Non Uniform Random Walks

Abstract : Given $\epsilon _i ∈ [0,1)$ for each $1 < i < n$, a particle performs the following random walk on $\{1,2,...,n\:\}$par If the particle is at $n$, it chooses a point uniformly at random (u.a.r.) from $\{1,...,n-1\}$. If the current position of the particle is $m (1 < m < n)$, with probability $\epsilon _m$ it decides to go back, in which case it chooses a point u.a.r. from $\{m+1,...,n\}$. With probability $1-\epsilon _m$ it decides to go forward, in which case it chooses a point u.a.r. from $\{1,...,m-1\}$. The particle moves to the selected point. What is the expected time taken by the particle to reach 1 if it starts the walk at $n$? Apart from being a natural variant of the classical one dimensional random walk, variants and special cases of this problemarise in Theoretical Computer Science [Linial, Fagin, Karp, Vishnoi]. In this paper we study this problem and observe interesting properties of this walk. First we show that the expected number of times the particle visits $i$ (before getting absorbed at 1) is the same when the walk is started at $j$, for all $j > i$. Then we show that for the following parameterized family of $\epsilon 's: \epsilon _i = \frac{n-i}{n-i+ α · (i-1)}$,$1 < i < n$ where $α$ does not depend on $i$, the expected number of times the particle visits $i$ is the same when the walk is started at $j$, for all $j < i$. Using these observations we obtain the expected absorption time for this family of $\epsilon 's$. As $α$ varies from infinity to 1, this time goes from $Θ (log n) to Θ (n)$. Finally we studythe behavior of the expected convergence timeas a function of $\epsilon$ . It remains an open question to determine whether this quantity increases when all $\epsilon 's$ are increased. We give some preliminary results to this effect.
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Cyril Banderier and Christian Krattenthaler. Discrete Random Walks, DRW'03, 2003, Paris, France. Discrete Mathematics and Theoretical Computer Science, DMTCS Proceedings vol. AC, Discrete Random Walks (DRW'03), pp.345-358, 2003, DMTCS Proceedings
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Nisheeth Vishnoi. Non Uniform Random Walks. Cyril Banderier and Christian Krattenthaler. Discrete Random Walks, DRW'03, 2003, Paris, France. Discrete Mathematics and Theoretical Computer Science, DMTCS Proceedings vol. AC, Discrete Random Walks (DRW'03), pp.345-358, 2003, DMTCS Proceedings. 〈hal-01183925〉

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