A stationary random graph of no growth rate

(1)

www.imstat.org/aihp Annales de l’Institut Henri Poincaré - Probabilités et Statistiques

2014, Vol. 50, No. 4, 1161–1164 DOI:10.1214/13-AIHP563

A stationary random graph of no growth rate

Ádám Timár

¹

Bolyai Institute, University of Szeged, Aradi v. tere 1, 6720 Szeged, Hungary. E-mail:[email protected] Received 22 August 2012; revised 19 March 2013; accepted 16 April 2013

Abstract. We present a random automorphism-invariant subgraph of a Cayley graph such that with probability 1 its exponential growth rate does not exist.

Résumé. Nous construisons un sous-graphe aléatoire invariant par automorphismes d’un groupe de Cayley qui n’a presque sûre- ment pas un taux de croissance exponentiel bien défini.

MSC:60C05; 05C80

Keywords:Unimodular graph; Stationary random graph; Growth rate

At the Banff workshop “Graphs, groups and stochastics” in 2011, Vadim Kaimanovich asked the following question:

Question 1. For an arbitrary(Γ, o)unimodular random rooted graph,does the exponential growth rate

nlim→∞logBΓ(o, n)/n

exist almost surely? HereB_Γ(o, n)denotes the ball of radiusnaroundoinΓ. We give a negative answer to the question.

Theorem 2. There is a transitive unimodular graphGwith a random invariant spanning subgraphΓ such that for any pointo∈Gand almost everyΓ

lim inf

n→∞ logB_Γ(o, n)/n=0 and lim sup

n→∞ logB_Γ(o, n)/n=c

withc >0.Moreover,there exists such aΓ that is a spanning tree with one end.

Unimodular random graphs are random rooted graphs with a certain stationarity property (see the next paragraph).

Unimodular random graphs provide a common framework to examples such as automorphism-invariant random subgraphs of transitive unimodular graphs (e.g., Cayley graphs), augmented Galton–Watson trees, Borel equivalence

1Research was supported by Sinergia grant CRSI22-130435 of the Swiss National Science foundation and by MTA Rényi “Lendület” Groups and Graphs Research Group. This research was partly realized in the frames of TÁMOP 4.2.4. A/1-11-1-2012-0001 “National Excellence Program,”

subsidized by the European Union and co-financed by the European Social Fund.

(2)

1162 Á. Timár

relations. . . . Many such examples arise as the local weak limit (or Benjamini–Schramm limit) of a sequence of finite graphs; a main open question by Aldous and Lyons is whether all of them arise this way, [1].

Suppose that the random rooted graph(G, o)has every degree bounded bydalmost surely, and define a new graph (G, o), by adding loops to vertices ofGin a way that makes itd-regular. Choose an edge uniformly of all edges inG incident too, and letxbeoif the chosen edge is a loop, otherwise letxbe the endpoint of the edge different fromo.

If the resulting distribution on the triples(G, o, x)is the same as the distribution of the triples(G, x, o), then we call (G, o)unimodular. The bound on the degrees can be removed, and several alternative definitions exist, see [1].

The exponential growth rate of an infinite graph is one of the important invariants that are defined as a limit. For other invariants, such as the speed of the random walk, the defining limit is known to exist almost surely for every unimodular random graph, using some subadditive inequality, [1]. Hence it may be surprising that a similar argument does not work for the existence of growth, as it does for every Cayley graph.

Proof of Theorem2.

Construction: LetΔbe the triangular lattice,L=Z2Zbe the Cayley graph of the lamplighter group, where we think about each vertex ofLas a pair(ξ, k), whereξ ∈ {0,1}^Zof finite support is the status of lamps andk∈Zis the position of the marker, and the generators defining edges are(10,0), (0,1), (0,−1)(hence a vertex is adjacent to vertices that result from switching the lamp at the location of the marker, or from making the marker step to the right or to the left). DefineG=Δ×L×K₂₈, whereK₂₈is the complete graph on vertex set{1,2, . . . ,28}. Here the productH₁×H₂of graphsH₁, H₂is defined on vertex setV (H₁)×V (H₂)with(x₁, x₂)adjacent to(y₁, y₂)if and only ifx_i=y_i orx_iis adjacent toy_iinH_i, fori=1,2.

Consider critical percolation on the vertices ofΔ, that is, delete each vertex with probability 1/2 independently from each other, and call the deleted verticesclosed, the remaining onesopen. LetC0andC1be the set of components (clusters) induced by closed and open vertices respectively;C:=C0∪C1. From percolation theory it is known that every component inCis finite. Furthermore, any componentCinC0is separated from infinity by a unique component CinC1to which it is adjacent to (meaning that there is a uniqueC∈C1such thatCis contained in a finite component ofΔ\Cand there is an edge betweenC andC), and conversely, any componentCinC1is separated from infinity by a unique componentCinC0. Hence there is a naturally defined oriented treeT onCas vertex set: let there be an edge pointing fromxtoy(x→y), if the clusteryseparatesxfrom infinity and they are adjacent (in particular, if one of them is closed then the other has to be open). Call the set of leaves in the treeL1, and recursively, defineLi as the set of verticesx inT with the property that the longest path oriented towardsxhas lengthi(measured in the number of vertices in it).

Fix sequencec₁c₂ · · ·to be defined later. Define a random variableX_i to be uniformly chosen in{1, . . . , ci} (i=1,2, . . .). Consider the equivalence relationRi onZwherexandyare equivalent if forα_i:=_i

j=1X_jj−1 k=1c_k we have[(x−α_i)/c₁· · ·c_i] = [(y−α_i)/c₁· · ·c_i](where[·]denotes the floor function). That is, the sequence ofZ- invariant partitions defined by the(Ri)is coarser and coarser, and eachRi consists of classes ofc₁· · ·c_i consecutive integers. This gives rise to a sequencePiof invariant coarser and coarser partitions ofL, where each class ofPihave sizec1· · ·c_i2^c¹^···^cⁱ: let points(ξ1, k1)and(ξ2, k2)be in the same class ifk1andk2are in the same class ofRi andξ1

andξ2only differ on this class.

LetSi be the set of subgraphs ofGinduced by the finite sets of the formδ×σ×K₂₈, whereδ∈Li andσ∈Pi. So each element ofSihas the form of the product of a percolation cluster ofΔinLi, a class of the partition ofPi and K₂₈. LetS_i⁰be the set of those elements inSi where theδ above is closed, and letS_i¹be the set of those where it is open. HenceSi=S_i⁰∪S_i¹; call the elements of

iSi cans. Cans in someS_i⁰will be calledtype0, those in someS_i¹ aretype1.

First, we will define the edges ofΓ that go between two distinct cans, then those that go inside one. Suppose that δ×σ×K28andδ×σ×K28are in

Si, and such thatσ⊂σandδ→δinT. Then choose a random edge ofG betweenδ×σ×K₂₈andδ×σ×K₂₈and add it toΓ. Denote the endpoint of this edge inδ×σ×K₂₈byv(δ, σ ).

Do it for every such pair independently. This way we have defined a tree on the set

iSiof cans.

What is left is to define edges within cans. For cans of type 1, let every edge ofGinduced in the can be inΓ. Thec_i will be chosen so that with high probability the diameter of this induced graph in the can in logarithmic in its size. If we wantedΓ to be a tree, we can choose uniformly a spanning tree of the canCat this point that is geodesic with respect to some uniformly chosen pointx, i.e. the distance of anyyfromxinCis the same as in this tree. The diameter of the resulting spanning tree is still logarithmic in the size with high probability (as a consequence of the

(3)

No growth rate 1163

triangle inequality), and this is the only thing we will use later. For cansδ×σ×K₂₈ of type 0, choose uniformly a Hamiltonian path starting fromv(δ, σ )that is contained in some Hamiltonian cycle induced by δ×σ ×K28 in G, independently from all other choices. To see that such a Hamiltonian path always exists, we need to show that a Hamiltonian cycle exists. Choose any spanning tree inδ×σ, take a closed depth-first walkv₁, . . . , v_m(withv₁=v_m) that visits every vertex ofδ×σ at least once and at most degree(Δ×L)=28 times, and then replace every copy of a vertexvinv1, . . . , vmby one or more of the vertices(v,1), (v,2), . . . , (v,28)∈δ×σ×K28in such a way that each of these occur exactly once in the resulting new cycle. Note that every Hamiltonian cycleO inδ×σ ×K₂₈ gives rise to exactly two Hamiltonian paths starting fromv(δ, σ ): delete one of the edges inO adjacent tov(δ, σ ).

This also shows that for anyx∈V (δ×σ×K₂₈), the distance betweenxandv(δ, σ )in one of these paths is at least

|δ×σ×K28|/2.

This finishes the construction ofΓ. It is easy to check that the resultingΓ is ergodic.

Verification: We sketch here why the upper and lower growth rates are different, before giving a rigorous proof in the rest. There are radiiR when the ballBR(o):=BΓ(o, R)of radiusR inΓ just exits a canC of type 0, in the direction of the can that separates it from infinity (meaning thatB_R(o)intersects only finite components ofΓ \C, butB_R₊₁intersects the infinite component as well). Then with high probability some constant proportion ofB_R(o)is withinC, so it has to contain some constant proportion of the Hamiltonian path inC, which will imply that its radius is also close to the volume ofC(this gives the claim about the lower growth rate). On the other hand, there are radii RwhenB_R(o)just exits a canCof type 1, in which case a significant proportion of the volume ofB_R(o)is contained inCwith high probability. Since the growth of the ball inCis exponential, the radius increase between entering and exitingCis at most the logarithm of the size of|C|. By the fast increase of thec_i we can conclude thatRis also about logarithmic in|BR(o)|(for this, it is enough to assume that the sum of the sizes of the smaller cans that are intersected by a minimalo–Cpath is of order logc_i with high probability). This gives the statement about the upper growth.

Now we work out the above argument in details. For a clusterδ∈C let intδ be the complement of the infinite component ofΔ\δ (that is, the union of all clusters thatδ separates from infinity, includingδ). Denote byπ_Δthe projection fromGtoΔandπH be the projection toL×K28=:H. From now on, condition on the eventE0:= {ois in a canC∈S1}. LetC=C₁, C₂, . . .be consecutive cans that an infinite simple path fromoinΓ visits (in particular, π_Δ(C₁), π_Δ(C₂), . . .determines a simple path inT oriented fromπ_Δ(C₁)to infinity). Note that by definition ifi < i thenC_i separatesCi from infinity inΓ. We will now specify how theci’s have to be chosen. Letc1=1. Fori >1, letc_i be a number such that with probability at least 1−2⁻ⁱ we have

logc_i≥28c1· · ·c_i₋12^c¹^···^cⁱ⁻¹intπ_Δ(C_i). (3) LetE_i be the event that (3) holds.

Letebe the edge connectingC_i toC_i₊₁inΓ, and letR be such that the ballB_o(R)of radiusR aroundoinΓ contains exactly one endpoint ofe(the one inCi). Then this ball is contained in the finite component ofΓ \e. The vertex set of the finite component ofΓ \earises as a union of the vertices in cansC, namely, it isV (

{C: π_H(C)⊂ π_H(C_i), π_Δ(C)⊂intπΔ(C_i)}). In particular,

Bo(R)≤ C: πH

C ⊂πH(Ci), πΔ

C ⊂intπΔ(Ci)=intπΔ(Ci)πH(Ci . (4) Now, in the case thatCi is of type 0, the restrictionCi|Γ is a pathP. By the choice ofP and our remark on the Hamiltonian paths, with probability≥1/2 the ballB_o(R)contains all of the points inP, hence|P|/2 gives a lower bound forR. We conclude that conditioned onE₀andE_i, with probability at least 1/2 there is anRsuch thatB_R(o) has radius

R≥ |C_i|/2=π_Δ(C_i)π_H(C_i)/2 and has volume

Bo(R)≤intπΔ(Ci)πH(Ci)≤ diam

intπΔ(Ci) +1 ²πH(Ci)≤πΔ(Ci)²πH(Ci),

where diam(intπ_Δ(C_i)) denotes the diameter of intπ_Δ(C_i), and the second inequality follows from the quadratic growth of Δ. We obtain that log|B_o(R)|/R gets arbitrarily close to 0 with probability at least 1/2 (and hence, by ergodicity, with probability 1). Obviously this remains true if we do not condition ono∈C∈S1.

(4)

1164 Á. Timár

Suppose finally, thatC_i is of type 1. Then the diameter ofΓ|Ciis logarithmic to its size, so conditioned onE_i we have for some constantathatR≤ |C₁| + · · · + |C_i₋₁| +alog|C_i| ≤28c1· · ·c_i₋₁2^c¹^···^cⁱ⁻¹|intπ_Δ(C_i)| +alog|C_i| ≤ 2alog|C_i| with probability at least 1/2, using (3). On the other hand with probability at least 1/2 the volume of B_o(R)∩C_i is at least|C_i|/2, since the choice ofeis uniform. We obtain that with some positive constantc

logBo(R)/R > c,

which holds for infinitely manyRalmost surely (by an argument similar to the end of last paragraph).

Acknowledgements

I thank Lewis Bowen, Russ Lyons and Gábor Pete for valuable discussions, and an anonymous referee for suggestions for improvement.

Reference

[1] D. Aldous and R. Lyons. Processes on unimodular random networks.Electron. Commun. Probab.12(2007) 1454–1508.MR2354165