A note on the local weak limit of a sequence of expander graphs

We show that the local weak limit of a sequence of finite expander graphs with uniformly bounded degree is an ergodic (or extremal) unimodular random graph. In particular, the critical probability of percolation of the limiting random graph is constant almost surely. As a corollary, we obtain an improvement to a theorem by Benjamini-Nachmias-Peres (2011) on locality of percolation probability for finite expander graphs with uniformly bounded degree where we can drop the assumption that the limit is a single rooted graph.


Introduction
Local weak convergence of a sequence of finite expander graphs has been studied in relation to locality of critical probability of percolation in [4]. Locality of critical probability for Bernoulli bond percolation on infinite graphs is in itself a very significant and well studied problem, beginning with Schramm's locality conjecture for transitive graphs; see [8] and [6] and the recent paper [7] and the references therein for some recent developments in this direction for infinite graphs. For finite graphs, such locality was shown to hold when the graphs were expanders and the sequence of graphs converged locally weakly to a fixed infinite rooted graph in [4]. Their argument, in turn, was an extension of the arguments used in [2], who studied critical probability for the emergence of a giant component (that is a connected linear sized component) in finite expander graphs under the assumptions of regularity and high-girth. In [3] it was shown that in any expander, every giant component of given proportion emerges in an interval of length o(1), that is, for any c ∈ (0, 1), the property that the random subgraph of an expander G = (V, E) after Bernoulli bond percolation contains a giant component of size c|V |, has a sharp threshold; removing the regularity and highgirth assumptions in [2].
We first recall local weak convergence of bounded degree finite graphs. A rooted graph is denoted Let G denote the space of isomorphism classes of all rooted connected graphs with degrees bounded by ∆, for some fixed ∆ > 0. For r = 1, 2, . . . and a graph G, let B G (o, r) denote the closed ball of radius r around the vertex o in the graph G. Define the distance between two isomorphic classes of rooted graphs (G 1 , o 1 ) and is a compact and separable metric space.
For each n ≥ 1, let G n be a finite graph and let U n be a uniformly chosen random vertex in G n and (G, ρ) be an infinite random rooted graph in G, that is, (G, ρ) is a sample from a Borel probability measure µ on G. We say that the sequence of finite graphs {G n } converges locally weakly to (G, ρ) (or to µ) if for every R > 0 and for every finite rooted graph (H, ρ ′ ), This is equivalent to saying that µ is the weak limit of the laws of (G n , U n ). This is a special case of the graph limits defined in [5]. Such measures µ satisfy a "spatial stationarity" property called unimodularity, which roughly says that if mass is redistributed in the graph, then the expected mass that leaves the root is equal to the expected mass the arrives at the root, see Definition 2.1 of [1] for an exact definition.
Also, we shall denote by C((G, o), 1 1+s ) the 1 1+s neighborhood of the graph (G, o) in the space (G, D), that is, the set of all rooted graphs whose s-balls are isomorphic to (B G (o, s), o).
Let I denote the σ-field of events in the Borel σ-field of G that are invariant under non-rooted isomorphisms. The class U of unimodular probability measures on G is convex. An element of U is called extremal if it cannot be written as a convex combination of other elements of U . It follows from Theorem 4.7 of [1] that a unimodular probability measure µ on G is extremal if and only if I is µ-trivial, that is, I contains only sets of µ-measure 0 or 1 (ergodicity).  In this short paper, we first show that when a sequence of finite expander graphs converges locally weakly to a random graph G, then G is an ergodic (or extremal) unimodular random graph. Any measurable function f : G → R is called rerooting-invariant if it is invariant under changes in the position of the root (that is, for any graph τ and any two vertices v, v ′ ∈ τ , f ((τ, v)) = f ((τ, v ′ )); we sometimes suppress the root in the notation and simply denote it as f (τ )). We have the following theorem.

For two sets of vertices
Theorem 1.1. Let G n be a sequence of finite graphs with a uniform Cheeger constant lower bound c > 0 and a uniform degree bound ∆ > 0, such that G n → (G, ρ) locally weakly, where (G, ρ) is a random infinite rooted graph. Then G is an ergodic (or extremal) unimodular random graph. That is, if f is any rerooting-invariant function, then f (G) is constant almost surely. In particular, p c (G) is constant almost surely.
As a corollary to the above theorem, we get the following improvement to Theorem 1.3 of [4]. Corollary 1.2. Let G n be a sequence of finite expander graphs with uniformly bounded degree as above, such that G n converges locally weakly to an infinite random rooted graph (G, ρ). If p < p c (G) (this is well defined as p c (G) is constant by Theorem 1.1), then for any constant α > 0 we have P(G n (p) contains a component of size at least α|G n |) → 0 as n → ∞, and if p > p c (G), then there exists some α = α(p) > 0 such that P(G n (p) contains a component of size at least α|G n |) → 1 as n → ∞ .

Proofs
We first prove Theorem 1.1.
Moreover, as is an open cover of H 1 and H 1 is compact, it has a finite subcover, and let R = max i R (τ 1,i ,r 1,i ) be the maximum as {(τ 1,i , r 1,i )} range over this finite subcover. Then for all (τ i , r i ) ∈ H i , Define for i = 1, 2 and G n = (V n , E n ), Choose and fix n large enough such that |A i,n | ≥ np 0 4 and let A i = A i,n . Now, recall the following theorem due to Menger (1927). Let G = (V, E) be a graph and A, B ⊆ V . Then the maximum number of edge disjoint A − to − B paths equals the minimum size of an A − to − B separating edge cut.
Let L denote the maximum number of edge disjoint paths between A 1 and A 2 , and E(S, S c ) denote the number of edges between S and S c . Applying Menger's theorem with the sets A 1 , A 2 , and using |A i | ≥ np 0 /4, and the fact that the Cheeger constant is bounded below by c > 0 from the assumption of our theorem, we get Hence, there are at least cnp 0 /4 edge-disjoint paths from A 1 to A 2 . Since there are at most ∆n/2 edges in G, at least half of these paths, i.e., at least cnp 0 /8 edge-disjoint paths between A 1 and A 2 have length at most K each. Take two vertices v i ∈ A i that are incident on such a path of length at most K. Since for some (τ 1 , r 1 ) ∈ H 1 , and for some (τ 2 , r 2 ) ∈ H 2 ; hence B τ 1 (u 1 , R) ∼ = B τ 2 (r 2 , R) for some u 1 such that d(r 1 , u 1 ) ≤ K. This contradicts the choice of R. This completes the proof.
Finally we end this section by proving Corollary 1.2.
Proof of Corollary 1.2. We first prove the case when p < p c (G). Fix any ε > 0. Let H ⊆ G be the set of all rooted graphs with percolation critical probability equal to p c (G). For every (τ, v) ∈ H, as p < p c (τ ), there exists R (τ,v) = R (τ,v) (ε) large enough, such that, Consider the neighborhood C((τ, v), ). Then for all graphs in this neighborhood, their R (τ,v) balls around the root are isomorphic to B τ (v, R (τ,v) ), and hence the equation (1) holds for all such graphs. Consider ∪ (τ,v)∈H C((τ, v), 1 1+R (τ,v) ), an open cover of H. Also H is second countable (since G is). Hence H admits a countable subcover, say C 1 , C 2 , . . ., Now as G n converges locally weakly to G, choose n large enough such that, if F denotes the finite set of all rooted graphs of radius R and degree bounded by ∆, then, where L is the law of U n . Hence, where the inequality in the fourth line follows from (2). Next, following the arguments in the proof of Theorem 1.3 in [4] verbatim, it follows that where C 1 (n) is the largest component of G n (p), which proves the first assertion of the corollary. Now we prove the case when p > p c (G). As in the proof of Theorem 1.3 of [4] for this case, fix some ε > 0, let p 1 > p c such that 1 − p = (1 − p 1 )(1 − ε), and consider G n (p 1 ). As p 1 > p c (G), for all (τ, v) ∈ H, the set of all rooted graphs with percolation probability equal to p c (G), we have, v)) > 0} = 1, hence there exists some δ > 0 and 0 < η ≤ 1 such that µ{(τ, v) ∈ H : f ((τ, v)) > δ} ≥ η > 0.
For v ∈ G n , let B ′ p 1 (v, R) denote the set of vertices in G n (p 1 ) which are connected to v in a p 1 -open path of length at most R. Thus for all n ≥ n 0 , L × P(|B ′ p 1 (U n , R)| ≥ R) ≥ L × P p 1 (U n ↔ ∂B Gn (U n , R)) ≥ δη/2.
Next following the arguments in the proof of Theorem 1.3 of [4] verbatim, the lemma follows.

Acknowledgement
This work was completed when the author was a graduate student at UC Berkeley and was supported by the Loève Fellowship, which he gratefully acknowledges. The author thanks Nike Sun for suggesting this problem and for some helpful discussions. He also thanks Tom Hutchcroft for very helpful feedback on an earlier version of the paper. The author thanks the anonymous referee whose careful reading and detailed comments helped improve the paper.