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Unformatted text preview: spins
have only three neighbors. There are several options for dealing with this: (i)
we consider the spins outside to be 0, or (ii) we could specify a ﬁxed boundary
condition such as all spins +.
The sum is largest in case (i) when all of the spins agree or in case (ii) when
P
all spins are +. These conﬁguration minimizes the energy H =
x,y ⇠x ⌘x ⌘y
but there many more conﬁgurations one with a random mixture of +’s and ’s.
It turns out that as increases the system undergoes a phase transition from
a random state with an almost equal number of +’s and ’s to one in which
more than 1/2 of the spins point in the same direction.
Z ( ) is di cult to compute so it is fortunate that only the ratio of the
probabilities appears in the MetropolisHastings recipe. For the proposed jump
distribution we let q (⇠ , ⇠ x ) = 1/(2L + 1)2 if the two conﬁgurations ⇠ and ⇠ x
di↵er only at x. In this case the transition probability is
⇢
⇡ (⇠ x )
p(⇠ , ⇠ x ) = q (⇠ , ⇠ x ) min
,1
⇡ (⇠ )
Note that the ratio...
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This document was uploaded on 03/06/2014 for the course MATH 4740 at Cornell University (Engineering School).
 Spring '10
 DURRETT
 The Land

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