If x agrees with k of its four neighbors the ratio is

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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 Metropolis-Hastings 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).

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