In the first picture for each rule, replacements are made at randomly chosen steps, while in the second picture, they are in a sense always made at the earliest possible step. … In the first picture for each rule, the replacements are performed essentially at random.

So in the end, if one manages to find the ultimate rules for the universe, my expectation is that they will give rise to networks that on a small scale look largely random. But this very randomness will most likely be what for example allows a definite and robust value of 3 to emerge for the dimensionality of space—even though all of the many complicated phenomena in our universe must also somehow be represented within the structure of the same network.

In most cases, the typical behavior produced by the model looks considerably more random than the data. And indeed at some level this is hardly surprising: for by using a probabilistic model one is in a sense starting from an assumption of randomness.

Random causal networks If one assumes that there are events at random positions in continuous spacetime, then one can construct an effective causal network for them by setting up connections between each event and all events in its future light cone—then deleting connections that are redundant in the sense that they just provide shortcuts to events that could otherwise be reached by following multiple connections.

Related results [to Central Limit Theorem] Gaussian distributions arise when large numbers of random variables get added together. If instead such variables (say probabilities) get multiplied together what arises is the lognormal distribution Exp[-(Log[x] - μ ) 2 /(2 σ 2 )]/(Sqrt[2 π ] x σ ) For a wide range of underlying distributions the extreme values in large collections of random variables follow the Fisher–Tippett distribution Exp[(x - μ )/ β ] Exp[-Exp[(x - μ )/ β ]]/ β related to the Weibull distribution used in reliability analysis. For large symmetric matrices with random entries following a distribution with mean 0 and bounded variance the density of normalized eigenvalues tends to Wigner's semicircle law 2Sqrt[1 - x 2 ] UnitStep[1 - x 2 ]/ π while the distribution of spacings between tends to 1/2( π x)Exp[1/4(- π )x 2 ] The distribution of largest eigenvalues can often be expressed in terms of Painlevé functions.

Apparent randomness is common in physiological processes such as twitchings of muscles and microscopic eye motions, as well as in random walks executed during foraging. My suspicion is that just as there appear to be small collections of cells—so-called central pattern generators—that generate repetitive behavior, so also there will turn out to be small collections of cells that generate intrinsically random behavior.

For each complete update of a rule 90 sequential cellular automaton, the pictures below show results with (a) left-to-right scan, (b) random ordering of all cells, the same for each pass through the whole system, (c) random ordering of all cells, different for different passes, (d) completely random ordering, in which a particular cell can be updated twice before other cells have even been updated once. … The following will update triples of cells in the specified order by using the function f : OrderedUpdate[f_, a_, order_]:= Fold[ReplacePart[ #1, f[Take[#1, {#2 - 1, #2 + 1}]], #2] &, a, order] A random ordering of n cells corresponds to a random permutation of the form Fold[Insert[#1, #2, Random[Integer, Length[#1]] + 1] &, {}, Range[n]]

But often—just as in so many other kinds of systems—the behavior is instead complex and seemingly quite random.

Tests of randomness The statistical tests that I have performed include the eight listed on page 1084 .

But things that are too random also do not normally seem to be associated with the exercise of a will. Thus for example continual random twitching in our muscles is not normally thought to be a matter of human will, even though some of it is the result of signals from our brains.