8.4
 An Exact Solution for the Dimension of a Random Walk on a Sierpinski
Gasket
We can derive the
exponent s in Eq. 3 directly
by using equations similar to Eq. 1.
Begin by applying a test to T_{C'B'}. Let us imagine
that we execute four separate trials for randomly walking from C^{'}
to B^{'}. From C^{'} there are four paths to
B^{'}. Refer to Figure 8.2(a). On
average, we expect that: on one trial we go directly to the leftmost B^{'}
in time T_{AB} (the same as the
time to go between the adjacent point A and B); on one trial we go directly
to the rightmost B^{'}; on one trial we go to the leftmost point
B in time T_{AB}, and then take time
T_{BB'} to arrive at a B^{'};
and, finally, on average one of the walks will first take us the rightmost point
B in time T_{AB}, and from there
it will take T_{BB'} to arrive at a B^{'}.
Express this analysis
as an equation:
In
short, our second equation relating the average internal
times is:
T_{C'B'} = T_{AB}+ 
T_{BB'}
2

. 

(8.17) 
Finally,
we apply the same logic to four trial random walks which are initiated
at a point B. On average: one walk will take time T_{AB} to arrive at the
other point B, and then the average time T_{BB'} to arrive at one
of the points B^{'}; one walk will take time
T_{AB} to move to point
A, and then time T_{AB'} to arrive at B^{'};
one walk will take time T_{AB} to arrive at C^{'},
and then time T_{C'B'} to arrive at a B^{'};
and, one walk go directly to B^{'} in time T_{AB}. Summing these four
ways to go, we have (with the parenthesized terms in order of the
above description):
4T_{BB'} 


(T_{AB}+ T_{BB'}) + (T_{AB}+ T_{AB'}) 



(T_{AB}+ T_{C'B'}) + (T_{AB}). 

(8.18) 

Simplifying
this equation:
3T_{BB'} = 4T_{AB}+ T_{AB'}+ T_{C'B'}. 

(8.19) 

Q8.17: Do your averaged quantities computed in Task 1 in HandsOn 8.1
satisfy these equations? Check the numbers. Do you get better agreement
using average times as found averaging over the data of all students?


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33  Computing the Resistance of the Sierpinski Gasket