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 TC'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 TAB (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 TAB , and then take time TBB' to arrive at a B'; and, finally, on average one of the walks will first take us the rightmost point B in time TAB , and from there it will take TBB' to arrive at a B'.

Express this analysis as an equation:
 4 TC'B' = the average length of time to execute four trials fromC' to B' = TAB (the time to go directly to the leftmost B') + TAB (the time to go directly to the rightmost B') + TAB (the time to go from C' to leftmost B (i.e., 1 step)) + TBB' (the average time to get to B' from B) + TAB (the time to go from C' to rightmost B (i.e., 1 step)) + TBB' (the average time to get to B' from B) = TAB + 2 TBB' . (8.16)

In short, our second equation relating the average internal times is: TC'B' = TAB + TBB' 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 TAB to arrive at the other point B, and then the average time TBB' to arrive at one of the points B'; one walk will take time TAB to move to point A, and then time TAB' to arrive at B'; one walk will take time TAB to arrive at C', and then time TC'B' to arrive at a B'; and, one walk go directly to B' in time TAB . Summing these four ways to go, we have (with the parenthesized terms in order of the above description):
 4 TBB' =
 ( TAB + TBB' ) + ( TAB + TAB' )
 +
 ( TAB + TC'B' ) + ( TAB ).
(8.18)
Simplifying this equation:
 3 TBB' = 4 TAB + TAB' + TC'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?

 Q8.18: Equations 1, 17, and 19 are a system of three simultaneous equations in the three unknowns TBB , TAB' , and TC'B' in terms of the known TAB . Try solving these equations to prove Eq. 4.

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