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Unformatted text preview: stings
algorithm has its roots in statistical physics. A typical problem is the Ising
model of ferromagnetism. Space is represented by a two dimensional grid ⇤ =
{ L, . . . L}2 . If we made the lattice three dimensional, we could think of the
atoms in an iron bar. In reality each atom has a spin which can point in some
direction, but we simplify by supposing that each spin can be up +1 or down
1. The state of the systems is a function ⇠ : ⇤ ! { 1, 1} i.e., a point in the
product space { 1, 1}⇤ . We say that points x and y in ⇤ are neighbors if y is
one of the four points x + (1, 0), x + ( 1, 0), x + (0, 1), x + (0, 1). See the
picture:
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y+
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+yxy
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Given an interaction parameter , which is inversely proportional to the temperature, the equilibrium state is
!
X
1
⇡ (x) =
exp
⇠x ⇠y
Z( )
x,y ⇠x
where the sum is over all x, y 2 ⇤ with y a neighbor of x, and Z ( ) is a constant
that makes the probabilities sum to one. At the boundaries of the square...
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 Spring '10
 DURRETT
 The Land

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