Equivalence of zero entropy and the Liouville property for stationary random graphs

Piaggio, Matí­as; Lessa, Pablo

doi:10.1214/16-ejp4650

Cited by 5 publications

(3 citation statements)

References 21 publications

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“…In the present setting of stationary random graphs, the implication =⇒ was established by Benjamini and Curien [10], and refined by Benjamini, Duminil-Copin, Kozma and Yadin [11]. The converse ⇐= was proved by Carrasco Piaggio and Lessa [20] (see also [13]), but under an additional growth assumption. Since this is the implication that we are going to use, we need to give more details.…”

Section: Spectral Radius Entropy and The Liouville Propertymentioning

confidence: 65%

“…where W 1 denotes the L 1 −Wasserstein distance on P 1 (V G , d G ), see (20) below. Note that the computation of κ G (x, y) amounts to solving a finite-dimensional linear optimization problem, and is therefore amenable to standard algorithmic techniques (see [22] for a beautiful interactive curvature calculator).…”

Section: Introduction 1non-negative Curvaturementioning

confidence: 99%

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Sparse expanders have negative curvature

Salez¹

2021

Preprint

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We prove that bounded-degree expanders with non-negative Ollivier-Ricci curvature do not exist, thereby solving a long-standing open problem suggested by Naor and Milman and publicized by Ollivier (2010). In fact, this remains true even if we allow for a vanishing proportion of large degrees, large eigenvalues, and negatively-curved edges. To establish this, we work directly at the level of Benjamini-Schramm limits, and exploit the entropic characterization of the Liouville property on stationary random graphs to show that non-negative curvature and spectral expansion are incompatible "at infinity". We then transfer this result to finite graphs via local weak convergence. The same approach also applies to the Bacry-Emery curvature condition CD(0, ∞), thereby settling a recent conjecture of Cushing, Liu and Peyerimhoff (2019).

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Section: Spectral Radius Entropy and The Liouville Propertymentioning

confidence: 65%

Section: Introduction 1non-negative Curvaturementioning

confidence: 99%

Sparse expanders have negative curvature

Salez¹

2021

Preprint

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“…In the rest of the article we show that Poisson-Delaunay random walks on Riemannian symmetric spaces (previously studied in [BPP14], [Paq17], and [CPL16]) may be fruitfully seen as distance stationary sequences. In particular, we show that one may obtain interesting results about their speed by using the Furstenberg type formula.…”

Section: Introductionmentioning

confidence: 89%

A Furstenberg type formula for the speed of distance stationary sequences

Carrasco,

Lessa,

Paquette

2017

Preprint

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We prove a formula for the speed of distance stationary random sequences. A particular case is the classical formula for the largest Lyapunov exponent of an i.i.d. product of two by two matrices in terms of a stationary measure on projective space. We apply this result to Poisson-Delaunay random walks on Riemannian symmetric spaces. In particular, we obtain sharp estimates for the asymptotic behavior of the speed of hyperbolic Poisson-Delaunay random walks when the intensity of the Poisson point process goes to zero. This allows us to prove that a dimension drop phenomena occurs for the harmonic measure associated to these random walks. With the same technique we give examples of co-compact Fuchsian groups for which the harmonic measure of the simple random walk has dimension less than one.

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Recurrence, Transience, Liouville and Speed

Curien

2023

Lecture Notes in Mathematics

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Equivalence of zero entropy and the Liouville property for stationary random graphs

Cited by 5 publications

References 21 publications

Sparse expanders have negative curvature

Sparse expanders have negative curvature

A Furstenberg type formula for the speed of distance stationary sequences

Recurrence, Transience, Liouville and Speed

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