A Phase Transition for Hyperbolic Branching Processes

Karpelevich, F. I.; Pechersky, E.; Suhov, Yuri

doi:10.1007/s002200050405

Cited by 16 publications

(19 citation statements)

References 10 publications

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“…В статье результаты работ [14] и [11] о хаусдорфовой размерно сти предельного множества однородного гиперболического ветвя щегося диффузионного процесса распространяются на случай не постоянного механизма деления. Точнее, мы рассматриваем не однородный ветвящийся диффузионный процесс на пространстве Лобачевского H d и предполагаем, что параметры процесса равно мерно сходятся к их предельным значениям на абсолюте c?H d .…”

Section: Branching Diffusions On H D With Variable Fission: the Hausdunclassified

“…Точнее, мы рассматриваем не однородный ветвящийся диффузионный процесс на пространстве Лобачевского H d и предполагаем, что параметры процесса равно мерно сходятся к их предельным значениям на абсолюте c?H d . В этих предположениях устанавливается формула для хаусдорфовой размерности h(A) предельного (случайного) множества Л С <?H d , которая согласуется с формулами, полученными в [14] и [11] для однородного случая. Метод основан на свойствах минимального решения уравнения Штурма-Лиувилля с потенциалом, принимаю щим два значения, и на элементах гармонического анализа на H d .…”

Section: Branching Diffusions On H D With Variable Fission: the Hausdunclassified

“…Statement (ii) of Theorem 2.1 can be deduced, following an argument developed originally in [14] and repeated in [11], from the following theorem. …”

Section: Remark Under Assumption V+ > 1 (D -L)mentioning

confidence: 99%

“…The main result of [14] and [11], for an HHBP on a Lobachevsky space, was as follows. More precisely, let p k {z) be the probability that in the course of fission at point z, the number of offspring produced equals fc, then x(z) = YlkLi hPk(z)-Define the local fission potential (LFP) by v(z) = X(z)(x(z) -1).…”

Section: Introduction Homogeneous Branching Processes On a D-dimensimentioning

confidence: 99%

“…have been considered first by Lalley and Sellke [14] in the two-dimensional case (the branching diffusion on the Lobachevsky plane H 2 ) and subsequently by Karpelevich, Pechersky, and Suhov [11] in a general case. Following [14], we call such process a homoge neous hyperbolic branching process (HHBP).…”

Section: Introduction Homogeneous Branching Processes On a D-dimensimentioning

confidence: 99%

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Branching diffusions on $H^d$ with variable fission: The Hausdorff dimension of the limiting set

Кельберт¹,

Kelbert²,

Сухов³

et al. 2006

Teor. Veroyatnost. i Primenen.

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Section: Branching Diffusions On H D With Variable Fission: the Hausdunclassified

“…Statement (ii) of Theorem 2.1 can be deduced, following an argument developed originally in [14] and repeated in [11], from the following theorem. …”

Section: Remark Under Assumption V+ > 1 (D -L)mentioning

confidence: 99%

Section: Introduction Homogeneous Branching Processes On a D-dimensimentioning

confidence: 99%

Section: Introduction Homogeneous Branching Processes On a D-dimensimentioning

confidence: 99%

See 3 more Smart Citations

Branching diffusions on $H^d$ with variable fission: The Hausdorff dimension of the limiting set

Кельберт¹,

Kelbert²,

Сухов³

et al. 2006

Teor. Veroyatnost. i Primenen.

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Limit Set of Branching Random Walks on Hyperbolic Groups

Sidoravičius

Wang

Xiang

2022

Comm Pure Appl Math

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Let Γ be a nonelementary hyperbolic group with a word metric d and ∂Γ its hyperbolic boundary equipped with a visual metric for some parameter . Fix a superexponential symmetric probability μ on Γ whose support generates Γ as a semigroup, and denote by ρ the spectral radius of the random walk Y on Γ with step distribution μ. Let ν be a probability on with mean . Let be the branching random walk on Γ with offspring distribution ν and base motion Y , and let be the volume growth rate for the trace of . We prove for that the Hausdorff dimension of the limit set Λ , which is the random subset of consisting of all accumulation points of the trace of , is given by . Furthermore, we prove that is almost surely a deterministic, strictly increasing, and continuous function of , is bounded by the square root of the volume growth rate of Γ , and has critical exponent 1/2 at in the sense that for some positive constant C. We conjecture that the Hausdorff dimension of Λ in the critical case is almost surely. This has been confirmed on free groups or the free product (by amalgamation) of finitely many finite groups equipped with the word metric d defined by the standard generating set. © 2022 Wiley Periodicals LLC.

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Percolation on Hyperbolic Graphs

Hutchcroft

2019

Geom. Funct. Anal.

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We prove that Bernoulli bond percolation on any nonamenable, Gromov hyperbolic, quasitransitive graph has a phase in which there are infinitely many infinite clusters, verifying a well-known conjecture of Benjamini and Schramm (1996) under the additional assumption of hyperbolicity. In other words, we show that p c < p u for any such graph. Our proof also yields that the triangle condition ∇ pc < ∞ holds at criticality on any such graph, which is known to imply that several critical exponents exist and take their mean-field values. This gives the first family of examples of one-ended groups all of whose Cayley graphs are proven to have mean-field critical exponents for percolation.

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A Phase Transition for Hyperbolic Branching Processes

Cited by 16 publications

References 10 publications

Branching diffusions on $H^d$ with variable fission: The Hausdorff dimension of the limiting set

Branching diffusions on $H^d$ with variable fission: The Hausdorff dimension of the limiting set

Limit Set of Branching Random Walks on Hyperbolic Groups

Percolation on Hyperbolic Graphs

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