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Home , Freeman Dyson

thesis Assume nothing wrote famously about the influence, or be useful for making predictions. unreasonable effectiveness of Wigner expressed the With modern technology, the automated in . His own ideas illustrate the point view that a good deal screening of enormous volumes of high- as well as any others. In 1956, at a conference dimensional data for interesting correlations on neutron in Gatlinburg, Tennessee, might be learned by has become a key tool of science, especially Wigner spoke on the levels of large, making a virtue of in molecular , environmental science, complex nuclei such as , for which economics and finance. data were just becoming available. He theoretical ignorance. But this practice faces a fundamental expressed the view that a good deal might problem. As the number of variables being be learned by making a virtue of theoretical Montgomery told Dyson that he’d been studied grows, the number of pairs of ignorance, and simply assuming random studying the pair-correlation function for variables among which correlations might values for the elements of the Hamiltonian the zeros of ζ(s), arriving at an estimate for be found grows even faster. In this case, matrix, which in quantum theory its asymptotic form of 1 − (sin(πx)/πx)2, straightforward calculation is almost certain determines the nuclear energy levels through which Dyson recognized immediately as the to detect correlations that look significant but its eigenvalues. pair-correlation function for the eigenvalues really aren’t. Suppose, for example, you have Wigner showed that this ‘simple of random matrices of the form studied by M input variables and N output variables — minded’ approach could establish baseline Wigner. To this day, it is not known why you might think of inflation, employment, expectations for the spacing of nuclear levels the same equation should turn up in the a stock market index and any other number in the absence of any other knowledge. distributions of both nuclear energy levels of economic quantities. Suppose you have “The question”, he noted, “is simply, what and the roots of ζ(s), and by implication in series for all of these quantities over are the distances of the characteristic the distribution of the prime numbers, but time T, and you look for correlations between values of a symmetric matrix with random computations suggest that the connection inputs and outputs. Then, even if these coefficients?” Wigner’s result, worked out is remarkably precise. variables were all independent with Gaussian in a few lines of algebra, gave a probability has computed the fluctuations, one would expect that the largest distribution of the form p(x) ~ xexp(−ax 2), locations of zeros of ζ(s) along the critical observed correlation — if you calculate them with x being the energy spacing and a = π/4, line, identifying as many as 1023 zeros and all — will be of the order (ln(MN))/T (see, for thereby pointing to a dearth of levels of finding a near-perfect agreement between example, J.-P. Bouchaud et al. Eur. Phys. J. B similar energy. The result contrasted sharply the predicted and measured correlations. The 55, 201–207; 2007), which gets large for any with what might have been the expected zeroes of ζ(s) effectively repel one another fixedT given enough variables to study. Poisson form, p(x) ~ exp(−x), for which much as do nuclear energy levels (see, for This has become known as the ‘curse of x = 0 would be a maximum rather than example, B. Hayes, American Scientist dimensionality’. Automation makes it easy a minimum. July–August 2003). to study everything, and too easy to find Wigner’s random-matrix approach was One intriguing possibility is that this meaningless patterns in doing so. Fortunately, indeed effective, as experiments over the connection points to some mysterious however, here too Wigner’s random-matrix next few years — especially those probing relationship between ζ(s) and the mathematics approach has proven useful, in this case for resonances in neutron scattering from of quantum theory. Even before the advent separating what is meaningful from what is uranium — showed a remarkably close fit to of quantum theory, nonsense. Correlations calculated between his predicted curve. Much more surprising, and George Pólya truly independent variables should produce perhaps, has been the enormous influence independently suggested — apparently a random M × N matrix. Studying the typical of random-matrix theory since then in inspired by some loose analogy to the spectral features of such random matrices areas ranging from pure mathematics to the discrete energy spectra recently discovered helps to establish baseline expectations for study of financial risk. All, one might say, by in atoms — that the zeroes of ζ(s) might be the correlations likely to be produced by assuming almost nothing and then working given by the eigenvalues of some unknown statistical fluctuations alone. In particular, out the consequences. Hermitian matrix. If so, then some Hermitian studies have shown that the eigenvalues of In the early 1970s, Freeman Dyson operator of the kind familiar to all physicists truly random matrices tend to be confined and Hugh Montgomery stumbled over may determine the positions of the Riemann within an interval with sharp edges. Hence, a connection between Wigner’s idea and zeroes and, with them, the distribution of the eigenvalues in empirical data found to stand pure mathematics. Montgomery had been primes — certainly a strange link between out and away from these edges should signify studying the famous , physics and mathematics. real and meaningful correlations. ∞ s defined asζ (s) = nΣ = 11/n , which has some zeros Random-matrix theory has now been Undoubtedly, further uses will be found located along the ‘critical line’ defined by applied widely in statistics, condensed- for Wigner’s random matrices, which in truth s = 1/2. It is unknown whether all zeros lie physics and elsewhere. But its practically were invented long before Wigner by people on this line — the assertion that they are is most important applications may be still to interested in correlations in empirical data. known as the — or how come, especially in extracting meaningful But Wigner’s 1956 stimulated the deeper the zeros are distributed. But it is known that from huge quantities of data. mathematical analysis of such matrices, with the distribution of such zeros can be linked The most obvious way to look for cause- repercussions that almost anyone would call to the distribution of prime numbers, and and-effect relationships in data, of course, surprising — even, perhaps, unreasonable. ❐ hence holds fundamental importance for is to identify correlations among variables, . which may point to mechanisms of causal MARK BUCHANAN nature physics | VOL 5 | DECEMBER 2009 | www.nature.com/naturephysics 855

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