The boundary that separates repetition on the left from randomness on the right moves an average of about 0.252 cells to the left at every step (compare page 949 ).

I suspect, however, that even though perturbations may determine the precise point at which turbulence begins, intrinsic randomness generation will dominate once turbulence has been initiated.

Self-gravitating systems The observed existence of structures such as galaxies might lead one to think that any large number of objects subject to mutual gravitational attraction might not follow the Second Law and become randomized, but might instead always form orderly clumps.

Hump m in the picture of sequence (c) shown is given by FoldList[Plus, 0, Flatten[Nest[Delete[NestList[Rest, #, Length[#] - 1], 2]&, Append[Table[1, {m}], 0], m]] - 1/2] The first 2 m elements in the sequence can also be generated in terms of reordered base 2 digit sequences by FoldList[Plus, 1, Map[Last[Last[#]]&, Sort[Table[{Length[#], Apply[Plus, #], 1 - #}& [ IntegerDigits[i, 2]], {i, 2 m }]]]] Note that the positive and negative fluctuations in sequence (f) are not completely random: although the probability for individual fluctuations in each direction seems to be the same, the probability for two positive fluctuations in a row is smaller than for two negative fluctuations in a row.

(The Schrödinger equation is like a diffusion equation in imaginary time, so the path integral for it can be thought of as like an enumeration of random walks. The idea of describing random walks with path integrals was discussed from the early 1900s.) … By the early 1980s there was then extensive work on lattice gauge theories in which the path integral (in Euclidean space) was approximated by randomly sampling discretized field configurations.

Starting in the 1980s after the development of lattice gauge theories, simulations of random surfaces and higher-dimensional spaces set up in this way were done—often using so-called dynamic triangulation based on random sequences of generalized Alexander moves from page 1038 .

For it is almost certain that experiments on, say, some specific cellular automaton whose rule has been picked at random from a large set will never have been done before.

Among those with roughly exponential growth but seemingly random fluctuations are s[s[s[s]]][s][s][s][k] , s[s[s]][s][s[s][s]][k] and s[s[s[s]][s][s]][k][s] .

At first it was thought that the visual system might be sensitive only to the overall autocorrelation of an image, given by the probability that randomly selected points have the same color.

Following work in the 1980s and 1990s by Mikhael Gromov and others, it is also known that for groups with randomly chosen underlying rules, the Cayley graph is usually either finite, or has a rapidly branching tree-like structure.