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## ← renorm 7 Conclusion Pt I

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Showing Revision 6 created 11/03/2017 by Charles Fiertz.

1. One of the major themes of this series of
2. this series of lectures has been going on
3. the data side from one level of
4. description to another.
5. From data to data prime with some kind of
6. coarse graining prescription and then
7. asking the question, okay, if this was
8. your model for the data at this scale what
9. is the corresponding model prime for the
10. data at this scale here and this
11. relationship was the one that we
12. understood as the renormalization
13. relationship. And that goes all the way
14. from how a Markov chain coarse grains and
15. flows indeed to the south manifold of the
16. higher dimensional space that the Markov
17. chains originally lived in. It applies
18. just as well to how electrodynamics
19. changes as you go from the finer
20. grain scale, where
21. let's say you can observe electrons on,
22. let's say a scale of 1 millimeter, up to
23. a scale of, let's say, a meter, and that
24. renormalization there, as I indicated
25. could be understood as changing, not the
26. laws of electrodynamics, but just one of
27. its parameters, the electron charge as you
28. moved to different distances. This
29. operation here we've left somewhat
30. ambiguous. In each of the talks I told you
31. a coarse graining operation that we were
32. going to use. And we did the Markov chains
33. and said "okay, you have some finite time
34. resolution" when we came to study the
35. icing model I said "okay, look here's how
36. we're going to decimate, we have our grid
37. and what we're going to do is we're going
38. to take every other particle as you go
39. along the grid in these directions we're
40. going to average over every other particle
41. like that, or rather, trace over every
42. particle like that." The one time where we
43. really started to ask which coarse
44. graining do we want to use was when we
45. came to do the CAs, we looked at Israeli
46. and Goldenfeld's work, where we found is
47. that they were simultaneously solving for
48. the model, the g, that came from the f,
49. but also, solving for the projection
50. function that took the supercells and
51. mapped them into groups, into single cell
52. examples. And so I'll draw an example here
53. of how Goldenfeld and Israeli's projection
54. might work in some case, in fact it takes
55. blank spaces to a blank cell, but if
56. there's one filled-in cell, it always
57. takes it to a filled-in cell at the coarse
58. grain level description. What Israeli and
59. Goldenfeld were doing were simultaneously
60. solving for these two objects. And when
61. they did that, one of the things that we
62. talked quite a bit about was that they
63. found that in fact, Rule 110 could indeed
64. be efficiently coarse-grained. And that's
65. kind of remarkable, right? It's sort of
66. like saying, "you know, like, yeah, you
67. know your clock speed is 5 GHz and you
68. have you know, memory of, you know, 16 GB
69. but actually I can do what you think you
70. want to do, I could do it in half the
71. memory and half the time." Now, when we
72. actually came to look at what the coarse
73. graining was doing for Rule 110, we were
74. much less impressed. And for example, one
75. of the kinds of coarse grainings that
76. Israeli and Goldenfeld discovered was the
77. Garden of Eden coarse graining, which
78. turned out to be incredibly trivial. What
79. it did was it took a certain subset of
80. supercells that could never be produced by
81. Rule 110, not in fact blocks of two, they
82. had to go to a longer set of blocks to
83. find them. But they found these Garden of
84. Eden supercells and then projected the
85. whole world into Garden of Eden versus not
86. Garden of Eden, you know post-fall, right?
87. And then by projecting them into those two
88. spaces, then they could actually map
89. Rule 110 onto Rule 0. And yet they
90. satisfied what they wanted, and which
91. seems like a natural thing to satisfy,
92. which is the commutation of the diagram.
93. Right, the diagram they wanted to commute
94. was if you evolve on the fine grain scale
95. and project. If you use the f operation
96. twice and then project, it's the same as
97. using the projection and then the g
98. operation once. So these commuted and yet,
99. the answer was somewhat unsatisfying.
100. In the case of the icing model, we had
101. this goal, our goal was secretly to figure
102. out what was going on with phase
103. transitions in the sort-of two-dimensional
104. grid where all of a sudden at some
105. critical point you found that the whole
106. system coordinated. And so in the end they
107. said "you know, look, this was not the
108. world's greatest coarse graining, because
109. you couldn't quite get a solution, but it
110. was good enough." Always what's happening
111. in each of these stories, the Markov
112. chain, the cellular automata, the icing
113. model, the Krohn-Rhodes Theorem is that
114. secretly we have some idea of what we want
115. the data to do for us, and therefore, we
116. have some idea of what we want this
117. projection operator to be. And in a subset