Complexity Explorer Santa Fe Institute

Introduction to Renormalization

Lead instructor: Simon DeDeo

Your progress is not being saved! Enroll now or log in to track your progress or submit homework.
Please return to the Courses page to find current offerings.

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 http://bit.ly/SFIrenorm, also located in the homework problem and solution sets in the final unit of this tutorial.