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.

It is known that to achieve this exactly, m must be at the least the number of either positive or negative eigenvalues of the distance matrix for the network, and can need to be as much as n - 1 , where n is the total number of nodes.

The length of the sequence at the n th step grows like λ n , where λ ≃ 1.3 is the root of a degree 71 polynomial, corresponding to the largest eigenvalue of the transition matrix for the substitution system.

For large t , the total number of elements typically grows like λ t , where λ is the largest eigenvalue of m; the relative numbers of elements of each color are given by the corresponding eigenvector.

Fractal dimensions [of additive cellular automata] The total number of nonzero cells in the first t rows of the pattern generated by the evolution of an additive cellular automaton with k colors and weights w (see page 952 ) from a single initial 1 can be found using g[w_, k_, t_] := Apply[Plus, Sign[NestList[Mod[ ListCorrelate[w, #, {-1, 1}, 0], k] &, {1}, t - 1]], {0, 1}] The fractal dimension of this pattern is then given by the large m limit of Log[k,g[w, k,k m + 1 ]/g[w, k, k m ]] When k is prime it turns out that this can be computed as d[w_, k_:2] := Log[k,Max[Abs[Eigenvalues[With[ {s = Length[w] - 1}, Map[Function[u, Map[Count[u, #] &, #1]], Map[Flatten[Map[Partition[Take[#, k + s - 1], s, 1] &, NestList[Mod[ListConvolve[w, #], k] &, #, k - 1]], 1] &, Map[Flatten[Map[{Table[0, {k - 1}], #} &, Append[#, 0]]] &, #]]] &[Array[IntegerDigits[#, k, s] &, k s - 1]]]]]]] For rule 90 one gets d[{1, 0, 1}] = Log[2, 3] ≃ 1.58 .

The statistical distribution of zeros was studied by Andrew Odlyzko and others starting in the late 1970s (following ideas of David Hilbert and George Pólya in the early 1900s), and it was found that to a good approximation, the spacings between zeros are distributed like the spacings between eigenvalues of random unitary matrices (see page 977 ).

For any rule, s n for large n will behave like κ n , where κ is the largest eigenvalue of m .

Page 147 showed how Sin[x] + Sin[ √ 2 x] has nested features, and these are reflected in the distribution of eigenvalues for ODEs containing such functions.

A slightly different but still related approach is to study the density of eigenvalues of the Laplace operator—which can also be thought of as measuring the number of solutions to equations giving linear constraints on numbers assigned to connected nodes.