Processes on Unimodular Random Networks

By David J. Aldous, Russell Lyons. 2006

534 citations recorded by the source corpus.

We investigate unimodular random networks. Our motivations include their characterization via reversibility of an associated random walk and their similarities to unimodular quasi-transitive graphs. We extend various theorems concerning random walks, percolation, spanning forests, and amenability from the known context of unimodular quasi-transitive graphs to the more general context of unimodular random networks. We give properties of a trace associated to unimodular random networks with applications to stochastic comparison of continuous-time random walk.

Subfields: Discrete Mathematics and Combinatorics, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/ec4f946ab604cc4c85e09dbc2bdd90a1957b9526

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Citations in the corpus

Invariant transports of stationary random measures and mass-stationarity (2009, background): “In fact, the papers [2] and [1] show that the mass-transport principle can be extended beyond this setting.”
On the growth of $L^2$-invariants for sequences of lattices in Lie groups (2012, background): “All the material here is well known; for more information, we refer the reader to [7], [10], [26].”
Ising models on locally tree-like graphs (2008, methodology): “ote in passing that our derivation of the formula (2.9) implicitly uses the ISING MODELS ON LOCALLY TREE-LIKE GRAPHS 7 fact that T(P,ρ,∞) possesses the involution invariance of [2]. As pointed out in [1], every local limit of ﬁnite graphs must have the involution invariance property (which clearly n”
A large deviation principle for Wigner matrices without Gaussian tails (2012, methodology): “This requires the analysis of large deviations for sparse random networks, and some use of the theory of local convergence for random networks that was recently developed by Benjamini and Schramm [6], Aldous and Steele [2], and Aldous and Lyons [1].”
Limits of locally–globally convergent graph sequences (2012, background): “Aldous and Lyons [1] conjectured that all involution-invariant measures arise as graph limits.”
Kesten's theorem for Invariant Random Subgroups (2012, background): “This phenomenon is pointed out by Aldous and Lyons in their starting paper on unimodular random networks [4].”
On the phase transition in random simplicial complexes (2014, background): “It can be easily verified that (G∗, d) is a separable and complete metric space, which comes as usual equipped with its Borel σ-algebra; see [2].”
A view from infinity of the uniform infinite planar quadrangulation (2012, background): “See [1, 6] for a general study of random graphs that are invariant under re-rooting along the simple random walk.”
Unimodular random trees (2012, background): “To prove that this is invariant, we first show involution invariance of the constructed marked tree and then appeal to Aldous and Lyons [2, Theorem 3.2] to get that it is an invariant percolation on Td .”
Large deviations of empirical neighborhood distribution in sparse random graphs (2013, methodology): “Sofic measures share a stationarity property called unimodularity [1].”
Recurrence of planar graph limits (2012, methodology): “(This concept is described with more details in [1].”
The measurable Kesten theorem (2011, background): “The notion has been introduced in [2]: for the definition, see Section 2.”
Hyperbolic and Parabolic Unimodular Random Maps (2016, background): “y the edge set of Gand setting !p(e) = 1 (U(e) p) for every e2Eand p2[0;1]. The critical probability of Gis dened by pc(G) := inf fp: P(!p has an innite connected component) = 1g: It is well-known [2, 63] that if (G;ˆ) is an ergodic unimodular random rooted graph, then for each p2[0;1] the number of”
A critical branching process model for biodiversity (2004, unknown): “See [5, 4] for more sophisticated examples.”