Exact sampling from non-attractive distributions using summary states

Animations

Warning: some of the pictures are rather reminiscent of Reservoir Dogs!

These animations show the summary state method in action for an Ising model on a periodic triangular lattice. Black and white indicate up and down spins, and red indicates an uncertain spin (i.e., ?).

Note that in ferromagnetic cases, the animation is the same as it would be using the Propp and Wilson algorithm, with red indicating spins that differ between the bounding states. The antiferromagnetic case is non-attractive, and hence is beyond the scope of the original Propp and Wilson algorithm.

f_tri_4.2.gif (1056K GIF): ferromagnetic, T=4.2
a_tri_small_5.0.gif (44K GIF): antiferrmagnetic, T=5.0, 18x18
a_tri_5.0.gif (1272K GIF): antiferromagnetic, T=5.0
a_tri_globalfield_3.0.gif (144K GIF): antiferrmagnetic, T=3.0, global magnetic field H=6

The following animations show the effect of introducing a single uncertain spin into a nearly equilibrated lattice of an antiferromagnetic Ising system. Spins were set up or down with equal probability, and the system was simulated forward in time for 100 iterations. Then a randomly chosen spin was made uncertain, and the simulation was continued. Each frame shows the result of one iteration, which involves updating every spin in the lattice. Note that the threshold for convergence of the algorithm is T=4.83.

diverge_a_2.0.gif (192K GIF): T=2.0, 100 frames
diverge_a_4.0.gif (488K GIF): T=4.0, 100 frames
diverge_a_4.8.gif (1567K GIF): T=4.8, 300 frames

Note: The delay between frames is set short, so if the animations seem to run sluggishly, you may want to download them to a local machine and then view them.

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Last modified: Mon May 8 11:04:13 2000