Unformatted text preview: ks that did not achieve
ﬁrst passage in 105 steps were prematurely terminated. Figure 1 shows the computed values of f (n)
for low values of n, while ﬁgure 2 shows a logarithmic plot highlighting the asymptotic behavior. For
large n, the curve becomes almost linear, and by applying regression over the range 103 ≤ n ≤ 105
we ﬁnd that f (n) ∝ n−1.50061 , which appears to match the theoretical result of f (n) ∝ n−3/2 for
the Bernoulli walk.
Figure 3 shows a plot of the survival probability S (n). Again, this curve appears to become
linear for large n, and by applying regression we ﬁnd S (n) ∝ n−0.500147 . Since we have a negative
exponent, we see that S (n) → 0 as n → ∞, and thus our expected probability of return is 1. 2.2 Biased Cauchy walk The GSL code was also run for d = 1.0 and d = −1.0. The same number of trials were used for
d = −1.0 as for the unbiased case, but 2 × 108 trials were used for d = 1.0, since many of these
walks took a great number of steps to complete, thus creating a larger computational overhead.
The computed f (n) for low values of n is shown in ﬁgure 1, while a log plot showing the asymptotic
behavior is shown in ﬁgure 2. We see that for large n, the curves in this ﬁgure become almost
linear. Applying linear regression over the range 103 ≤ n ≤ 105 shows that f (n) ∝ n−1.75027 for
d = −1.0, and f (n) ∝ n−1.25015 for d = 1.0.
The survival probability function S (n) for these cases is shown in ﬁgure 3. Again, these curves
appear linear as n increases, and by applying linear regression we ﬁnd that S (n) ∝ n−0.750051 for
d = −1.0 and S (n) ∝ n−0.250044 for d = 1.0. Thus we expect that the probability of return is
always 1, even for the case of positive drift, although some of these walks may take a very long
time to return. Nevertheless, this ﬁts with our intuition, since Cauchy walkers are capable of taking
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This note was uploaded on 01/23/2014 for the course MATH 18.366 taught by Professor Martinbazant during the Fall '06 term at MIT.
- Fall '06