Hyperbolic and Parabolic Unimodular Random Maps

Angel, Omer; Hutchcroft, Tom; Nachmias, Asaf; Ray, Gourab

doi:10.1007/s00039-018-0446-y

Cited by 41 publications

(54 citation statements)

References 61 publications

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“…More generally, for unimodular, planar graphs, other notions of hyperbolicity (including p c < p u ) have been studied in [4] and proved to be equivalent to each other. It might be interesting to study the relation with our setting: if it is true that any hyperbolic (in the sense of [4]) unimodular map contains a supercritical Galton-Watson tree, then our results of Section 2 apply. On the other hand, it is clear that every unimodular planar map containing a supercritical Galton-Watson tree is hyperbolic in the sense of [4].…”

Section: Lemma 22mentioning

confidence: 99%

Supercritical causal maps: geodesics and simple random walk

Budzinski¹

2019

Electron. J. Probab.

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We study the random planar maps obtained from supercritical Galton-Watson trees by adding the horizontal connections between successive vertices at each level. These are the hyperbolic analog of the maps studied by Curien, Hutchcroft and Nachmias in [15], and a natural model of random hyperbolic geometry. We first establish metric hyperbolicity properties of these maps: we show that they admit bi-infinite geodesics and satisfy a weak version of Gromov-hyperbolicity. We also study the simple random walk on these maps: we identify their Poisson boundary and, in the case where the underlying tree has no leaf, we prove that the random walk has positive speed. Some of the methods used here are robust, and allow us to obtain more general results about planar maps containing a supercritical Galton-Watson tree. Figure 1: The circle packing of a causal triangulation constructed from a Galton-Watson tree with geometric offspring distribution of mean 3/2. This was made with the help of the software CirclePackPoisson boundary. The second goal of this work is to study the simple random walk on C(T ) and to identify its Poisson boundary. First note that C(T ) contains as a subgraph the supercritical Galton-Watson tree T , which is transient, so C(T ) is transient as well. We recall the general definition of the Poisson boundary. Let G be an infinite, locally finite graph, and let G ∪ ∂G be a compactification of G, i.e. a compact metric space in which G is dense. Let also (X n ) be the simple random walk on G started from ρ. We say that ∂G is a realization of the Poisson boundary of G if the following two properties hold:• (X n ) converges a.s. to a point X ∞ ∈ ∂G,• every bounded harmonic function h on G can be written in the formwhere g is a bounded measurable function from ∂G to R.We denote by ∂T the space of infinite rays of T . If γ, γ ∈ ∂T , we write γ ∼ γ if γ = γ or if γ and γ are two "consecutive" rays in the sense that there is no ray between them. Then ∼ is a.s. an equivalence relation for which countably many equivalence classes have cardinal 2 and all the others have cardinal 1. We write ∂T = ∂T / ∼. There is a natural way to equip C(T ) ∪ ∂T with a topology that makes it a compact space. We refer to Section 3.1 for the construction of this topology, but we mention right now that ∂T is homeomorphic to the circle, whereas ∂T is homeomorphic to a Cantor set. The space C(T ) ∪ ∂T can be seen as a compactification of the infinite graph C(T ). We show that this is a realization of its Poisson boundary.Theorem 2 (Poisson boundary of C(T )). Almost surely:1. the limit lim(X n ) = X ∞ exists and its distribution has full support and no atoms in ∂T , 2. ∂T is a realization of the Poisson boundary of C(T ).Note that, by a result of Hutchcroft and Peres [19], the second point will follow from the first one.Positive speed. A natural and strong property shared by many models of hyperbolic graphs is the positive speed of the simple random walk. See for example [24] for supercritical and [13,5] for the PSHIT or their half-planar a...

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Section: Lemma 22mentioning

confidence: 99%

Supercritical causal maps: geodesics and simple random walk

Budzinski¹

2019

Electron. J. Probab.

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“…It will also be useful to consider a more general version of this construction, in which we let the sizes of the tori grow as a specified function of the height. Let d ≥ 2, let n ≥ 1, and let Figure 1: The canopy tree of one-dimensional tori T 1 (3,2). The grey edges give a 3-to-2 correspondence between levels l and l + 1 for each l ≥ 0.…”

Section: Trees Of Torimentioning

confidence: 99%

“…(1). (v 1 , v 2 , W 1 ) and (v 1 , v 2 , W ) are both type 1, v 2 = v 2 , and v 1 , v 1 are adjacent in G, and W, W are adjacent in D, or (2). v 1 = v 1 , W = W , and v 2 and v 2 are adjacent in S.…”

Section: Proof Of Theorem 14mentioning

confidence: 99%

Counterexamples for percolation on unimodular random graphs

Angel¹,

Hutchcroft²

2018

Unimodularity in Randomly Generated Graphs

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We construct an example of a bounded degree, nonamenable, unimodular random rooted graph with p c = p u for Bernoulli bond percolation, as well as an example of a bounded degree, unimodular random rooted graph with p c < 1 but with an infinite cluster at criticality. These examples show that two well-known conjectures of Benjamini and Schramm are false when generalised from transitive graphs to unimodular random rooted graphs.

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“…This was extended by [1,Example 9.6] to show that every unimodular random rooted plane graph satisfying a mild finiteness condition admits a natural unimodular probability measure on the plane duals; in fact, the root of the dual can be chosen to be a face incident to the root of the primal graph. The recent paper [2] makes a systematic study of unimodular random planar graphs, synthesizing known results and introducing new ones, showing a dichotomy involving 17 equivalent properties.…”

Section: Introductionmentioning

confidence: 99%

A stationary planar random graph with singular stationary dual: dyadic lattice graphs

Lyons

White

2019

Probab. Theory Relat. Fields

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Dyadic lattice graphs and their duals are commonly used as discrete approximations to the hyperbolic plane. We use them to give examples of random rooted graphs that are stationary for simple random walk, but whose duals have only a singular stationary measure. This answers a question of Curien and shows behaviour different from the unimodular case. The consequence is that planar duality does not combine well with stationary random graphs. We also study harmonic measure on dyadic lattice graphs and show its singularity.

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Hyperbolic and Parabolic Unimodular Random Maps

Cited by 41 publications

References 61 publications

Supercritical causal maps: geodesics and simple random walk

Supercritical causal maps: geodesics and simple random walk

Counterexamples for percolation on unimodular random graphs

A stationary planar random graph with singular stationary dual: dyadic lattice graphs

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