Tomasz Necio

Theoretical Physics Proseminar 2019-11-04

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- 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*

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

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*)

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.

**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

**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

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 $p(H) = \prod_{ij} p(H_{ij}) \propto e^{-\frac{1}{2}\mathrm{Tr}H^2}$

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)**

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!**

The remaining ensemble is called **Gaussian Orthogonal Ensemble (GOE)**

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

- Edward Lorentz

**Conditions for chaos:**

- Exponential sensitivity to change in initial conditions
- Limited available "space"

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

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!

Classically chaotic, ergodic systems

Spectra of

chaotic quantum systemsshow 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

**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!

Quantum-like experimental realisation

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

**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**

- Random matrix theory is in its core
**a very simple idea**that gives spectacular insights into the statistics of many complex systems - It is mostly known from
**nuclear physics**where it predicted the distribution of energy spacings in atomic nuclei **Bohigas-Giannoni-Schmit conjecture**states that random matrices describe quantum systems that would be chaotic in the classical regime- 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!

Random matrix theory
Tomasz Necio
University of Warsaw