Unexpected Appearances of Random Matrices

Statistics

The investigation of random matrices was initiated by the early works of Wishart [1] in 1928, when he introduced the probability distribution of random matrices now bearing his name. The Wishart distribution generalizes the sum of squares of normal variables (a.k.a. {\chi^2}-distribution) and can be used for the maximum-likelihood estimation of the covariance matrix of i.i.d. vectors drawn from a multivariate normal distribution. The Wishart ensemble {W_N(V,n)} has {n} degrees of freedom and is characterized by a fixed {N\times N} real positive definite matrix {V}. It consists of real symmetric, positive definite matrices of size {N\times N} with the probability density function

\displaystyle P(X)=\bigg[2^{\tfrac{nN}{2}}\pi^{\tfrac{N(N-1)}{4}}\det(V)^{\tfrac{n}{2}}\prod_{k=1}^N\Gamma\bigg(\frac{n-k+1}{2}\bigg)\bigg]^{-1}\det(X)^{\tfrac{n-N-1}{2}}e^{-\tfrac{1}{2}\mathrm{tr}(V^{-1}X)}, \ \ \ \ \ (1)

where {\Gamma} denotes the gamma function {\Gamma(t)=\int_0^\infty x^{t-1}e^{-x}\,dx}. In the special case {N=1}, {V=1} the density (1) takes the form that of the {\chi^2}-distribution.

Nuclear Physics

The first surprising employment of random matrices was in nuclear physics. It was Wigner in his 1951 paper [2], who proposed “a new kind of statistical physics” based on random matrices to model certain properties of excited states of heavy nuclei. His intent was to gain some meaningful information about the high-energy states of complex systems by considering ensembles of different Hamiltonians, which are related by their ‘symmetry’. The theoretical footing of Wigner’s idea is Bohr‘s Compound-nucleus model from 1936 [3], which was designed to explain nuclear reactions, such as neutrons bombarding a nucleus of not too small atomic weight, as two-stage events:

  1. Union of the colliding bodies into a single unit, the so-called compound nucleus. This takes about {10^{-21}} seconds.
  2. Disintegration of the unstable compound into an ejected small particle and a product nucleus.

Between the two events there is a relatively long period of time, typically {10^{-19}} to {10^{-15}} seconds. To demonstrate his model Bohr made a toy model (see Figure 1 below). A number of billiard balls in a shallow basin represent the target nucleus. The struck ball enters the basin hitting other particles and quickly distributing its energy among them. If the basin and the particles are regarded as perfectly smooth and elastic, there will be a point when some particles close to the basin’s edge receive so much energy that they exit the compound.

Bohr's toy model for compound-nucleus scattering.
Bohr’s toy model for compound-nucleus scattering.

At a first glance, Wigner’s proposal seems baffling for (at least) two reasons. First, the energy levels and eigenstates of a quantum mechanical system is completely determined by its Hamiltonian, therefore one could ask: “How could a statistical approach be of any use?” One reason is that energy levels of highly excited states of heavy atoms become so dense that precise information about individual levels cannot be obtained using exact methods. Second, the usage of an ensemble of Hamiltonians to describe a single system. This is in opposition with standard statistical mechanics, where one considers copies of identical physical systems, all governed by the same Hamiltonian but differing in initial conditions, and calculates thermodynamic functions by averaging over this ensemble. As Dyson explains in [4]:

“We picture a complex nucleus as a ‘black box’ in which a large number of particles are interacting according to unknown laws. The problem then is to define in a mathematically precise way an ensemble of systems in which all possible laws of interaction are equally probable.”

Wigner considered {N\times N} real symmetric matrices {H=(H_{ij})} with the algebraically independent entries {H_{ij}}, {i\leq j} being also statistically independent normal variables with the probability distribution

\displaystyle P(H)dH=C\ e^{-\tfrac{1}{2}\mathrm{tr}(H^2)}\prod_{i\leq j}dH_{ij}, \ \ \ \ \ (2)

where {C} is a normalization constant. This is called the Gaussian Orthogonal Ensemble (GOE). The word ‘orthogonal’ is meant to indicate that the distribution (2) is invariant under conjugation with orthogonal matrices, that is {H\rightarrow RHR^\top} with {R^\top=R^{-1}}. Various spectral statistics of this ensemble can be calculated, such as the joint eigenvalue distribution, nearest neighbour distribution, correlation functions, etc. When compared with experimental data, the effectiveness of GOE is striking (see Figure 2).

Nearest neighbour distribution for the Nuclear Data Ensemble comprising 1726 level spacings of different nuclei with the same spin and parity.
Nearest neighbour distribution for the Nuclear Data Ensemble comprising 1726 level spacings of different nuclei with the same spin and parity.

In 1962, Dyson showed [4] that there are three invariant ensembles of random matrices: orthogonal, unitary, and symplectic. Again, orthogonal, unitary, and symplectic refers to the symmetry groups. Those with Gaussian components are called GOE, GUE, and GSE. The joint distribution of their eigenvalues can be written as

\displaystyle P_\beta(\lambda)=C_\beta\exp\bigg(-\frac{1}{2}\beta\sum_{j=1}^N\lambda_j^2\bigg)\prod_{1\leq j<k\leq N}|\lambda_j-\lambda_k|^\beta, \ \ \ \ \ (3)

with {\beta=1} for orthogonal, {\beta=2} for unitary, and {\beta=4} for symplectic. For more on the topic of random matrices in physics see, for example Mehta’s book [5] or the 7th John von Neumann Lecture given by Wigner [6].

Number Theory

An astounding connection was discovered between analytic number theory and random matrices in 1972, when Montgomery gave a talk in Princeton at the Institute for Advanced Study about his results on the non-trivial zeros of the Riemann zeta function

\displaystyle \zeta(z)=\sum_{n=1}^\infty n^{-z}. \ \ \ \ \ (4)

He explained how the zeros on the critical line, viz. {z_k=1/2+\mathrm{i}x_k} such that {\zeta(z_k)=0}, tend to repel each other and that he conjectured a formula for the distribution of the gaps between them. After the talk during teatime Montgomery spoke to Dyson, who didn’t attend the talk. Here is Montgomery’s recollection of the conversation:

“Freeman Dyson was standing across the room. I had spent the previous year at the Institute and I knew him perfectly well by sight, but I had never spoken to him. Chowla said: ‘Have you met Dyson?’ I said no, I hadn’t. He said: ‘I’ll introduce you.’ I said no, I didn’t feel I had to meet Dyson. Chowla insisted, and so I was dragged reluctantly across the room to meet Dyson. He was very polite, and asked me what I was working on. I told him I was working on the differences between the non-trivial zeros of Riemann’s zeta function, and that I had developed a conjecture that the distribution function for those differences had integrand {1-[\sin(\pi r)/\pi r]^2}. He got very excited. He said: ‘That’s the form factor for the pair correlation of eigenvalues of random Hermitian matrices!’ “

Pair correlation of zeros of {\zeta}. Solid line: GUE prediction. Scatter plot: empirical data based on zeros {z_n}, {10^{12}+1\leq n\leq 10^{12}+10^5}.

For a more detailed description, see [7].

Dynamical Systems and Quantum Chaos

A significant boost to random matrix theory was the discovery its connection with classical and quantum chaos. In the late 70’s and early 80’s the quantum spectra of conservative systems, which have chaotic classical analogues, was explored (e.g. Sinai billiard). Sequences of energy levels of the same symmetry class were generated numerically. In 1984, Bohigas, Giannoni, and Schmit [8] produced a statistically significant set of data consisting of more than 700 eigenvalues of the Sinai billiard. They applied methods of statistical analysis (invented by Dyson and Mehta for problems in nuclear physics) and found that nearest neighbour spacings of levels were distributed like that of the GOE.

Nearest neighbour spacing distribution for the Sinai biliard. The histogram comprises about 1000 consecutive eigenvalues.
Nearest neighbour spacing distribution for the Sinai biliard. The histogram comprises about 1000 consecutive eigenvalues.

They formulated the conjecture that quantum systems with time-reversal symmetry, whose classical counterparts are chaotic, have spectral statistics of the GOE. In the absence of time-reversal symmetry GOE is replaced by GUE. In contrast, it was conjectured by Berry and Tabor in 1977 [9] that if the corresponding classical dynamics is completely integrable, then the nearest neighbour spacing is equal to the waiting time between consecutive events of a Poisson process. Both of these conjectures are still open, but their statement is strongly supported by a vast number of examples.

Consult [10] for an extensive review about random matrix theories and their applications in quantum physics.

References

  1. J. Wishart, The generalised product moment distribution in samples from a normal multivariate population, Biometrika 20A (1-2): 32-52, 1928.
  2. E.P. Wigner, On the statistical distribution of the widths and spacings of nuclear resonance levels, Mathematical Proceedings of the Cambridge Philosophical Society 47, 790-798, 1951.
  3. N. Bohr, Neutron capture and nuclear constitution, Nature 137, 344-348, 1936.
  4. F.J. Dyson, Statistical Theory of the Energy Levels of Complex Systems, I, II & III, J. Math. Phys. 3, 140-175, 1962.
  5. M.L. Mehta, Random Matrices, Vol. 142, Academic press, 2004.
  6. E.P. Wigner, Random Matrices in Physics, SIAM Review 9(1) 1-23, 1967.
  7. P. Bourgade, Tea time in Princeton, Harvard College Math. Review 4, Spring 2012.
  8. O. Bohigas, M.J. Giannoni, and C. Schmit, Characterization of Chaotic Quantum Spectra and Universality of Level Fluctuation Laws, Phys. Rev. Lett. 52(1), 1984.
  9. M.V. Berry and M. Tabor, Level clustering in the regular spectrum, Proc. Roy. Soc. A 356, 375-394, 1977.
  10. T. Guhr, A. Müeller-Groeling, and H.A. Weidenmüeller, Random Matrix Theories in Quantum Physics: Common Concepts, Phys. Rept. 299, 189-425, 1998; arXiv:cond-mat/9707301
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