Discrete Random Walks, DRW'03
Cyril Banderier and Christian Krattenthaler (eds.)
DMTCS Conference Volume AC (2003), pp. 345358
author:  Nisheeth Vishnoi 

title:  Non Uniform Random Walks 
keywords:  Non uniform random walk 
abstract: 
Given
ε
for each
i
∈ [0,1)
1 < i < n,
a particle performs the following random walk on
{1,2,...,n}
:If the particle is at
n
, it chooses a point uniformly at random (u.a.r.)
from
{1,...,n1}.
If the current position of the particle is
m
(
1<m<n
), with probability
ε
it decides to go back, in which case it chooses a
point u.a.r. from
m
{m+1,...,n}
. With probability
1ε
it decides to go forward, in which case it chooses a
point u.a.r. from
m
{1,...,m1}
. 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 problem arise 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
ε
's:
ε
where
i
= (ni) / (ni+ α · (i1)) ,
1<i<n
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
ε
's. As
α
varies from infinity to 1, this time goes from
Θ(log n)
to
Θ (n).
Finally we study the behavior of the expected convergence time as a function of
ε
. It remains an open question to determine whether
this quantity increases when all
ε
's are increased. We give some preliminary results to
this effect.

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reference:  Nisheeth Vishnoi (2003), Non Uniform Random Walks, in Discrete Random Walks, DRW'03, Cyril Banderier and Christian Krattenthaler (eds.), Discrete Mathematics and Theoretical Computer Science Proceedings AC, pp. 345358 
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