Literature

• M. L. Mehta, Random matrices
• K. Życzkowski et al., Level Spacing Distribution Revisited
• J. Tworzydło, lecture Computer modeling of physical phenomena
• D. Ullmo, Bohigas-Giannoni-Schmit conjecture, Scholarpedia
• Hans A. Weidenmüller, Random Matrices in Physics
• Wolfram Mathematica Documentation: Random Matrices
• Giacomo Livan, Marcel Novaes, Pierpaolo Vivo Introduction to Random Matrices, Theory and Practice
• T. Tao Topics in Random Matrix Theory

Talk outline

1. Introduction – history and basic idea behind random matrix theory
2. Atomic nuclei deciphered – first application of random matrices in physics: statistical properties of energy levels in atomic nuclei explained with so-called Gaussian ensembles
3. Quantum chaos – an amazing conjecture linking random matrix theory with chaotic systems
4. Summary

Energy levels in nuclei

Idea

Random matrix theory

Study of eigenvalues and eigenvectors of matrices,
whose elements are random variables.

• 1930's: first appearance – mathematical statistics
• 1950's: Wigner – application to nuclear physics
• 1960's: Dyson – huge contributions
• 1980's: Bohigas, Giannoni, Schmit – conjecture linking chaos in quantum regime with random matrices
• 1980's-now: explosive growth of new applications, such as number theory (Montgomery, Dyson)

Idea

In quantum mechanics systems are characterised by their Hamiltonians, represented by Hermitian matrices, and their states and energies are eigenvectors and eigenvalues of these matrices.

Idea

Statistical ensemble of matrices, each corresponding to some system. Expectation: very few systems' properties will deviate from that of the average

(Ensemble: possible states of the system + their probability distribution)

Even if global properties of the system (e.g. energy) differ, we expect their local statistical properties (e.g. spacings between energy levels) will look similar to those of the ensemble average

Idea

Statistical ensemble of matrices, each corresponding to some system. Expectation: very few systems' properties will deviate from that of the average

→ like in statistical physics

Gaussian ensembles

Let's consider an $N \times N$ complex matrix, whose matrix elements we draw independently from a Gaussian distribution (effectively $2N^2$ random variables), e.g. $p(H_{ij}) = \frac{1}{2\pi} e^{-\frac{1}{2}|H_{ij}|^2}$

Independence of matrix elements → probability density of getting an arbitrary matrix $H$ is $p(H) = \prod_{ij} p(H_{ij}) \propto e^{-\frac{1}{2}\mathrm{Tr}H^2}$

Gaussian ensembles

Gaussian unitary ensemble

We want to model quantum-mechanical systems via their hamiltonians – which must be hermitian ($H = H^\dagger$).

The remaining part of the ensemble we call Gaussian Unitary Ensemble (GUE)

Gaussian unitary ensemble

Symmetries

When 'hermitian' is not enough

Over-all symmetries of the system additionally constrain the form of $H$.

Example: rotational symmetry $\implies$ angular momentum operators can always be chosen in a way that makes the hamiltonian real symmetric ($H = H^T$)

We can throw out all non-real matrices from GUE to make the model better!

Symmetries

The remaining ensemble is called Gaussian Orthogonal Ensemble (GOE)

Gaussian orthogonal ensemble

Chaos

When the present determines the future, but the approximate present does not approximately determine the future.

• Edward Lorentz

Conditions for chaos:

1. Exponential sensitivity to change in initial conditions
2. Limited available "space"

Example: the three-body problem is chaotic if the masses can't escape

Ergodicity

Time average is the same as ensemble average

Intuition: a chaotic system can eventually end up at any configuration, and this probability does not change in time → ensemble average best describes the unknown behaviour of the system

→ statistical physics again!

Dynamical billiards

Classically chaotic, ergodic systems

Example: Bunimovich stadium

Bohigas-Giannoni-Schmit conjecture

Spectra of chaotic quantum systems show the same local properties as those of the respective Gaussian ensemble

Conjecture supported by various numerical studies, and consistency with results from semi-classical approach

Justification

• Chaos: we don't know which state the system is in
• Unknown state → random state; our representation of states: matrices
• Ergodicity: average over the ensemble is most likely to correspond with the real system

→ similar argumentation as in the beginning, intuition from statistical physics!

Dynamical billiards

Microwave billiard

Quantum-like experimental realisation

Quantum version of a dynamical billiard, realised with microwave resonant cavities

Other applications

• Riemann hyphotesis: Hilbert–Pólya conjecture states that $\zeta$ zeros correspond to energy levels of a dynamical system. Later empirical data showed agreement with GUE ensemble (Montgomery, Dyson, Odłyżko)!

• Quantum scattering: modeled with other types of Gaussian matrices – Dyson's circular ensembles, evolution operators instead of hamiltonians, hermiticity → unitarity

• Mathematical statistics: estimation of covariance in multivariate statistics
• Optimisation of memorisation in the training of neural networks
• Modeling synaptic connections
• Modeling geographical distribution of trees in Scandinavia, and administrative departments in France
• Portfolio diversification

Summary

1. Random matrix theory is in its core a very simple idea that gives spectacular insights into the statistics of many complex systems
2. It is mostly known from nuclear physics where it predicted the distribution of energy spacings in atomic nuclei
3. Bohigas-Giannoni-Schmit conjecture states that random matrices describe quantum systems that would be chaotic in the classical regime
4. Random matrix theory has many different applications in seemingly distinct areas, from neuron modeling to number theory, and a lot of potential applications still awaits their discovery

Thank you!