Complexity Explorer Santa Few Institute

Introduction to Renormalization

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4.6 Ising Model Simulations » Quiz Solution

1. When beta is at the non-zero, but finite, fixed point, what happens when you coarse-grain:

A. the little islands where nodes all share the same value get smaller and smaller, and it looks like television snow.

B. fluctuations away from a uniform +1 or uniform -1 state get smaller and smaller, and the system is dominated by one state or the other.

C. little islands shrink, but there are larger islands of correlation that scale down to take their place. There are fluctuations on all scale, in a uniform "fractal" pattern.

D. none of the above.


Answer: (C). At this critical point, the system maintains the same level of correlation (beta value) no matter how far you zoom out. We'll see this in action in the final lecture of this unit.


2. The three models in Douglas Ashton's simulation look pretty similar in terms of their correlations at first. What happens to make them look different?

A. we evolve them forward dynamically, and they move to different configurations.

B. they look the same in a little zoomed in patch, but if we zoom out and look on larger scales, we can see that they have very different large-scale properties

C. we alter them by hand, showing the effect of changing beta.

D. none of the above


Answer: (B). In each of the three cases, Douglas has run a simulation and just taken one image -- basically, sampling the joint probability distribution for each of the three models. There's no dynamical evolution shown here, and nobody is changing the models by hand to become more different. Locally, the models look very similar. But if we zoom out to larger scales -- i.e., if we repeatedly do a decimation transformation, dropping half of the points out and moving the others closer together (so more and more of the image can fit on the screen), we see that they're very different on large scales. The effective beta parameter goes either to zero (totally correlation), or to infinity (totally decorrelated), or (if precisely poised at the critical temperature) it remains the same, self-similar on all scales.