this setting.Our approach is based on weighted equidistribution results for countable state shift spaces.Suppose that G is amenable, that µ is non-degenerate and that (G, µ) is aperiodic.We let λ denote the common value given by Theorem 1.1.As above, µ ξ (g) = λ −1 h(g)µ(g), where h(g) = e ⟨ξ,π(g)⟩ .
We consider the sequence space Σ = S N µ and let σ : Σ → Σ be the shift map: . ., n}.We give Σ the topology generated by cylinder sets (which are both open and closed).
We denote by ν ξ the Bernoulli measure on Σ given by s=(s1,...,sn)∈Λn where we use the notation s ∞ ∈ Σ to mean the one-sided infinite concatenation of s = (s 1 , . . ., s n ) and δ s∞ denotes the Dirac measure at this point.We remark that we also have m n = 1 µ * n (e) s=(s1,...,sn)∈Λn µ(s 1 ) • • • µ(s n )δ s∞ but we do not use this formula.We will need to explicitly evaluate the measures m n on cylinder sets.Lemma 6.1.For a cylinder set [u 1 , . . ., u k ] we have that, for n > k, , Proof.This is a straightforward calculation.For n > k,
Proof.We know that lim sup n→∞ (µ * n ξ (e)) in particular the limit exists and is equal to the limsup.□ In order to show the required convergence for the m n , we introduce some ideas and terminology from thermodynamic formalism and large deviation theory.A function φ : Σ → R is called locally Hölder continuous if (6.1) sup for some C > 0 and 0 < θ < 1, for all n ≥ 1.For a locally Hölder continuous function φ : Σ → R, we define the Gurevič pressure P G (φ) by (The original definition given by Sarig in [10] is somewhat different, and only requires, (6.1) to hold for n ≥ 2, but, by Corollary 1 of [11], the above formula gives the Gurevič pressure in our setting.)We now fix ), so that, in particular, P G (φ) = 0. Let χ be the indicator function of some cylinder.We can easily calculate from the definition that, for t ∈ R, P G (φ + tχ) ≤ max{0, |t|} for all t ∈ R. Hence, by Corollary 4 of [11], t → P G (φ + tχ) is real analytic for t ∈ R and, by Theorems 6.12 and 6.5 of [12], (6.2) dP (φ + tχ) dt t=0 = χ dν ξ .
(The same discussion remains true if χ is replaced with any bounded locally Hölder function but indicator functions of cylinders are sufficient for our purposes.)For s ∈ S n ν , let τ s,n denote the orbital measure Following, Theorem 7.4 of [12], we have the following large deviations bound.Proposition 6.3.Given ϵ > 0, there exists C > 0 and η(ϵ) > 0 such that Proof.The proof is standard but we include it for completeness.We consider s ∈ S n µ such that χ dτ s,n > χ dν ξ + ϵ and χ dτ s,n < χ dν ξ − ϵ separately.For t > 0, we have Using P G (φ) = 0 and (6.2), we see that, for sufficiently small t > 0, we have Similarly, for t < 0, lim sup and, for sufficiently small t < 0, this upper bound is negative.Combing these estimates gives the result.□ Since Λ n ⊂ S n µ , we have the following immediate corollary.Corollary 6.4.For ϵ > 0, we have We can now establish the ratio limit theorem for amenable groups.as n → ∞. □