# Random matrices

Random matrices are, loosely speaking, matrices which elements are random numbers. For example, each element may be drawn independently from a Gaussian (normal) distribution. In more complex situations, whole matrices are drawn from a statistical ensemble, and each matrix is treated as an object having a certain probability of occurrence.

Random matrices were introduced to physics by E. Wigner who considered the problem of calculating the energy spectrum of atomic nuclei. He approximated the complicated (and unknown at that time) Hamiltonian of the nucleus by a random matrix and observed that the resulting distribution of spacings between energy levels agreed well with experiments. More recently, random matrix theory has been applied to quantum chaos, signal processing and wireless communication, quantum gravity, and stock markets.

In my work, I studied hermitian and non-hermitian matrices with correlated elements. This was motivated by financial applications, in particular by the problem of “portfolio optimization”: how to optimally invest money on a stock market? According to Markovitz portfolio theory, this problem is related to the problem of finding the spectrum of a covariance matrix for price changes of different stocks. However, the covariance matrix obtained from real data is “noisy”. Here enters the theory of random matrices: it explains how to find the relation between “real” and “experimental” correlation matrices, which can be used for cleaning the covariance matrix and extracting genuine correlations. Here is a few articles I have written (with Z. Burda, A.T. Goerlich, R. Janik and J. Jurkiewicz) on this subject, see also the right column for a few highlights.

# Random graphs and complex networks

Graphs are objects which consist of nodes connected by links. A simple, two dimensional grid is an example of a regular graph. In random graphs, the way the nodes are connected is to some extent random. Maximally random graphs (with links distributed completely at random) were first studied by Erdös, Rényi and their followers as part of mathematical graph theory, but in recent years Albert, Barabási, Watts, Strogatz, and others have shown that many real-world networks have properties of both regular and maximally random graphs.

These so called “complex networks” (CN) are neither completely regular as for example a 2d lattice, nor completely random as Erdös and Rényi graphs. Various models have been proposed to explain common properties of many CN such as power-law degree distributions, small diameter (“small-worlds”), and high clustering. The degree distribution tells how many neighbours a randomly chosen node has. Most real-world networks have many nodes with relatively high connectivity, and the probability of finding a node with a given degree (number of nearest neighbours) follows a power-law. The small-world effect means that CN are “short” – the average distance between any pair of nodes is very small compared to the number of nodes/links in the network. High clustering means that there are many highly interconnected cliques of nodes.

In my research, I was interested in developing a general theory of complex networks, based on such concepts of statistical mechanics as statistical ensemble and partition function. Here you can find my publications related to networks and random graphs. Some results are briefly mentioned in the right panel.

# Zero-range processes and related models

Many physical processes are far from equlibrium, which means that macroscopic currents or flows are present in the system. The zero-range process (ZRP) is a very simple model of such processes. In ZRP, particles hop between nodes of a network with rates that depend only on the number of particles at the departure node. ZRP is interesting because, despite being a non-equilibrium process, it has a simple, mathematically accessible steady state, which makes it possible to calculate many things analytically.

I worked on the phenomenon of condensation in ZRP: for some choices of the hopping rate and/or the structure of the network, a finite fraction of particles “condenses” at a single node if the density of particles exceeds a critical value. For regular lattices, the condensate takes place at a randomly chosen node – the symmetry of the lattice is spontaneously broken. In a series of papers with L. Bogacz, Z. Burda, and W. Janke we showed that if symmetry is explicitly broken (e.g. the network is irregular), condensation happens at the most connected node.

I also considered simultaneous evolution of particles and networks, and extensions of ZRP to more complicated hopping rates depending on the state of neighbouring nodes (with J. Sopik, H. Meyer-Ortmanns and W. Janke). In this latter case, the condensate often extends to more than one site and form a “droplet” (see right). The extension of this droplet depends on the balance between local (zero-range) and nearest-neighbour interactions.

Recently, I have discovered (with M. Evans) a new “explosive” condensation in which the process of condensate’s formation accelerates in time (video on the right). This is probably the first example of instantaneous gelation (known so far only from mean-field models of coagulation) in a spatially extended system. This strange behaviour is caused by the hopping rate which increases with the number of particles at both departure and arrival sistes. Surprisingly, the steady state has the same simple mathematical structure as in the zero-range process.

I have also co-authored (with P. Mottishaw and M. Evans) a paper on exclusion processes on the Bethe lattice.

# Some other problems

**Random walks****.** I and my colleagues (Z. Burda, J. Duda and J.-M. Luck) investigated a new type of random walk in which all paths of the same length between the two given points are assumed to have the same probability. The stationary distribution of this maximal entropy random walk is localized in the presence of lattice defects, which generally leads to the absence of diffusion on random networks. This closely resembles the quantum phenomenon of Anderson localization but is purely classical in nature. Our work has been discussed in the literature in the context of community finding or link prediction in complex networks.

**Spin glasses. **I developed (with Z. Burda) a new, perturbative method of counting the number of metastable states in Ising spin glasses. We obtained results for regular two-and three-dimensional lattices and random regular graphs. All these results are much more accurate than those obtained before in computer simulations.

**Effective models of causal dynamical triangulations (CDT)****.** CDT is a computer-simulation approach to quantum gravity. I have published (with Z. Burda and L.Bogacz) a paper about a simple statistical-physics model that demonstrates how the quantum de-Sitter universe emerges in a phenomenon similar to ZRP condensation.

Density of eigenvalues (spectrum) in the complex plane of a product of two random matrices is very simple and highly universal: r(z)=1/(2p|z|). A similar, universal result holds for a product of more than two matrices (paper).

I have calculated (with I. M. Sokolov) finite-size corrections to the power-law degree distribution p(k) (see picture above) which are important for understanding processes such as percolation and epidemic spreading.

I have also developed (with L. Bogacz, Z. Burda and W. Janke) a computer program based on sampling the space of random graphs, which allows for generating networks with desired features such as the power-law degree distribution.

Condensation in ZRP: particles (initially distributed at random) form clusters and finally produce a condensate. The hopping rate u(m)=1+3/m where m is the number of particles at the departure site

The condensate becomes spatially extended particles located on neighbouring sites can interact (see papers).

“Explosive” condensation: the rate at which a cluster of particles moves through the system increases with its size, similarly to a rain drop falling through the mist.

Localization of the probability (white areas in the plot) of finding a random walker maximizing the entropy of trajectories in the presence of defects (randomly removed links of a 2d lattice are indicated by pairs of dots).

Various regular graphs for which we calculated the number of metastable states in spin glasses with Gaussian distribution of couplings. The number N of such states grows exponentially with the number of spins n, N = exp(A*n). A non-trivial problem is to find the coefficient A. This is what we did here.