Stat 150
Stochastic Processes
Spring 2009
Lecture 18: Markov Chains: Examples
Lecturer:
Jim Pitman
•
A nice collection of random walks on graphs is derived from random movement
of a chess piece on a chess board.
The state space of each walk is the set of
8
×
8 = 64 squares on the board:
a
b
c
d
e
f
g
h
1
2
3
4
5
6
7
8
Each kind of chess piece at a state
i
on an otherwise empty chess board has
some set of all states
j
to which it can allowably move.
These states
j
are
the
neighbours of
i
in a graph whose vertices are the 64 squares of the board.
Ignoring pawns, for each of king, queen, rook, bishop and knight, if
j
can be
reached in one step from
i
, this move can be reversed to reach
i
from
j
. Note
the pattern of black and white squares, which is important in the following
discussion.
The King
interior
corner
edge
8 possible moves
3 possible moves
5 possible moves
•
•
•
1
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Lecture 18: Markov Chains: Examples
2
In the graph,
i
←→
j
means that
j
is a king’s move from
i
, and
i
is a king’s move from
j
.
Observe that
N
(
i
) := #( possible moves from
i
)
∈ {
3
,
5
,
8
}
. From general
discussion of random walk on a graph in previous lecture, the reversible
equilibrium distribution is
π
i
=
N
(
i
)
Σ
where
Σ :=
X
j
N
(
j
) = (6
×
6)
×
8 + (4
×
6)
×
5 + 4
×
3
Question
: Is this walk regular? Is it true that
∃
m
:
P
m
(
i, j
)
>
0 for all
i, j
?
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 Spring '09
 Pitman,Jim
 Markov Chains, Markov chain

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