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

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

## This document was uploaded on 03/06/2014 for the course MATH 4740 at Cornell University (Engineering School).

Ask a homework question - tutors are online