Ergodic theory on Galton—Watson trees: speed of random walk and dimension of harmonic measure

1995 ◽  
Vol 15 (3) ◽  
pp. 593-619 ◽  
Author(s):  
Russell Lyons ◽  
Robin Pemantle ◽  
Yuval Peres
Keyword(s):  
Random Walk ◽  
Ergodic Theory ◽  
Harmonic Measure ◽  
Branching Process ◽  
Main Tool ◽  
Simple Random Walk ◽  
Family Tree ◽  
First Order ◽  
The Family ◽  
Rate Of Escape

AbstractWe consider simple random walk on the family treeTof a nondegenerate supercritical Galton—Watson branching process and show that the resulting harmonic measure has a.s. strictly smaller Hausdorff dimension than that of the whole boundary ofT. Concretely, this implies that an exponentially small fraction of thenth level ofTcarries most of the harmonic measure. First-order asymptotics for the rate of escape, Green function and the Avez entropy of the random walk are also determined. Ergodic theory of the shift on the space of random walk paths on trees is the main tool; the key observation is that iterating the transformation induced from this shift to the subset of ‘exit points’ yields a nonintersecting path sampled from harmonic measure.

scholarly journals Asymptotic entropy of transformed random walks

2016 ◽  
Vol 37 (5) ◽  
pp. 1480-1491 ◽  
Author(s):  
BEHRANG FORGHANI
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Ergodic Theory ◽  
Stopping Time ◽  
Stopping Times ◽  
Discrete Groups ◽  
Differential Entropy ◽  
Random Walks On Groups ◽  
Rate Of Escape ◽  
Asymptotic Entropy

We consider general transformations of random walks on groups determined by Markov stopping times and prove that the asymptotic entropy (respectively, rate of escape) of the transformed random walks is equal to the asymptotic entropy (respectively, rate of escape) of the original random walk multiplied by the expectation of the corresponding stopping time. This is an analogue of the well-known Abramov formula from ergodic theory; its particular cases were established earlier by Kaimanovich [Differential entropy of the boundary of a random walk on a group. Uspekhi Mat. Nauk38(5(233)) (1983), 187–188] and Hartman et al [An Abramov formula for stationary spaces of discrete groups. Ergod. Th. & Dynam. Sys.34(3) (2014), 837–853].

scholarly journals Parking on a Random Tree

2018 ◽  
Vol 28 (1) ◽  
pp. 23-45 ◽  
Author(s):  
CHRISTINA GOLDSCHMIDT ◽  
MICHAŁ PRZYKUCKI
Keyword(s):  
Phase Transition ◽  
Branching Process ◽  
Empty Space ◽  
Alternative Proof ◽  
Family Tree ◽  
Objective Method ◽  
Analytic Combinatorics ◽  
The Family ◽  
Unique Path ◽  
Discontinuous Phase

Consider a uniform random rooted labelled tree on n vertices. We imagine that each node of the tree has space for a single car to park. A number m ≤ n of cars arrive one by one, each at a node chosen independently and uniformly at random. If a car arrives at a space which is already occupied, it follows the unique path towards the root until it encounters an empty space, in which case it parks there; if there is no empty space, it leaves the tree. Consider m = ⌊α n⌋ and let An,α denote the event that all ⌊α n⌋ cars find spaces in the tree. Lackner and Panholzer proved (via analytic combinatorics methods) that there is a phase transition in this model. Then if α ≤ 1/2, we have $\mathbb{P}({A_{n,\alpha}}) \to {\sqrt{1-2\alpha}}/{(1-\alpha})$, whereas if α > 1/2 we have $\mathbb{P}({A_{n,\alpha}}) \to 0$. We give a probabilistic explanation for this phenomenon, and an alternative proof via the objective method. Along the way, we consider the following variant of the problem: take the tree to be the family tree of a Galton–Watson branching process with Poisson(1) offspring distribution, and let an independent Poisson(α) number of cars arrive at each vertex. Let X be the number of cars which visit the root of the tree. We show that $\mathbb{E}{[X]}$ undergoes a discontinuous phase transition, which turns out to be a generic phenomenon for arbitrary offspring distributions of mean at least 1 for the tree and arbitrary arrival distributions.

A Discrete Analogue of a Theorem of Makarov

1993 ◽  
Vol 2 (2) ◽  
pp. 181-199 ◽  
Author(s):  
Gregory F. Lawler
Keyword(s):  
Random Walk ◽  
Hausdorff Dimension ◽  
Harmonic Measure ◽  
Simple Random Walk ◽  
Discrete Analogue ◽  
Connected Subset ◽  
Image Position ◽  
Dimension One

A theorem of Makarov states that the harmonic measure of a connected subset of ℝ2 is supported on a set of Hausdorff dimension one. This paper gives an analogue of this theorem for discrete harmonic measure, i.e., the hitting measure of simple random walk. It is proved that for any 1/2 < α < 1, β < α − 1/2, there is a constant k such that for any connected subset A ⊂ ℤ2 of radius n,where HA denotes discrete harmonic measure.

Eight Drives and the Family Tree

1967 ◽  
Vol 12 (1) ◽  
pp. 6-8
Author(s):  
SUSAN DERI
Keyword(s):  
Family Tree ◽  
The Family

The family tree

1997 ◽  
Author(s):  
Frank Landy
Keyword(s):  
Family Tree ◽  
The Family

The size of the region occupied by one type in an invasion process

1976 ◽  
Vol 13 (02) ◽  
pp. 355-356 ◽  
Author(s):  
Aidan Sudbury
Keyword(s):  
Random Walk ◽  
Simple Random Walk ◽  
Rectangular Lattice ◽  
Invasion Process

Particles are situated on a rectangular lattice and proceed to invade each other's territory. When they are equally competitive this creates larger and larger blocks of one type as time goes by. It is shown that the expected size of such blocks is equal to the expected range of a simple random walk.

Stability in 𝐿𝑞-norm for inverse source parabolic problems

2020 ◽  
Vol 28 (6) ◽  
pp. 797-814
Author(s):  
Elena-Alexandra Melnig
Keyword(s):  
Parabolic Equations ◽  
Main Tool ◽  
Parabolic Systems ◽  
Carleman Estimates ◽  
Parabolic Problems ◽  
Lebesgue Spaces ◽  
First Order ◽  
Lipschitz Estimates ◽  
Systems Of Parabolic Equations

AbstractWe consider systems of parabolic equations coupled in zero and first order terms. We establish Lipschitz estimates in {L^{q}}-norms, {2\leq q\leq\infty}, for the source in terms of the solution in a subdomain. The main tool is a family of appropriate Carleman estimates with general weights, in Lebesgue spaces, for nonhomogeneous parabolic systems.

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