Co-spectral radius for countable equivalence relations

We define the co-spectral radius of inclusions $\mathcal{S}\leq \mathcal{R}$ of discrete, probability measure-preserving equivalence relations, as the sampling exponent of a generating random walk on the ambient relation. The co-spectral radius is analogous to the spectral radius for random walks on $G/H$ for inclusion $H\leq G$ of groups. For the proof, we develop a more general version of the 2-3 method we used in another work on the growth of unimodular random rooted trees. We use this method to show that the walk growth exists for an arbitrary unimodular random rooted graph of bounded degree. We also investigate how the co-spectral radius behaves for hyperfinite relations, and discuss new critical exponents for percolation that can be defined using the co-spectral radius.


Introduction
Let Γ be an infinite group generated by a finite symmetric set S and let G = Cay(Γ, S) be its Cayley graph.We would like to measure the size of certain subsets C of Γ, in a translation invariant way, using the graph structure of G.As a measuring tool, we will use the lazy 1 random walk (g n ) on G, starting at the identity, through the sampling probabilities p n,C = P(g n ∈ C).
We are mostly interested in the case when C has zero density and consider the sampling exponent When H is a subgroup of Γ, the limit defining ρ(H) will exist and be equal to the spectral radius of the random walk on the quotient Schreier graph Sch(Γ, H, S).Indeed, the covering map Γ → Γ/H gives a bijection between walks returning to H on Γ and walks returning to the root on

Sch(Γ, H, S).
There is considerable literature on this notion [40,36,28,7,39], starting with the amenability criterion of Kesten [30,3,4].For arbitrary subsets of Γ, the sampling exponent will not exist in general, but the picture changes, when it is defined by a Γ-invariant stochastic process. 1 Laziness is convenient, but it's not necessary.See Section 3.3 for details.
Theorem 1.1.Let Γ be a countable group and consider an i.i.d.percolation on a Cayley graph of Γ.Then almost surely, for every connected component C of the percolation, the limit exists.
This result is most interesting when the percolation clusters are infinite and there are infinitely many of them a.s.It is easy to see that once ρ(C) exists, it is independent of the starting point of the walk and so, by the indistinguishability theorem of Lyons and Schramm [34,Theorem 3.3], it will be a constant on infinite clusters, depending only on the percolation parameter.Note that when Γ is amenable, the above phase does not exist for i.i.d.percolations, but by Kesten's theorem [30], in this case, ρ(C) equals 1 for any subset C anyways.That is, ρ is not a suitable measuring tool for amenable groups.For non-amenable groups, it is a well-known conjecture that the non-uniqueness phase exists for any Cayley graph of the group [9].
We establish the above theorem in a much wider generality, using the framework of countable measure preserving equivalence relations.Note that our most general results in this direction (see Theorem 3.2, as well as Section 5 for the translation between relations and percolation) do not even involve an ambient group anymore, but for the introduction, we stick to group actions.
Let (X, µ) be a standard Borel probability space and let Γ act on (X, µ) by µ-preserving maps.
We define the orbit relation A subrelation S of R is a Borel subset of R that, as a relation on X, is an equivalence relation.
Let S be a subrelation of R. For x, y ∈ X such that (x, y) ∈ R and for a natural n let the sampling probabilities be p n,x,[y] S = P((g n x, y) ∈ S) where, as before, (g n ) is the lazy random walk on G starting at the identity of Γ.That is, we walk from x and take the probability that we hit the S-class of y.
Theorem 1.2.Let Γ be a countable group acting by measure preserving maps on (X, µ) and let S be a subrelation of the orbit relation of the action.Then for µ-almost every y ∈ X, for every x ∈ Γy ρ S (x, [y] S ) = lim n→∞ p n,x,[y] S 1/n .exists and is independent of x.Moreover, if S is either normal or ergodic, then ρ S is almost surely constant.
Note that one can also state an equivalent, stochastic form of Theorem 1.2 using the notion of an Invariant Random Partition, that is, a random partition of the group that is invariant in distribution under the shift action.Invariant random partitions are defined in [38,Section 8.1].They are natural stochastic generalizations of subgroups, as individually shift invariant partitions are exactly coset partitions with respect to a subgroup.The reformulation says that for any invariant random partition of a countable group, all the partition classes have well defined sampling exponents.We chose to state Theorem 1.2 in the relation language because that is the internal language of its proof.We develop the invariant random partition language in the paper [2].
Theorem 1.2 does not seem to follow from the usual arguments.The local environment may look quite different from different points of a class.In algebraic terms, there is no natural quotient object on which the group would act.This is a major deviance from the subgroup case, where this homogeneity holds and trivially makes p n,H supermultiplicative in n.As a result, we were not able to derive the existence of the sampling exponent with the usual tools, including the standard or Kingman ergodic theorems.For the case of unimodular random trees, Theorem 1.1 follows from an argument we call the 2-3 method.In order to prove Theorem 1.2 we introduce a generalized version of 2-3 method, which we expect to have more applications.We recall the rough idea behind the 2-3 method is to establish a local submultiplicative nature of the sequence and then yield the existence of the limit by a density argument.The reader can form a quick impression on this method by reading the text after the statement of Theorem 1.4.
It turns out that the sampling exponent still admits a spectral interpretation, and equals a norm of a natural Markov-type operator acting on a quotient object of sorts (see Theorem 3.2).Because of that and to keep consistency with the subgroup case, we call the sampling exponent ρ S (x, [y] S ) the co-spectral radius of the class [y] S in the paper.This operator approach also has interesting connections to prior work in percolation theory.For example, our methods recover a lemma due to Schramm on the connectivity decay of random walks in critical percolation which has been used in later results in percolation theory [31], [21,Lemma 6.4], [22,Section 3].We refer the reader to the discussion preceding Section 3.2 for more details.
This Markov-type operator is defined on a Hilbert space which is constructed by integrating the bundle of Hilbert spaces ℓ 2 ([x] R /S) into one Hilbert space.While we will not need it for this work, this Hilbert space can be naturally related to the Jones' basic construction of the inclusion L(S) ≤ L(R) of von Neumann algebras of the corresponding equivalence relations, as well as the L 2 -space of a natural measure space occurring in [15] (see Definition 1.4 of that paper).We expect that our new methods will have applications to the operator algebraic setting.We refer the reader to the discussion preceding Section 3.1.
We now state the 2-3 method theorem in its most general form that leads to Theorem 1.2.
Theorem 1.3.Let R be a discrete, measure-preserving equivalence relation over a standard probability space (X, µ), and fix π ∈ L 1 (X, µ) with π(x) ∈ (0, ∞) for almost every x ∈ X.Let exists and is positive almost surely.Further: In the special case when f k (x, y) is the probability of transition from x to y by a standard random walk in k steps, the first part of the theorem can be seen as a large deviations estimate, in the sense that it controls the density of starting points where the random walk sampling probability deviates from what is suggested by the co-spectral radius.In fact, it is natural to ask whether our main result holds in the "annealed" sense, that is, when we take expected value of the sampling probabilities before we take the n-th root.A warning comment here is that in this case we have to consider the event of returning to the class of the starting point, because equivalence classes often can not be individually identified in a measurable way (this is what is called indistinguishability in percolation theory, which is equivalent to the ergodicity of a subrelation in the measured language).
In any case, this "annealed" version is much simpler to prove than our main result and most of the effort in the paper is spent to establish the pointwise (or "quenched") version.Additionally, we show that the "annealed" version is the essential supremum of the "quenched" version and that in many cases, the "quenched version" is a.s.constant.See Theorem 3.2 for a precise relation between the "annealed" and "quenched" versions.As a sample application, in the case of Bernoulli bond percolation it follows from the indistinguishability result of Lyons-Schramm [34] and our work that the "annealed" version and the "quenched" version agree on infinite clusters.
Remark 1.The following was pointed out to us by the anonymous referee.As in Theorem 1.1 consider an i.i.d percolation on the Cayley graph of a group Γ.Let X n be a random walk on Γ with X 0 = e and with transition probabilities whose support generates Γ.We let P(X 0 ↔ X n ) be the probability that X n is in the connected component of e in this percolation, and we let C be the connected component of e in this percolation.
Theorem 1.3 implies in the context of Theorem 1.1 that almost surely, where C is the cluster of X 0 .The subadditive ergodic theorem shows for, e.g., Bernoulli percolation that almost surely So in order to get the correct almost-sure decay of the connection probability conditioned on the random walk, one must take the limit of the expectation of the logarithm.Our results says that if one instead conditions on the cluster C, then one does not need to take the expectation of the logarithm.In this sense, our result may be thought of as saying that the major contribution to is from contributions which are of "typical size" (up to subexponential factors).
For the expectation E[P(X 0 ↔ X n |X n )] this is not the case, and often rare events contribute substantially to the expectation.
Then A is a self-adjoint operator and log A = lim n→∞ The usual technique used to establish the rate of growth in ergodic theory is the Kingman sub-additive theorem.We weren't able to find any action or equivalence relation with a submultiplicative cocycle that would control the number of walks in G, so we couldn't use it to solve the problem.As w n (o) can be naturally expressed as an inner product, one is also tempted to use spectral theory, but this also did not work for us.Instead we use the mass transport principle to show that the inequalities hold with overwhelming probability, as n gets large.To see why this is useful, imagine that we know that these inequalities hold always and with the implicit constant 1.Then, the sequence w 2 p 3 q (o) is sup-multiplicative, so the limit lim p,q→∞ 1 2 p 3 q log w 2 p 3 q (o) exists.The function n → log w n (o) is Lipschitz and the set 2 p 3 q is dense on the logarithmic scale, so we can deduce that the limit lim n→∞ 1 n log w n (o) also exists.
1.2.Outline of the paper.Section 2 introduces the necessary background for the paper, including a discussion of measure-preserving equivalence relations and how to reduce the study of percolation clusters to equivalence relations.Section 3 contains a proof of the co-spectral radius and its basic properties.In that section we also include some background on representations of equivalence relations, so that in subsection 3.1 we may identify the co-spectral radius with an operator norm.
In Section 3.2 we give a proof of the general 2-3 method which gives us the pointwise existence of the co-spectral radius as a special case.In Section 3.3 we show that the co-spectral radius is almost surely constant if the subrelation is ergodic or normal.In Section 4, we show that the co-spectral radius agrees with the spectral radius when the subrelation is hyperfinite and give a counterexample to the converse (i.e. a Kesten's theorem for subrelations) using monotone couplings of IRS's.In Section 5 we use the co-spectral radius to define new critical exponents for percolation.Finally, in Section 5.2 we use the 2-3 method to establish the existence of walk growth and relate it to an operator norm.
Remark.Note that this paper and parts of [1] and [2] first appeared on arXiv as one long text.
Following explicit suggestions of helpful referees, we decided to separate the work into the three papers, also to make the results more accessible to their natural audiences.
Acknowledgements.Much of the work by the last named author was done on visits to the Rényi institute in Budapest.He would like to thank the Rényi Institute for its hospitality.We would like to thank Gabor Pete for helpful conversations.We thank the anonymous referee for their numerous comments, which greatly improved the paper.

Background and Notation
A standard probability space is pair (X, µ) where X is a standard Borel space, and µ is the completion of a Borel probability measure on X.We say that E ⊆ X is measurable if it is in the domain of µ.An equivalence relation over (X, µ) is a Borel subset R ⊆ X × X so that the relation ∼ on X given by x ∼ y if (x, y) ∈ R is an equivalence relation.For x ∈ X, we let We will continue to use µ for the completion of µ.If R is discrete, we say that it is measurepreserving if the map R → R given by (x, y) → (y, x) is measure-preserving.Equivalently, this just means that the mass-transport principle holds: For a group Γ, and S ⊆ Γ, we use S for the subgroup of Γ generated by S. If Γ is a countable group, and Γ (X, µ) is a measure-preserving action, then R Γ,X = {(x, gx) : g ∈ Γ} is a discrete, measurepreserving equivalence relation.We use the notation S ≤ R to mean that S is a subequivalence relation of R, namely a subset of R which is also an equivalence relation over (X, µ).We often abuse terminology and say that S is a subrelation of R, and leave it as implicit that S should also be an equivalence relation.If (X, µ) and E ⊆ X is measurable we let If E has positive measure, then R| E is a measure-preserving relation over the probability space (E, µ(E∩•) µ(E) ).We let [R] be the group of all bimeasurable bijections φ : X → X so that φ(x) ∈ [x] R for almost every x ∈ X.We identify two elements of [R] if they agree almost everywhere.We have This is a complete, separable, translation-invariant metric on [R] and this turns [R] into a Polish group.We use Prob([R]) for the Borel probability measures on [R].Since [R] is a Polish group, the space Prob([R]) can be made into a semigroup under convolution: so if ν 1 , ν 2 ∈ Prob([R]), then Let G be a graph.We write V (G), E(G) for the vertex and the edge set of G. Let v ∈ V (G) and r ∈ N. The r-ball around v is denoted by B G (v, r) and the r-sphere by S G (v, r).
We use Vinogradov's notation and write f ≪ g if |f | is bounded by a constant times |g|.
For a Banach space V , we let B(V ) be the space of continuous, linear operators T : V → V .For T ∈ B(V ), we set: At various times we will have to appeal to spectral theory of bounded self-adjoint operators on a Hilbert space.Since most standard references on this theory assume Hilbert spaces are complex, in order to make these applications most transparent all Hilbert spaces will be assumed complex throughout the paper.
2.1.Translation between percolation and equivalence relations.Some of our results are stated in the percolation theory language but all our proofs will be based on measured equivalence relations and graphings.An invariant (edge) percolation on a unimodular random graph (G, o) is a random triple (G, o, P ) where P is a subset of edges of G and the distribution of the triple is invariant under the re-rooting equivalence relation.The following proposition associates a p.m.p 2 measured equivalence relations to a percolation is such a way that Theorem 1.1 can deduced from Theorem 1.2.
Proposition 2.1.Let (G, o) be a unimodular random graph with an invariant percolation P .There exists a p.m.p countable equivalence relation (Ω # , ν # , R) with a generating graphing (ϕ i ) i∈I and a sub-relation S ⊂ R such that (1) The rooted graph (G ω , ω) with the vertex set [ω] R and the edge set (2) The law of the pairs (P o , G) where P o is the connected component of the percolation P ⊂ G is the same as the law of ( Proof.We follow closely the construction in [6, Example 9.9].Let Ω be the space of pairs ((G, o), S) where (G, o) is a rooted graph of degree at most d and S is a subset of edges.The distribution of the percolation P is naturally a probability measure on Ω, let us call it µ.Let Ω # be a the set of triples ((G, o), S, λ), where (G, o), S are as before and λ is a two coloring of the vertices of (G, o).Let µ # be the distribution of the random triple ((G, o), P, Λ), where Λ is an i.i.d coloring.The space (Ω # , µ # ) is equipped with a natural finite graphing Φ in which ((G, o), S, λ), ((G ′ , o ′ ), S ′ , λ ′ ) are connected if and only if G = G ′ , S = S ′ , λ = λ ′ and o ′ is a neighbour of o.The graphing Φ spans the re-rooting measured equivalence relation R, which preserves µ # .For each point ω ∈ Ω # , the equivalence class [ω] R is equipped with a bounded degree graph structure G ω .The resulting random rooted graph (G ω , ω) has the same law as (G, o) and the second coordinate has the same law as the percolation P .
To construct the sub-relation S, we select a sub-graphing Φ ′ ⊂ Φ where ((G, o), S, λ), ((G ′ , o ′ ), S ′ , λ ′ ) are connected if and only if G = G ′ , S = S ′ , λ = λ ′ and o ′ is a neighbour of o connected by an edge in S. In this way the connected component of P containing the root is given by the S-equivalence class of w in G w .

Existence of the co-spectral radius for subrelations
Let R be an ergodic probability measure-preserving equivalence relation over a standard probability space (X, µ) and let ν ∈ Prob([R]) be countably supported and symmetric, i.e. ν({φ}) = For n ∈ N, (x, y) ∈ R, we let p ν n,x,y (resp.p ν n,x,S ) be the probability that the random walk corresponding to ν starting at x is at y after n steps (resp. in [x] S after n steps).By direct calculation, If ν is clear from the context (which is the usually the case), we will use p n,x,S , p n,x,x instead of p ν n,x,S , p ν n,x,x .We are interested in the existence of the co-spectral radius of S inside R which is, by definition, the limit In particular, we will show that this limit exists almost surely.While there are easy examples where this limit genuinely depends upon x (see Example 6) we will show that in many cases it is almost surely constant and is the norm of a self-adjoint operator on a Hilbert space naturally associated to S ≤ R.This is, of course, motivated by the case of an inclusion of groups ∆ ≤ Γ.Here the existence of the co-spectral radius, as well as the fact that it is the norm of the corresponding Markov operator on ℓ 2 (Γ/∆) is a nontrivial, but well known, fact.In the case Γ is finitely-generated and ν is the uniform measure on a finite generating set of Γ we are looking at a random walk on a Schreier graph and it is easier to see existence of this limit using the natural action of Γ on Γ/∆.Even in the case ν is not the uniform measure on a finite subset of Γ the action Γ Γ/∆ naturally enters into the very definition of the Markov operator.In the relation case this a priori presents a problem.
Because S is a subrelation of R, for x ∈ X we can divide [x] R into S-equivalence class.We let [x] R /S be the space of S-equivalence classes in [x] R .The field of spaces [x] R /S is analogous to Γ/∆, and so we may consider ℓ 2 ([x] R /S) as analogous to ℓ 2 (Γ/∆).However, there is no obvious natural action of [R] on [x] R /S and this makes it difficult to see how one would define a Markov operator, and thus the co-spectral radius.We proceed to explain how to navigate this difficulty by collecting the field of Hilbert spaces ℓ 2 ([x] R /S) together in a natural object.
Definition 3.1.Let (X, µ) be a standard probability space, then a measurable field of Hilbert spaces over X is a family (H x ) x∈X of separable Hilbert spaces, together with a family Meas(H x ) ⊆ x∈X H x so that: • there is a sequence (ξ ∞, where we identify two elements of Meas(H x ) if they agree outside a set of measure zero.We put an inner product on and this gives ⊕ X H x the structure of a Hilbert space.
We shall typically drop "over X" in "a measurable field of Hilbert space over X" if X is clear from the context.For later use, if (H x ) x is a measurable field of Hilbert spaces, then given A ⊆ X measurable and ξ ∈ Meas(H x ), we let 1 A ξ ∈ Meas(H x ) be defined by In our case, we can give the family (ℓ facts about direct integral imply this collection of measurable vectors satisfy the above axioms (see As mentioned above, there is no obvious natural action of [R] on [x] R /S.However, we do have a natural unitary representation of [R] on L 2 (R/S).Define We will not need it for this paper, but this can be regarded as a representation of R itself (a precise definition will be given in [2]).For our purposes, we simply note that we have natural Markov operators defined on L 2 (R/S).Namely, for a countably supported ν ∈ Prob([R]), we define Here we are mildly abusing notation and using ν(φ) for ν({φ}), this will not present problems since ν is atomic.
Theorem 3.2.Let R be a measure-preserving equivalence with countable orbits over a standard probability space (X, µ), and let ν ∈ Prob([R]) be atomic.Suppose that the support of ν generates R. Fix S ≤ R. exists.Moreover, (ii) The pointwise limit exists almost surely, and (iii) Suppose that the partial one-sided normalizer of S ≤ R acts ergodically (see Definition 3.8 for the definition).Then ρ S ν is almost surely constant, and by (ii) equals ρ(R/S, ν).In particular, this applies if S is normal or ergodic.
We will often drop the ν from ρ S ν if it is clear from context, and simply write ρ S .Let X, µ, R be as in Theorem 3.2.Suppose that y ∈ X and that the limit defining ρ S (y) exists.Given and so the limit exists and equals ρ S (y).If ν is assumed lazy, then and so exists and equals ρ S (x).Thus Theorem 3.2 recovers Theorem 1.2.
Remark 2. Co-spectral radius for normal subrelations also occurs in [10,Lemma 6.7], however in that context the subrelation is both normal and ergodic, which gives a well-defined quotient group.
If the subrelation is normal, there is a quotient groupoid [15] however our situation is general enough (encompassing when the subrelation is ergodic or when it is normal) that we cannot appeal directly to the group case as in [10].We remark that the space L 2 (R/S) is closely related to the relation S that appears in [15, Definition 1.4] (we caution the reader the roles of S, R are reversed in [15] relative to our work and so this relation is denoted R there), which is a measure-preserving Explicitly, We proceed to explain how they are related.Since we not explicitly use the connection between L 2 (R/S) and L 2 ( S), we will only sketch the details.
In [2], we will explain how to give Y the structure of a standard Borel space and equip it with a natural σ-finite measure.This measure will be defined in such a way to make L 2 (Y ) naturally unitarily isomorphic to L 2 (R/S).We also have an isometric embedding given by Part of the significance of the relation S for the results in [15] is that certain properties of the inclusion S ≤ R (e.g. the index, normality) are reflected in terms of properties of the inclusion R ≤ S.
An alternative explanation for this can be given by von Neumann algebras: let L(S), L(R) be the von Neumann algebras of the equivalence relations S, R as defined in [14] (the analogous notation there is M (S), M (R)).We then have a natural inclusion of von Neumann algebras L(S) ≤ L(R).
The von Neumann algebra L( S) can be realized as the basic construction M = L(R), e L(S) , in the sense of Jones [25, Section 3], of L(S) ≤ L(R).For the interested reader, we remark that under this correspondence, the space L 2 (R/S) corresponds to the following subspace of L 2 ( M ) where u φ are the canonical unitaries in L(R) corresponding to the elements of [R] and e L(S) is the Jones projection corresponding to the inclusion L(S) ≤ L(R) (see [15]).Moreover, the action of [R] naturally acts on H by conjugating by u φ , and this action is isomorphic to the action of [R] on L 2 (R/S) we define above.We refer to [25], [11, Appendix F], for the appropriate definitions, which we will not need in this work.
We now proceed to prove (i) of the above Theorem, whose proof is almost entirely operator theory.
3.1.Proof of Theorem 3.2 (i).The essential idea behind Theorem 3.2 (i) is that we have a natural vector in L 2 (R/S) which is given by the measurable field ξ x = δ [x] S .By a direct calculation, we have that λ S (ν) 2n ξ, ξ = p 2n,x,S dµ(x).
We then have to show that Because ν is symmetric, the operator λ S (ν) is self-adjoint and the existence of the limit on the right-hand side follows from the spectral theorem [ Proof.Fix a countable subgroup Γ ≤ [R] so that Γx = [x] R for almost every x.We start by proving the following claim.
for almost every x (the sum above converges since the E φ are disjoint).Since Γ is countable, this proves that x → ξ x ([ψ(x)] S ) is measurable, and this proves the claim.

Having shown the claim, for
for almost every x ∈ X, and also which is a measurable function of x.The lemma now follows from countability of Γ and [37, Lemma IV.8.10].
We use the following well known Lemma (see e.g.[24, Equation 2.8] for a proof), which is the main way we will relate the operator norm of the Markov operator λ S (ν) to the growth of the matrix coefficients λ S (ν) 2n ξ, ξ .Lemma 3.4.Let H be a Hilbert space and T ∈ B(H) self-adjoint.Let K be an index set, and let In order to apply this in the context of a direct integral of Hilbert spaces, the following density criterion will be useful.
Lemma 3.5.Let (X, µ) be a standard probability space and let (H x ) x∈X be a measurable family of Hilbert spaces over X and set H = X H x .Suppose we have a sequence such that Proof.Suppose that η ∈ H and that η, 1 A ξ n = 0 for every n ∈ N and every measurable A ⊆ X.
Then for every measurable A ⊆ X and every n ∈ N we have Since this holds for every A, applying this with A being {x ∈ X : Re(i j η x , ξ n,x ) > 0} for j = 0, 1, 2, 3, and taking real and imaginary parts of the above integral shows that that for every n ∈ N we have that η x , ξ n,x = 0 for almost every x ∈ X.By countability, for almost every x ∈ X we have η x , ξ n,x = 0 for all n ∈ N. Since H x = span{ξ n,x : n ∈ N} for almost every x ∈ X, we deduce that η x = 0 for almost every x ∈ X, i.e. η = 0 as an element of H. Thus we have shown that the only vector in H orthogonal to {1 A ξ n : n ∈ N, A ⊆ X is measurable} is the zero vector, and this implies that Proof of Theorem 3.2 (i).Let ξ ∈ L 2 (R/S) be the measurable vector field given by ξ By the spectral theorem, there is a probability measure η on [− λ S (ν) , λ S (ν) ] so that λ S (ν) 2n ξ, ξ = t 2n dη(t).
From this, we see that lim n→∞ λ S (ν) 2n ξ, ξ 1 2n exists and is the L ∞ -norm of t with respect to η. Combining these results we see that lim n→∞ p 2n,x,S dµ(x) 1/2n exists.Call this limit ρ(R/S, ν) as in the statement of the Theorem.
We now turn to the proof that ρ(R/S, ν) = λ S (ν) .It follows from the logic in the preceding paragraph that ρ(R/S, ν) ≤ λ S (ν) .Let Γ be the subgroup of [R] generated by the support of ν.
Since ν generates R, we have [x] R = Γx for almost every x ∈ X.By Lemma 3.5, Choose k ∈ N so that c = ν * k ({φ}) > 0. Then for every n ∈ N we have that Integrating both sides we obtain Thus, This proves (1), and so completes the proof of Theorem 3.2 (i).
As mentioned in the introduction, Theorem 3.2 and our later work in 4 can be used to recover a result due to Schramm used in [31], [21,Lemma 6.4], [22,Section 3].Indeed, give a percolation of a connected, regular, transitive graph G by Section 2.1 one can build an inclusion S ≤ R of relations on a standard probability space as well as a symmetric probability measure ν on [R] so that in the notation of [21, Lemma 6.4], we have For p ∈ (0, 1), consider Bernoulli(p) edge percolation, where each edge is kept with probability p, let S p ≤ R p be the corresponding inclusion of equivalence relations.Set p c = inf{p : a.e.connected component in Bernoulli(p) percolation is finite}.
In the setup of the proof of Theorem 3.2 (i), we have that Since λ S (ν) is self-adjoint, the spectral theorem tells us that p 2n,x,S dµ(x) is increasing in n and Theorem 3.2 (i) characterizes its supremum as λ S (ν) .Additionally, for every n ∈ N we by Cauchy-Schwarz that p 2n,x,S dµ(x) With minor modifications, our results work for random walks on finite cost graphings (see Example 2 of Section 3.2).In this manner, we can recover the same estimate of Schramm when G is a connected, locally finite, transitive graph.As mentioned in the introduction, in the context of percolation all our proofs can be rephrased without equivalence relations and can be done in the language of percolation theory.
3.2.The 2-3 method and Proof of Theorem 3.2 (ii).We now explain how to deduce the existence of the pointwise limit defining the co-spectral radius.We first state the general Theorem behind this existence and then explain why it applies to our setting, as well as to more general situations.For notation, if f, g : R → [0, ∞] are measurable, then we define their convolution to be the function f * g : R → [0, ∞] given by Given a measurable π : X → C, we say that we just say f is symmetric.
Theorem 3.6.Let R be a discrete, measure-preserving equivalence relation over a standard probability space (X, µ), and fix π ∈ L 1 (X, µ) with π(x) ∈ (0, ∞) for almost every x ∈ X.Let ) for almost every x ∈ X and every k ∈ N we have 0 exists and is positive almost surely.Further: The name 2-3 refers to the way we are proving Theorem 3.6.In the proof we have two separate steps where we show that "typically" y f 2k (x, y) ∼ ( y f k (x, y)) 2 and ( y f 3k (x, y)) ∼ ( y f k (x, y)) 3 .Since 2, 3 generate a multiplicative semi-group which is asymptotically dense on the logarithmic scale, we are able to deduce that the exponential growth of ( y f k (x, y)) has a definite rate.
Before jumping into the proof of Theorem 3.6, let us list several examples where it applies.For all of these examples, fix a discrete, measure-preserving equivalence relation R over a standard probability space (X, µ) and fix a S ≤ R.
It is direct to check that our hypotheses apply in this case with π = 1, D(x) = p 2,x,x .In this case and we recover the existence of the pointwise co-spectral radius ρ S ν .Moreover, item (ii) of Theorem 3.6 as well as Theorem 3.2 (i) imply that So we recover the operator norm of the Markov operator λ S (ν) as the L ∞ -norm of ρ S ν .
Example 2. Suppose that ν : R → [0, 1] is measurable and that y∈[x] R ν(x, y) = 1 for almost every x ∈ X.Moreover, assume that there is a π : X → (0, ∞) with π ∈ L 1 (X, µ) so that ν is π-symmetric.Consider the Markov chain on [x] R with transition probabilities ν(x, y), and for k ∈ N and (x, y) ∈ R, let p k,x,y be the probability that the random walk corresponding to this Markov chain starting at x is at y after k steps.Set and p 2k,x,S = y∈[x] S p 2k,x,y .Note that if f, g : R → [0, ∞] are π-symmetric, then so is f * g.Since So we deduce the existence of 2k,x,S .
A good example to keep in mind is the following.Recall that the full pseudogroup, denoted of R is by definition the set of bimeasurable bijections φ : dom(φ) → ran(φ) satisfying • dom(φ), ran(φ) are measurable subsets of X, we let φ −1 be the element of [[R]] with dom(φ −1 ) = ran(φ), ran(φ −1 ) = dom(φ), and so that whose vertex set is [x] R and whose edge set is φ∈Φ {{y, φ α (y)} : We say that Φ is a graphing if G Φ,x is connected for almost every x ∈ X.We define the cost of a graphing to be This definition is due to Levitt in [32] and the cost of a relation (which is by defining the infimum of the cost of its graphings) was further systematically studied in [19,18].If Φ is a finite cost graphing, then for almost every x ∈ X define ν(x, y) = we have a well-defined co-spectral radius for the simple random walk associated to finite cost graphings.
Another good example is the following.Consider a symmetric ν ∈ Prob([R]), and E ⊆ X a measurable set with µ(E) > 0. Assume that for almost every x ∈ E, we have that y∈[x] R ∩E p x,y > 0 (e.g. this holds if the random walk is lazy). .
As above this defines a Markov chain on [x] R ∩E with transition probabilities given by ν| E .We have k,x,y be the probability that the random walk starting at x with these transition probabilities is at y after k steps.Setting 2k,x,y , we deduce the almost sure existence of the "conditional" co-spectral radius In this case for x ∈ E we have where p 2k,x,S,E is the probability that the random walk corresponding to ν starting at x is in [x] S ∩E after 2k steps.All of our hypotheses apply in this case with X replaced with E, R replaced with R| E , and π = 1.So we recover the existence of the pointwise local co-spectral radius 2k,x,S,E , at least for x ∈ E. See Section 3.3 for more details.In that section we will show more generally that lim k→∞ p 1/2k 2k,x,S,E exists for almost every x ∈ X, see Corollary 3.11.This specific example will be important for us when we show that the spectral radius is almost surely constant in the case that S is either normal or ergodic.
Example 4. Fix a lazy, symmetric ν ∈ Prob([R]), and a measurable E ⊆ X with µ(E) > 0. Define for (x, y) ∈ R: Again, it is direct to check that the hypotheses of Theorem 3.6 apply with π = 1.Note that laziness implies that p 2k,x,S| E ≥ p(x, x) 2k > 0 for every x ∈ E and we always have Thus (c) of Theorem 3.6 holds.So we deduce the existence of the "restricted" co-spectral radius It can be shown by the same method of proof as Theorem 3.2 (i) that lim k→∞ p 2k,x,S| E dµ(x) So we again recover the norm of a corner of the Markov operator as the essential supremum of the restricted co-spectral radius.
Example 5. Let ε > 0 and let η : R → [ε, +∞) be a measurable bounded symmetric function.Define for (x, y) ∈ R: Theorem 3.6 yields almost sure existence of the growth exponent ρ η := lim k→∞ y∈ Let Φ ⊂ R be a symmetric graphing generating R. If we put η = 1 Φ , we get the existence of the growth exponent for the number of length 2k trajectories starting at x in the graph induced by Φ on the equivalence class of [x] R .In particular, for every unimodular random graph (G, o), the number of walks of length 2k starting from o has an exponential rate of growth almost surely.Note that such a result is definitely not true for any rooted bounded degree graph.This last example is discussed in more detail in Section 5.2.
Having explained why Theorem 3.6 implies the existence of the pointwise co-spectral radius, we now turn to the proof of Theorem 3.6.The following is the main technical lemma behind the proof.
and define . Then: (i) for every k ∈ N and almost every x ∈ X, (ii) for every k ∈ N: Note that in order to make sense of φ k , ψ k we are using hypothesis (c).
Proof.(i) By Cauchy-Schwarz, for almost every x ∈ X, where in the last step we use the hypothesis (b).Using Cauchy-Schwarz again: We can estimate the second sum using our hypothesis on f k : with all of the above inequalities and equalities holding for almost every x ∈ X.So we have shown that and rearranging proves the desired inequality.
(ii): By the mass transport principle and π-symmetry: Similarly, We now prove Theorem 3.6.It will be helpful to pass to limits along subsets of N which are "not too sparse" in a multiplicative sense.We say that A ⊆ N is asymptotically dense on the logarithmic We leave it as an exercise to the reader to show that if A ⊆ N is asymptotically dense on the logarithmic scale, and if (a k ) ∞ k=1 is a sequence of nonnegative real numbers for which there is a uniform C > 0 with Proof of Theorem 3.6.Adopt notation as in Lemma 3.7.Set For p, q ∈ N ∪ {0}, k ∈ N we have, by Lemma 3.7 and induction, that where By Lemma 3.7 and the fact that π ∈ L 1 (X, µ), both k k −2 πφ k , k k −2 πψ k converge almost everywhere.Since π(x) > 0 almost surely, we see that for almost every x ∈ X, there is a k 0 (depending upon x) so that for k ≥ k 0 we have φ k (x) ≤ k 2 , ψ k (x) ≤ k 2 .So for almost every x ∈ X, and for all k ≥ k 0 , and all p, q ∈ N ∪ {0}: Fix k 1 ≥ k 0 .Since A = {2 p 3 q k 1 : p, q ∈ N} is asymptotically dense on the logarithmic scale, we have by (2) and hypothesis (d): Letting k 1 → ∞, we see that f ≥ f almost everywhere, and this proves that f exists almost everywhere.Note that (3) applied with (i): By (2), we have for p, k ∈ N and almost every x ∈ X: where in the last step we use the arithmetic-geometric mean inequality.Lemma 3.7 implies letting p → ∞ and applying Fatou's Lemma proves (i).
(ii): Part (i) shows that We now prove the reverse inequality.Fatou's Lemma implies that for all r ∈ N: where in the last step we use Holder's inequality for k > r. .
Letting r → ∞ and using that 0 < π(x) for a.e.x and that π ∈ L 1 (X, µ) shows that Normal subrelations, the proof of Theorem 3.2 (iii).In this subsection, we prove that the co-spectral radius ρ S (x) is almost surely constant when S is either ergodic normal (the case when S is ergodic is fairly direct, see Corollary 3.11 (a)).We prove a common generalization of the cases where S is normal or ergodic, for which we need partial one-sided normalizers.
Definition 3.8.Let (X, µ) be a standard probability space and S ≤ R discrete, measure-preserving equivalence relations on (X, µ).We define the partial one-sided normalizers to be the set of φ ∈ [[R]] so that for almost every x ∈ dom(φ) we have φ([x] S ∩ dom(φ)) ⊆ [φ(x)] S .We use P N R (S) for the set of partial one-sided normalizers of S inside of R.
One example to keep in mind for intuition is the following: suppose that R is the orbit equivalence relation of a measure-preserving action of G on (X, µ).Suppose that H ≤ G, and let S = {(x, ax) :

R (S).
More generally, φ g E ∈ P N (1) R (S) for every g ∈ G in the normalizer of H.We can thus think of the partial one-sided normalizers as a generalization of normalizers to the setting of the full pseudogroup.
Another source of elements in the partial normalizer is the following.Recall that the set of endomorphisms of R over S, denoted End R (S) are the measurable functions φ : X → X so that for almost every x ∈ X the following two conditions hold: Such functions need not be injective modulo null sets, and when they fail to be injective modulo null sets are not measure-preserving.Indeed, one can use the mass-transport principle to show that φ ∈ End R (S) we have that (we will not use this fact so will not give a full proof of it).However, as we will see shortly (see Lemma 3.10) for each φ ∈ End R (S) we may find a partition modulo null sets (E i ) i∈I of X into countably many sets so that φ| E i is injective for each i ∈ I.
This last example in fact provides the motivation for our consideration of the one-sided partial normalizers, as it is one way to define normal subrelations.
Definition 3.9 (See Theorem 2.2 of FSZ).Let R be a discrete, measure-preserving equivalence relation on a standard probability space (X, µ).A subrelation From the above definition, one can imagine attempting to prove that the co-spectral is almost surely constant for a normal subrelation by trying to understand how it varies after applying endomorphisms of S ≤ R.However, as alluded to above, endomorphisms can have undesirable properties (namely lack of injectivity and failure to be measure-preserving) and the possibility of these undesirability properties make them ill-suited for our proofs.A good tradeoff for our purposes is to drop being everywhere defined and gain being measure-preserving and injective.This is precisely the purpose of the one-sided partial normalizers.
We have a natural way for elements of P N (1)

This allows us to say what it means for P N
(1) R (S) to act ergodically.It simply means that if f ∈ L p (X, µ) and α φ (f ) = 1 ran(φ) f for every φ ∈ P N (1) R (S), then f is essentially constant.We leave it as an exercise to the reader to follow the usual arguments to verify that this is equivalent to saying there exists a countable C ⊆ P N (1) R (S) so that if a measurable subset of X is invariant under C, then its measure is either 0 or 1.Note that P N for almost every x ∈ X.
Proof.By [15, Theorem 2.2], we may find a countable D ⊆ End R (S) so that for almost every x ∈ X. Fix a countable subgroup H ⊆ [S] so that [x] S = Hx for almost every x ∈ X.Then for almost every x ∈ X we have that Fix a countable subgroup G ⊆ [R] so that [x] R = Gx for almost every x ∈ X.Then for each φ ∈ D we may write, up to sets of measure zero,

By the preceding lemma and the inclusion [S] ⊆ P N
(1) R (S) the condition that P N (1) R (S) act ergodically on (X, µ) is satisfied if either S is ergodic or S is normal and R is ergodic.We will show that ρ S is almost surely invariant under the partial one-sided normalizers, and is thus essentially constant when P N R (S) are only partially defined, we will need to relate the co-spectral to the modification given by Example 3 by restricting to a subset.It will not be hard to show from Theorem 3.6 that ρ S E (x) as defined in Example 3 exists for almost every x ∈ X.To deduce the existence of ρ S E (x) for almost every x ∈ X, as well as some of its basic properties, we need to recall the S-saturation of a measurable set.Given a discrete, measure-preserving equivalence relation S on a standard probability space and a measurable E ⊆ X, the S-saturation of E is the (unique up to measure zero sets) measurable subset E ⊆ X satisfying the following properties: is a countable subgroup which generates S, then a model for E can be given by It will also be helpful to relate ρ S E to a quantity more operator-theoretic in nature.For a measurable set E ⊆ X of positive measure, and x ∈ X we let We also let so it follows from the spectral theorem that the limit defining ρ E (R/S, ν) exists.We now proceed to show that lim k→∞ p 1 2k 2k,x,S,E exists.
Corollary 3.11.Let (X, µ) be a standard probability space, and let S ≤ R be discrete, probability measure-preserving equivalence relations defined over X.Let ν ∈ Prob([R]) be symmetric, countably supported, and generate R. Fix a measurable E ⊆ X with positive measure.
(a) For almost every x ∈ X we have that exists and is almost surely S-invariant.
Proof.Note that the sequence of functions satisfies the hypothesis of Theorem 3.6 with R replaced by R| E .This proves that ρ S E (x) exists for almost every x ∈ E. Since Theorem 3.6 also proves (b).
To prove that ρ S E (x) exists for almost every x, let E be the S-saturation of E. Note that ρ S E (x) = 0 for almost every x ∈ E c .For almost every x ∈ E, we have that ρ S E (y) exists for all y ∈ [x] S ∩ E. Fix such an x ∈ E, and let y ∈ [x] S ∩ E. Then there is an ℓ ∈ N with p ℓ,x,y > 0. So for all k ≥ 2ℓ, This shows that for almost every x ∈ E we have lim inf The proof that lim sup is similar.
It will be helpful to study how ρ E (R/S, ν) varies as a function of E. We can embed measurable subsets (modulo null sets) of (X, µ) into L 1 (X, µ) via identifying each set with its indicator function.
We thus have a natural distance on measurable sets modulo null sets by We show that ρ E (R/S, ν) is semicontinuous as function of E. For later use, it will also be useful to know that ρ E (R/S, ν) as a function of S. We now define a topology on subrelations of R making this precise.
For R a discrete, probability measure-preserving equivalence relation on (X, µ), φ ∈ [[R]], and For a given subrelation S ≤ R, a finite set Ω ⊆ [[R]], and ε > 0, let O Ω,ε (S) consist of all subrelations We define a topology on subrelations of R (modulo null sets) by declaring that the family O Ω,ε (S) form a neighborhood basis of S. Suppose that I is countable, and that Φ = In this case, given (α i ) i∈I ∈ (0, 1] I with i α i = 1, we can define a metric on subrelations of R by It can be check that this metric induces the topology defined above (this metric is moreover complete, though we will not need this).We now prove our promised semicontinuity.
Lemma 3.12.Let (X, µ) be a standard probability space and R a discrete, measure-preserving equivalence relation define over X.Let ν ∈ Prob([R]) be symmetric, countably supported, and generate R. The map (E, S) → ρ E (R/S, ν) is lower semicontinuous (where the domain is all pairs of positive measure, measurable subsets E of X and subrelations S of R).
and the spectral theorem we have .
It thus suffices to prove that ] be defined by dom(id E ) = E and id E (x) = x for every x ∈ E. Then E p 2k,x,S,E dµ(x) can be rewritten as Note that each term in this sum is a continuous as a function of (E, S).Since φ∈supp(ν * 2k ) ν * 2k (φ) = 1, it follows that the sum converges uniformly.Thus is continuous.
For technical reasons, in order to show that ρ S is invariant under the partial normalizer of S inside of R, it will be helpful to reduce to the case that ν is lazy.We will briefly need to adopt notation for how the co-spectral radius depends upon the measure.So for S ≤ R, X as in Theorem 3.14 and a countably supported ν ∈ Prob([R]), we use ρ S ν (x) for the co-spectral radius at x defined using ν.Similarly, for n ∈ N we use p ν n,x,S for the probability that the random walk starting at x associated to ν is in [x] S after n steps.
It will be helpful to use the following lemma which shows that the local co-spectral radius associated to ν is uniformly close to the local co-spectral radius associated to (1 − t)ν + tδ id as t → 0. Lemma 3.13.Let (X, µ) be a standard probability space, and S ≤ R discrete, probability measurepreserving equivalence relations defined over X. for every measurable set E ⊆ X of positive measure, we have that ρ E (R/S, ν t ), ρ E (R/S, ν) are the L ∞ -norms of s → s, s → (1 − t)s + t with respect to η.In particular, for all measurable E ⊆ X of positive measure To prove the lemma, fix c > 0 and let Note that E is S-invariant, by S-invariance of ρ S ν and ρ S νt .Assume, for the sake of contradiction, that µ(E) > 0. Then by the preceding paragraph we have On the other hand, the definition of E forces Since c > 1, we obtain a contradiction.Thus almost everywhere.A similar proof shows that ρ S ν − ρ S νt ≤ ct(1 + ρ) almost everywhere.Thus for all c > 1, and the proof is completed by letting c → 1.
The above reduction to the lazy case and the semicontinuity in Lemma 3.12 allow us to give a general formula for the local co-spectral radius in terms of the co-spectral radius.We will ultimately use to prove invariance under partial normalizers of S ≤ R by restrict to sets where we have uniform lower bound on transition probabilities p k,x,φ(x) for φ ∈ P N R (S).
Theorem 3.14.Let (X, µ) be a standard probability space, and let S ≤ R be discrete, probability measure-preserving equivalence relations defined over X, and fix a symmetric and countably supported ν ∈ Prob([R]).If E ⊆ X has positive measure, then where E is the S-saturation of E.
Proof.By Lemma 3.13, we may assume that ν is lazy.Observe that by S-invariance of E, we have that ρ S 1 E = ρ S E almost everywhere.Hence, it is enough to show that ρ S E = ρ S E almost everywhere.
Clearly ρ S E ≤ ρ S E so it suffices to show that reverse inequality holds almost everywhere.For k ∈ N, Then E k is an increasing sequence of sets with E k ⊇ E. By symmetry and laziness of ν we know that k E k = E up to sets of measure zero.By Lemma 3.12, it suffices to show that for almost every k ∈ N we have that ρ S E k ≤ ρ S E almost surely.For m ∈ N, set E k,m = {x ∈ X : p 2k,x,S,E > 1/m}.Then E k,m are increasing and Taking 2l th roots of both sides and letting l → ∞ shows that Since ρ S E , ρ S E k,m exist almost everywhere, this completes the proof.
We now have amassed enough results to show that the co-spectral radius does not increase after applying partial normalizing elements.
Proposition 3.15.Let (X, µ) be a standard probability space, and let S ≤ R be discrete, probability measure-preserving equivalence relations defined over X, and fix a symmetric countably supported ) which generates R. Then for every φ ∈ P N R (S) we have for almost every x ∈ dom(φ).
Proof.First assume that ν is lazy.Fix φ ∈ P N R (S), replacing X with a conull set we may assume that φ([x] S ∩ dom(φ)) ⊆ [φ(x)] S for every x ∈ X.Since ν is lazy and generating, for almost every x ∈ dom(φ) we have that p k,x,φ(x) > 0 for all large k.So, up to sets of measure zero, Since ν is lazy, this union is increasing and thus it follows that we may find a sequence δ n → 0 of positive numbers and an increasing sequence of integers r n so that Set F n = {x ∈ dom(φ) : p rn,x,φ(x) > δ n }, and E n = φ(F n ).By Borel-Cantelli, for almost every x ∈ dom(φ) we have that x ∈ F k for all sufficiently large k.By Theorem 3.14, for almost every x ∈ dom(φ) the follow conditions are satisfied for all sufficiently large k: Fix an x ∈ dom(φ) which satisfies the above three conditions, and fix k such that the above three bulleted items hold.Then for all l ∈ N, Since φ([x] S ∩ dom(φ)) ⊆ [φ(x)] S , and φ(x) ∈ E k , x ∈ F k , we have for all l ≥ r k : where in the second to last step we use symmetry and the fact that x ∈ F k .Thus: almost everywhere.The general case follows from the lazy case by using Lemma 3.13.
We are now ready to prove that the co-spectral radius does not change under the partial one-sided normalizers.
Corollary 3.16.Let (X, µ) be a standard probability space, and let S ≤ R be discrete, probability measure-preserving equivalence relations defined over X, and fix a symmetric and countably supported ν ∈ Prob([R]) which generates R. Suppose that P N R (S) acts ergodically on (X, µ).Then ρ S is almost surely constant (in particular, by Lemma 3.10 this applies if S is normal in R and R is ergodic).
Proof.Let C be as in Lemma 3.10.By the preceding proposition and countability, for every t ∈ [0, 1] E t = {x ∈ X : ρ S (x) ≥ t} is invariant under C, and so by ergodicity of R has measure 0 or 1 for every t ∈ [0, 1].If s = sup{t : µ(E t ) = 1}, then ρ S ≥ s almost everywhere, and ρ S (x) < s + 1 n for almost every x and every n ∈ N. Thus ρ S (x) = s for almost every x.
We close this section with an example illustrating the fact that ρ S (x) may fail to be essentially constant if we only assume that R is ergodic.Thus we need to assume something special about the inclusion S ≤ R. Example 6.Let R be an ergodic, discrete, measure-preserving equivalence relation on (X, µ).Let E ⊆ X be a measurable set of positive measure.Let We claim the following holds.Claim: (i) for almost every x ∈ E c , we have ρ S (x) = ρ(R, ν), (ii) for almost every x ∈ E, we have ρ S (x) = 1.
In particular, if R is not hyperfinite, then by [10, Lemma 2.2.] there is a ν ∈ Prob([R]) so that ρ(R, ν) < 1 (this also follows from condition (GM) in [26] being equivalent to hyperfiniteness) and this gives an example where the co-spectral radius is not essentially constant.
(i): For almost every x ∈ E c we have ρ S (x) = ρ T (x), so this follows by the preceding paragraph.
(ii): For almost every x ∈ E we have ρ S (x) = ρ R E (x), with notation as in Corollary 3.11.So this follows from Theorem 3.14.
Since S is hyperfinite, we can write S = n S n where S n ≤ R is an increasing sequence and S n is an equivalence relation where almost every equivalence class is finite.Note that S n converges to S in the topology defined by Lemma 3.12.Thus by Lemma 3.12, it is enough to show that ρ(R/S n , ν) ≤ ρ(R, ν) for every n ∈ N.For an integer ℓ, let E ℓ = {x ∈ X : |[x] Sn | ≤ ℓ}.Then E k are an increasing sequence of measurable subsets whose union is conull.By Theorem 3.2, we know that ρ(R/S n , ν) = ρ Sn ∞ .So it suffices to show that ρ Sn 1 E ℓ ∞ ≤ ρ(R, ν) for all ℓ, n.
Fix integer n, ℓ ∈ N. We may then choose φ Then: . For each j ∈ {0, • • • , ℓ} such that x ∈ dom(φ j ), we may choose a non-negative integer t j so that p t j ,x,φ j (x) > 0. Thus p 2k,x,φ j (x) ≤ p 2(k+t j ),x,x p −2 t j ,x,φ j (x) . Thus for almost every x.Hence In the group context, the analogous statement is that if Γ is a countable, discrete group and ν ∈ Prob(Γ) is symmetric with supp(ν) = Γ, then for any amenable H ≤ Γ we have that ρ(Γ, ν) = ρ(Γ/H, ν).It is known that the converse fails, namely there are cases of Γ, ν and nonamenable H ≤ Γ so that ρ(Γ, ν) = ρ(Γ/H, ν).It is a theorem of Kesten [30,Theorem 2] that the converse is true if we assume in addition that H is normal.This was generalized to invariant random subgroups by Abért-Glasner-Virág (see [3]).Normal subgroups and IRS's can both be realized as a special case of normal subrelations.So it is natural to ask if R is an ergodic, probability measure-preserving, discrete relation and if ν ∈ Prob([R]) is symmetric and generating, and if S ⊳ R has ρ(R, ν) = ρ(R/S, ν), do we necessarily have that S is hyperfinite?We will show that the answer is "no" in general, but it might be helpful first to study a special case.
In general, a normal subrelation S ⊳ R can be expressed in terms of the partial one-sided normalizers of S generating R (recall our discussion for Section 3. We will often use probabilistic language and think of the Sub(Γ)-valued random variables K, H with distribution η 1 , η 2 as the coupled IRS's.Thus we will often say "let K ≤ H be a monotone joining of IRS's H, K".
In our context, given S ≤ R and Γ ≤ N R (S) we get a monotone joining of IRS's by considering the pushforward of µ under the map Γ → Sub(Γ) × Sub(Γ) given by x → (Stab Γ (x), H x ).We can also reverse this construction.Recall that Γ = F 2 ⋊ N with γ ∈ F 2 acting on N by γ((g λ ) λ )γ −1 = (g λ ) γλ .For any subset Σ ⊂ F 2 let N Σ be the subgroup λ∈Σ G.By construction, N Σ is a normal subgroup of N and for every γ ∈ F 2 we have γN Σ γ −1 = N γΣ .This defines a Γ-equivariant map A percolation on F 2 is random subset of F 2 with distribution invariant under the left translations.
For any percolation P ∈ {0, 1} F 2 the group N P is an invariant random subgroup of Γ.Let p < q ∈ (0, 1) and let (P, Q) ∈ {0, 1} F 2 × {0, 1} F 2 be a Γ-invariant coupling of two Bernoulli percolations of parameters p and q respectively such that P ⊂ Q a.s.This can be arranged by first choosing P as Bernoulli percolation of parameter p and then declaring Q to be the union of P and an independent copy of a Bernoulli percolation with parameter q−p 1−p .In this way we construct an invariant random couple of subgroups N P ⊂ N Q .
The set Q \ P is infinite a.s.so the quotient N Q /N P is the direct sum of infinitely many copies of G.In particular N P is almost surely not co-amenable in N Q .Let S be a finite generating set for Γ.We claim that ρ(Γ/N P , S) = ρ(Γ/N Q , S) = ρ(Γ/N, S).This will follow quite quickly from the semi-continuity properties of the co-spectral radius.Since P is a Bernoulli percolation, for any n ∈ N we will almost surely find γ n ∈ F 2 such that γP contains the R-ball around the identity.The sequence of subgroups N γnP converges to N in Sub(Γ).On the other hand ρ(Γ/N γnP , S) = ρ(Γ/N P , S), because N γnP = γ n N P γ −1 n .By Lemma 4.6 we conclude that ρ(Γ/N P , S) ≥ ρ(Γ/N, S).The reverse inequality is clear so ρ(Γ/N P , S) = ρ(Γ/N, S).In the same way we show that ρ(Γ/N Q , S) = ρ(Γ/N, S).Choose R > 0 such that the support of f is contained in B Λ∞ (R).For big enough n we will have an isomorphism between labeled graphs ι n : B Λn (R + 1) ≃ B Λ∞ (R + 1).Let f n = f • ι n be the pullback of f to ℓ 2 (Γ/Λ n ).Then, We deduce that lim inf n→∞ ρ(Γ/Λ n , S) ≥ ρ(Γ/Λ ∞ , S) − ε.We finish the proof by taking ε → 0.

Co-spectral radius and percolation
Let (G, o) be a transitive graph with the root at o (resp.unimodular random graph).A invariant percolation P is a random subset of edges of G, such that the distribution of P is invariant under graph automorphisms (resp.invariant under the re-rooting equivalence relation).but the definitions below can be easily adapted to all ergodic unimodular random graphs.
• p u is defined as the infimum of p ∈ [0, 1] such that B p has a unique infinite connected component a.s.
• p c is defined as the infimum of p ∈ [0, 1] such that B p has an infinite connected component a.s.
• For x, y ∈ V (G) let τ p (x, y) denote the probability that x and y are connected in B p .The exponent p exp is the supremum of all p such that τ p (x, y) decays exponentially in d(x, y).
• Consider the operator T p acting on compactly functions on V (G) defined by τ p (x, y)φ(y).
The exponent p [2→2] is the supremum of p such that T p defines a bounded operator ℓ 2 (V (G)) → They implicitly depend on the random walk, we always choose the standard one.
(1) Let p Ram (for Ramanujan) be the supremum of p such that ρ Bp = ρ G .
(2) Let p cK (for co-Kaimanovich) be the infimum of p such that ρ Bp = 1.
The reason for the term "co-Kaimanovich" is as follows.For spectral radius (as opposed to co-spectral radius) of relations R, having ρ(R, ν) = 1 for some countably supported ν ∈ Prob([R]) whose support generates R is not equivalent to hyperfiniteness of R as discussed in detail in work

2020
Mathematics Subject Classification.60K35; 37A30, 37A20, 46N30, 60G10.M. Abert acknowledges support from the KKP 139502 project, the ERC Consolidator Grant 648017 and the Lendulet Groups and Graphs research group.B. Hayes gratefully acknowledges support from the NSF grants DMS-1600802, DMS-1827376, and DMS-2000105.M. Fraczyk was partly supported by the Dioscuri Programme initiated by the Max Planck Society, jointly managed with the National Science Centre (Poland), and mutually funded by the Polish Ministry of Science and Higher Education and the German Federal Ministry of Education and Research.
ξ1/2n , where in the last step we use self-adjointness of λ S (ν).The construction of the inclusion S ≤ R forces that the spectral radius of G is equal to the co-spectral radius of T ≤ R where T = {(x, x) : x ∈ X} is the trivial relation.For Bernoulli(p) percolation with p < p c , finiteness of the clusters tells us that [x] Sp is finite for almost every x ∈ X.We will later show (see Proposition 4.1) that this implies that ρ(R/S p , ν) = λ Sp (ν) ≤ λ T (ν) = ρ(R, ν).Using the standard monotone coupling of Bernoulli percolation[33, Section 5.2], it is direct to see that λ Sp (ν) is semicontinuous and that λ Sp c (ν) ≤ λ T (ν) .Thus we obtain the estimate of Schramm for connected, transitive graphs.

( 1 )
R (S) automatically acts ergodically ifS is ergodic, because [S] ⊆ P N (1)R (S).The following lemma shows that P N(1)R (S) also acts ergodically if S is normal and R is ergodic.Lemma 3.10.Suppose that S is normal in R. Then there is a countable set C ⊆ P N(1)

Theorem 5 . 1 . 2 . 5 . 1 .
Let P be an invariant bond percolation on a unimodular random graph (G, o) of degree at most d.Let C be the connected component of o in the percolation P .Let X n be the standard random walk on G starting at o.The limit ρ P := lim n→∞ P(X n ∈ C) 1/2n exists almost surely.Proof.The fist step is to use Proposition 2.1 to construct an associated p.m.p equivalence relation with a graphing and a sub-relation that will allow us to apply Theorem 1.2.We now borrow the notation from the section 3 and Proposition 2.1.Let (G, o), P be an instance of the percolation and let x = ((G, o), P, λ) ∈ Ω # with an i.i.d.random coloring λ.The graph (G x , x) is isomorphic to (G, o) almost surely.Since C = [x] S almost surely, the probability of returning to the connected component of P containing the root o at time n is the same as p ν n,x,S .We have lim n→∞ P(X n ∈ C) 1/2n = lim n→∞ p ν 2n,x,S 1/2n .The limit on the right hand side exists for µ # almost all x, by virtue of Theorem 1.Critical values from spectral radius.Using Theorem 5.1, we can define two new critical values of the Bernoulli bond percolation that fit nicely with the existing classical exponents like p c , p u , p [2→2] and p exp .We recall their definitions below.The B p denotes the Bernoulli bond percolation with parameter p ∈ [0, 1].For simplicity we restrict to transitive rooted graph (G, o),

of Kaimanovich [ 27 ]
. This work of Kaimanovich greatly clarified a common misconception in the literature, and gave precise and tractable criteria to verify when a relation is hyperfinite in terms of the data of several spectral radii.E.g. this work shows that spectral radius 1 is equivalent to having ρ(R, ν) = 1 for every countably supported ν ∈ Prob([R]) whose support generates R. It is thus reasonable to call a relation R Kaimanovich if there is a ν ∈ Prob([R]) whose support generates R and has ρ(R, ν) = 1.Since co-spectral radius heuristically plays the role of spectral radius in a quotient, this motivates the term co-Kaimanovich.We now compare how these critical values are related.Proposition 5.3.(1)p Ram ≤ p cK ,(2) p cK ≤ p u (3) p exp ≤ p cK , (4) p [2→2] ≤ p Ram ,We remark that (4) is equivalent to a theorem of Hutchcroft[21, Proposition 6.4], but as our proof is short we include the proof for completeness.Proof.(1) If G is amenable, then 1 ≥ ρ Bp ≥ ρ G ≥ 1.If G is non amenable, then ρ G < 1.The inequality follows now from the obvious monotonicity of ρ Bp in p.

( 2 )
Let p > p u .Let S ≤ R be the inclusion of equivalence relations constructed in Proposition 2.1.The infinite connected component of B p in the percolated graph corresponds to an S-invariant 12, Theorems IX.2.2 and IX.2.3].The limit on the right-hand side is also dominated by λ S (ν) .We prove a general Hilbert space theorem which, after a small amount of work, will prove the reverse inequality.First, let us alleviate any concerns about the measurable structure defined on (ℓ 2([x] R /S)) x∈X .Lemma 3.3.Let S ≤ R be discrete measure-preserving equivalence relations over a standard probability space (X, µ).Define a family Meas(ℓ 2 m , so again by Lemma 3.12, it suffices to show that ρ S E k,m ≤ ρ S E almost surely.Fix an x ∈ E k,m so that both ρ S E (x), ρ S E k,m (x) exist.Then for every l ∈ N: [38]rem 4.4 (Theorem 8.15 of[38]).Let ζ ∈ Prob(Sub(Γ) × Sub(Γ)) be a monotone joining of IRS's η 1 , η 2 .Then there is a standard probability space (X, µ), a probability measure-preserving action Γ (X, µ) and a normal subrelation S of the orbit equivalence relation of Γ (X, µ) with the following property.We have that Γ ≤ N R (S) and if we setH x = {g ∈ Γ : (gx, x) ∈ S},and define Θ :X → Sub(Γ) × Sub(Γ) by Θ(x) = (Stab Γ (x), H x ), then ζ = (Θ) * (µ).So by Proposition 4.2 our question on co-spectral radii translates to the following.Suppose that K ≤ H is a monotone joining of IRS's, under what conditions do we have that ρ(Γ/H, ν) = ρ(Γ/K, ν) almost surely?Let us first that this is always true when K is coamenable in H.Proposition 4.5.Suppose that K ≤ H ≤ Γ are countable, discrete groups and that ν ∈ Prob(Γ).If K is coamenable in H, then ρ(Γ/H, ν) = ρ(Γ/K, ν).Proof.To say that K is coamenable in H means that the trivial representation of H is weakly contained in the quasi-regular representation of H on ℓ 2 (H/K).By induction of representations, it follows that the quasi-regular representation of Γ on ℓ 2 (Γ/H) is weakly contained in the quasi- are the quasi-regular representations.The reverse inequality is direct to argue, so this completes the proof.Notice that if Γ, H x , Stab(x) are as in Theorem 4.4, then having Stab(x) be coamenable in H x does not guarantee that S is hyperfinite.So from Theorem 4.4 and Proposition 4.5, we get not co-amenable, and yet still ρ(Γ/K, ν) = ρ(Γ/H, ν) < 1 for every finitely supported ν ∈ Prob(Γ) whose support generates Γ. Example 7. Counterexample to the converse of Proposition 4.5.Let G be a non-amenable finitely generated group and let Γ = F 2 ≀ G be the wreath product of F 2 with G. Let N := F 2 G.
an example of an equivalent relation R, a ν ∈ Prob([R]) and a normal S ⊳ R, so that S is not hyperfinite, and yet ρ(R, ν) = ρ(R/S, ν).So a naive generalization of Kesten's theorem does not hold in this context.In fact, there is even a monotone joining of IRS's K ≤ H of Γ so that K ≤ H is almost surely