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Complexity Analysis
Green Function
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Random Walks in the Quarter Plane Absorbed at the Boundary : Exact and Asymptotic
Random Walks in the Quarter Plane Absorbed at the Boundary : Exact and Asymptotic,Kilian Raschel
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Random Walks in the Quarter Plane Absorbed at the Boundary : Exact and Asymptotic
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Citations: 4
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Kilian Raschel
Nearest neighbor
random walks in the quarter plane that are absorbed when reaching the boundary are studied. The cases of positive and zero drift are considered. Absorption probabilities at a given time and at a given site are made explicit. The following asymptotics for these random walks starting from a given point $(n_0,m_0)$ are computed : that of probabilities of being absorbed at a given site $(i,0)$ [resp. $(0,j)$] as $i\to \infty$ [resp. $j \to \infty$], that of the distribution's tail of absorption time at xaxis [resp. yaxis], that of the Green functions at site $(i,j)$ when $i,j\to \infty$ and $j/i \to \tan \gamma$ for $\gamma \in [0, \pi/2]$. These results give the
Martin boundary
of the process and in particular the suitable Doob $h$transform in order to condition the process never to reach the boundary. They also show that this $h$transformed process is equal in distribution to the limit as $n\to \infty$ of the process conditioned by not being absorbed at time $n$. The main tool used here is complex analysis.
Published in 2009.
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(4)
Martin boundary of a killed random walk on $\Z_+^d$
(
Citations: 1
)
Irina IgnatioukRobert
Published in 2009.
The tMartin boundary of reflected random walks on a halfspace
Irina IgnatioukRobert
Published in 2009.
Green functions and Martin compactification for killed random walks related to SU(3)
Kilian Raschel
Published in 2009.
Passage time from four to two blocks in the voter model
Kilian Raschel
Published in 2009.