Complexity Explorer Santa Few Institute

Introduction to Renormalization

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2.4 A puzzle: origin of the slippy counter » Quiz Solution

What is funny about the two-state slippy counter when epsilon is less than 1/2 (but greater than zero)?

A. the fixed point is no longer found in a one-dimensional subspace.

B. it could not have been derived from a coarse-grained version of a two-state Markov Chain at a more rapid timescale.

C. in this parameter range, the counter can not be described as Markov chain.

D. (A) and (B)


Answer: (B). In order for the system to oscillate A, B, A, B... at the coarse-grained timescale, there needs to be a more complex process operating at the fine-grained timescale that "keeps track" of whether we're at an even or odd-numbered time step. (A) is incorrect because, as long as epsilon is greater than zero, T will itself coarse-grain just fine: it will end up somewhere on the P(A|A)=P(A|B) line. (C) is incorrect because while there’s no good two-state microtheory for the finer-grained version, the counter at this coarse-graining scale is just fine — we can (and do!) write down the transition matrix. If you want to dive into the microtheory problem further, take a look at the problem set questions for Unit #2 at, also located in the homework problem and solution sets in the final unit of this tutorial.