Random Walk and Other Lattice Models - City University of ...

[Pages:45]Random Walk and Other Lattice Models

Christian Benes

Brooklyn College Math Club

Brooklyn College Math Club 04-23-2013

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Outline

1 Lattices 2 Random Walk 3 Percolation

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Lattices

Lattices: Z, Z2

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Lattices

Lattices: Z, Z2 and Z3

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Lattices

More lattices: triangular and honeycomb

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Lattices

More lattices: Archimedean Lattices

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Random Walk

Random Walk

Let Xi be independent random variables satisfying

P {Xi = 1} = p, P {Xi = -1} = 1 - p. Then Sn =

n i =1

Xi

is

called

a

simple random walk on the one-dimensional integer lattice. We will

define S0 = 0 (meaning that the random walk starts at the origin).

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Position of Walker

Random Walk

14

12

10

8

6

4

2

0

-2

-4

0

10

20

30

40

50

60

70

80

90

100

Number of Steps

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