Unformatted text preview: om walk of a knight on a chess board. A chess
board is an 8 by 8 grid of squares. A knight moves by walking two steps in one
direction and then one step in a perpendicular direction. • • • •
⇥ • •
• • By patiently examining all of the possibilities, one sees that the degrees of
the vertices are given by the following table. Lines have been drawn to make
the symmetries more apparent.
2
3
4
4
4
4
3
2 3
4
6
6
6
6
4
3 4
6
8
8
8
8
6
4 4
6
8
8
8
8
6
4 4
6
8
8
8
8
6
4 4
6
8
8
8
8
6
4 3
4
6
6
6
6
4
3 2
3
4
4
4
4
3
2 The sum of the degrees is 4 · 2 + 8 · 3 + 20 · 4 + 16 · 6 + 16 · 8 = 336, so the
stationary probabilities are the degrees divided by 336.
This problem is boring for a rook which has 14 possible moves from any
square and hence a uniform stationary distribution. In exercises at the end of
the chapter, we will consider the other three interesting examples: king, bishop,
and queen. 1.6.3 Reversibility Let p(i, j ) be a transition probability with stationary distribut...
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This document was uploaded on 03/06/2014 for the course MATH 4740 at Cornell.
 Spring '10
 DURRETT
 The Land

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