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zero. Therefore we have
�∞
dl ilζ −l2 /2
φ(ζ, �) → φ(ζ ) =
e
−∞ 2π
2 = 2
2.1 e−ζ /2
√
.
2π Breakdown of the CLT for decaying walks
The PDF of X∞ for a = 0.99 Appendix A shows a simple C++ code to simulate 105 walkers from the decaying PDF. Figure
1 shows the PDF of x component of this distribution, and ﬁgure 2 shows a comparison to the M. Z. Bazant – 18.366 Random Walks and Diﬀusion – Problem Set 3 Solutions 3 0.09
0.08
0.07 P∞ (x) 0.06
0.05
0.04
0.03
0.02
0.01
0
30 20 10 0 10 20 30 x
Figure 1: Simulated PDF of X∞ for a = 0.99 based on 109 trials.
formulae calculated in question 1. We see that the two curves match to a very high degree of
accuracy, particularly in the central region. We know however that this match will not continue
�
forever: since X∞ is bounded, its PDF will be uniformly zero outside of some region. However a
Gaussian features nonzero probabilities everywhere. 2.2 The exact PDF of X∞ for a = 1/2 �
�
To ﬁnd the PDF of X∞ for a = 1/2 we ﬁrst consider the change of variables X = (x, y ) → R = (r, s)
given by
r=
s= 1+x+y
2 1 + x − y .
2 �
In the transform variables, the walk starts from R0 = (1/2, 1/2) and the PDF of the nth step is
given by
1
pn (r, s) = (δr,−1/2n+1 + δr,1/2n+1 )(δs,−1/2n+1 + δs,1/2n+1 ).
4
From this from, it is clear that the variables r and s are independent. We see that the N th step of
the r component can be written as
N RN 1 � 2In − 1
=+
2
2n+1
n=1 M. Z. Bazant – 18.366 Random Walks and Diﬀusion – Problem Set 3 Solutions 0.1 4 Simulation
Gaussian 0.01 P∞ (x) 0.001
0.0001
1e05
1e06
1e07
1e08
30 20 10 0 10 20 30 x
Figure 2: Simulated PDF of X∞ for a = 0.99 based on 109 trials on a log scale, compared with a
Gaussian curve with a variance calculated using the formula derived in question 1.
where the In ’s are independent random variables which take values 0 and 1 with equal probability.
This can be rewritten as
RN N
N
� In
1 � 1 −
+
2
2n+1
2n
n=1
n=1
�
�
N
� In
1
1 1 − 21
N
−2
+
22
2n
1 − 1
2 = = n=1 N
� In
1 +...
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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
 MartinBazant

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