Green function estimates and their applications to the intersections of symmetric random walks

Abstract This is the first of a series of two papers dealing with local limit theorems in relatively hyperbolic groups. In this first paper, we prove rough estimates for the Green function. Along the way, we introduce the notion of relative automaticity which will be useful in both papers and we show that relatively hyperbolic groups are relatively automatic. We also define the notion of spectral positive recurrence for random walks on relatively hyperbolic groups. We then use our estimates for the Green function to prove that $p_n\asymp R^{-n}n^{-3/2}$ for spectrally positive-recurrent random walks, where $p_n$ is the probability of going back to the origin at time n and where R is the inverse of the spectral radius of the random walk.

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Green function for the diffusion limit of one-dimensional continuous time random walks

Journal of Molecular Liquids ◽

10.1016/j.molliq.2004.02.015 ◽

2004 ◽

Vol 114 (1-3) ◽

pp. 165-171 ◽

Cited By ~ 21

Author(s):

W.T. Coffey ◽

D.S.F. Crothers ◽

D. Holland ◽

S.V. Titov

Keyword(s):

Green Function ◽

Random Walks ◽

Continuous Time ◽

Diffusion Limit ◽

One Dimensional ◽

Continuous Time Random Walks

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Gillis' Random Walks on Graphs

Journal of Applied Probability ◽

10.1239/jap/1110381390 ◽

2005 ◽

Vol 42 (1) ◽

pp. 295-301 ◽

Cited By ~ 1

Author(s):

Nadine Guillotin-Plantard

Keyword(s):

Random Walk ◽

Green Function ◽

Random Walks ◽

Regular Graph ◽

Simple Random Walk ◽

Square Lattice ◽

Random Walker ◽

Unit Step ◽

Random Walks On Graphs ◽

The Green Function

We consider a random walker on a d-regular graph. Starting from a fixed vertex, the first step is a unit step in any one of the d directions, with common probability 1/d for each one. At any later step, the random walker moves in any one of the directions, with probability q for a reversal of direction and probability p for any other direction. This model was introduced and first studied by Gillis (1955), in the case when the graph is a d-dimensional square lattice. We prove that the Gillis random walk on a d-regular graph is recurrent if and only if the simple random walk on the graph is recurrent. The Green function of the Gillis random walk will be also given, in terms of that of the simple random walk.

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Gillis' Random Walks on Graphs

Journal of Applied Probability ◽

10.1017/s0021900200000255 ◽

2005 ◽

Vol 42 (01) ◽

pp. 295-301 ◽

Cited By ~ 1

Author(s):

Nadine Guillotin-Plantard

Keyword(s):

Random Walk ◽

Green Function ◽

Random Walks ◽

Regular Graph ◽

Simple Random Walk ◽

Square Lattice ◽

Random Walker ◽

Unit Step ◽

Random Walks On Graphs ◽

The Green Function

We consider a random walker on ad-regular graph. Starting from a fixed vertex, the first step is a unit step in any one of theddirections, with common probability 1/dfor each one. At any later step, the random walker moves in any one of the directions, with probabilityqfor a reversal of direction and probabilitypfor any other direction. This model was introduced and first studied by Gillis (1955), in the case when the graph is ad-dimensional square lattice. We prove that the Gillis random walk on ad-regular graph is recurrent if and only if the simple random walk on the graph is recurrent. The Green function of the Gillis random walk will be also given, in terms of that of the simple random walk.

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