Rényi entropy of zeta urns.

We calculate analytically the Rényi entropy for the zeta-urn model with a Gibbs measure definition of the microstate probabilities. This allows us to obtain the singularities in the Rényi entropy from those of the thermodynamic potential, which is directly related to the free-energy density of the model. We enumerate the various possible behaviors of the Rényi entropy and its singularities, which depend on both the value of the power law in the zeta urn and the order of the Rényi entropy under consideration.

Random allocation models in the thermodynamic limit.

We discuss the phase transition and critical exponents in the random allocation model (urn model) for different statistical ensembles. We provide a unified presentation of the statistical properties of the model in the thermodynamic limit, uncover relationships between the thermodynamic potentials, and fill some lacunae in previous results on the singularities of these potentials at the critical point and behavior in the thermodynamic limit. The presentation is intended to be self-contained, so we carefully derive all formulas step by step throughout. Additionally, we comment on a quasiprobabilistic normalization of configuration weights, which was considered in some recent studies.

Perfect cycles in the synchronous Heider dynamics in complete network.

We discuss a cellular automaton simulating the process of reaching Heider balance in a fully connected network. The dynamics of the automaton is defined by a deterministic, synchronous, and global update rule. The dynamics has a very rich spectrum of attractors including fixed points and limit cycles, the length and number of which change with the size of the system. In this paper we concentrate on a class of limit cycles that preserve energy spectrum of the consecutive states. We call such limit cycles perfect. Consecutive states in a perfect cycle are separated from each other by the same Hamming distance. Also the Hamming distance between any two states separated by k steps in a perfect cycle is the same for all such pairs of states. The states of a perfect cycle form a very symmetric trajectory in the configuration space. We argue that the symmetry of the trajectories is rooted in the permutation symmetry of vertices of the network and a local symmetry of a certain energy function measuring the level of balance and frustration of triads.

Cleaning large-dimensional covariance matrices for correlated samples.

We elucidate the problem of estimating large-dimensional covariance matrices in the presence of correlations between samples. To this end, we generalize the Marcenko-Pastur equation and the Ledoit-Péché shrinkage estimator using methods of random matrix theory and free probability. We develop an efficient algorithm that implements the corresponding analytic formulas based on the Ledoit-Wolf kernel estimation technique. We also provide an associated open-source Python library, called shrinkage, with a user-friendly API to assist in practical tasks of estimation of large covariance matrices. We present an example of its usage for synthetic data generated according to exponentially decaying autocorrelations.

Towards the Heider balance: Cellular automaton with a global neighborhood.

We study a simple deterministic map that leads a fully connected network to the Heider balance. The map is realized by an algorithm that updates all links synchronously in a way depending on the state of the entire network. We observe that the probability of reaching a balanced state increases with the system size N. Jammed states become less frequent for larger N. The algorithm generates also limit cycles, mostly of length 2, but also of length 3, 4, 6, 12, or 14. We give a simple argument to estimate the mean size of basins of attraction of balanced states, and we discuss the symmetries of the system including the automorphism group as well as gauge invariance of triad configurations. We argue that both symmetries play an essential role in the occurrence of cycles observed in the synchronous dynamics realized by the algorithm.

Wealth Rheology.

We study wealth rank correlations in a simple model of macroeconomy. To quantify rank correlations between wealth rankings at different times, we use Kendall's τ and Spearman's ρ, Goodman-Kruskal's γ, and the lists' overlap ratio. We show that the dynamics of wealth flow and the speed of reshuffling in the ranking list depend on parameters of the model controlling the wealth exchange rate and the wealth growth volatility. As an example of the rheology of wealth in real data, we analyze the lists of the richest people in Poland, Germany, the USA and the world.

Universality of local spectral statistics of products of random matrices.

We derive exact analytical expressions for correlation functions of singular values of the product of M Ginibre matrices of size N in the double scaling limit M,N→∞. The singular value statistics is described by a determinantal point process with a kernel that interpolates between Gaussian unitary ensemble statistic and Dirac-delta (picket-fence) statistic. In the thermodynamic limit N→∞, the interpolation parameter is given by the limiting quotient a=N/M. One of our goals is to find an explicit form of the kernel at the hard edge, in the bulk, and at the soft edge for any a. We find that in addition to the standard scaling regimes, there is a transitional regime which interpolates between the hard edge and the bulk. We conjecture that these results are universal, and that they apply to a broad class of products of random matrices from the Gaussian basin of attraction, including correlated matrices. We corroborate this conjecture by numerical simulations. Additionally, we show that the local spectral statistics of the considered random matrix products is identical with the local statistics of Dyson Brownian motion with the initial condition given by equidistant positions, with the crucial difference that this equivalence holds only locally. Finally, we have identified a mesoscopic spectral scale at the soft edge which is crucial for the unfolding of the spectrum.

Modelling Excess Mortality in Covid-19-Like Epidemics.

We develop an agent-based model to assess the cumulative number of deaths during hypothetical Covid-19-like epidemics for various non-pharmaceutical intervention strategies. The model simulates three interrelated stochastic processes: epidemic spreading, availability of respiratory ventilators and changes in death statistics. We consider local and non-local modes of disease transmission. The first simulates transmission through social contacts in the vicinity of the place of residence while the second through social contacts in public places: schools, hospitals, airports, etc., where many people meet, who live in remote geographic locations. Epidemic spreading is modelled as a discrete-time stochastic process on random geometric networks. We use the Monte-Carlo method in the simulations. The following assumptions are made. The basic reproduction number is R0=2.5 and the infectious period lasts approximately ten days. Infections lead to severe acute respiratory syndrome in about one percent of cases, which are likely to lead to respiratory default and death, unless the patient receives an appropriate medical treatment. The healthcare system capacity is simulated by the availability of respiratory ventilators or intensive care beds. Some parameters of the model, like mortality rates or the number of respiratory ventilators per 100,000 inhabitants, are chosen to simulate the real values for the USA and Poland. In the simulations we compare 'do-nothing' strategy with mitigation strategies based on social distancing and reducing social mixing. We study epidemics in the pre-vacine era, where immunity is obtained only by infection. The model applies only to epidemics for which reinfections are rare and can be neglected. The results of the simulations show that strategies that slow the development of an epidemic too much in the early stages do not significantly reduce the overall number of deaths in the long term, but increase the duration of the epidemic. In particular, a hybrid strategy where lockdown is held for some time and is then completely released, is inefficient.

Ageing of complex networks.

Many real-world complex networks arise as a result of a competition between growth and rewiring processes. Usually the initial part of the evolution is dominated by growth while the later one rather by rewiring. The initial growth allows the network to reach a certain size while rewiring to optimize its function and topology. As a model example we consider tree networks which first grow in a stochastic process of node attachment and then age in a stochastic process of local topology changes. The ageing is implemented as a Markov process that preserves the node-degree distribution. We quantify differences between the initial and aged network topologies and study the dynamics of the evolution. We implement two versions of the ageing dynamics. One is based on reshuffling of leaves and the other on reshuffling of branches. The latter one generates much faster ageing due to nonlocal nature of changes.

Eigenvector statistics of the product of Ginibre matrices.

We develop a method to calculate left-right eigenvector correlations of the product of m independent N×N complex Ginibre matrices. For illustration, we present explicit analytical results for the vector overlap for a couple of examples for small m and N. We conjecture that the integrated overlap between left and right eigenvectors is given by the formula O=1+(m/2)(N-1) and support this conjecture by analytical and numerical calculations. We derive an analytical expression for the limiting correlation density as N→∞ for the product of Ginibre matrices as well as for the product of elliptic matrices. In the latter case, we find that the correlation function is independent of the eccentricities of the elliptic laws.

Beyond the Hypercube: Evolutionary Accessibility of Fitness Landscapes with Realistic Mutational Networks.

Evolutionary pathways describe trajectories of biological evolution in the space of different variants of organisms (genotypes). The probability of existence and the number of evolutionary pathways that lead from a given genotype to a better-adapted genotype are important measures of accessibility of local fitness optima and the reproducibility of evolution. Both quantities have been studied in simple mathematical models where genotypes are represented as binary sequences of two types of basic units, and the network of permitted mutations between the genotypes is a hypercube graph. However, it is unclear how these results translate to the biologically relevant case in which genotypes are represented by sequences of more than two units, for example four nucleotides (DNA) or 20 amino acids (proteins), and the mutational graph is not the hypercube. Here we investigate accessibility of the best-adapted genotype in the general case of K > 2 units. Using computer generated and experimental fitness landscapes we show that accessibility of the global fitness maximum increases with K and can be much higher than for binary sequences. The increase in accessibility comes from the increase in the number of indirect trajectories exploited by evolution for higher K. As one of the consequences, the fraction of genotypes that are accessible increases by three orders of magnitude when the number of units K increases from 2 to 16 for landscapes of size N ∼ 106 genotypes. This suggests that evolution can follow many different trajectories on such landscapes and the reconstruction of evolutionary pathways from experimental data might be an extremely difficult task.

Quaternionic R transform and non-Hermitian random matrices.

Using the Cayley-Dickson construction we rephrase and review the non-Hermitian diagrammatic formalism [R. A. Janik, M. A. Nowak, G. Papp, and I. Zahed, Nucl. Phys. B 501, 603 (1997)], that generalizes the free probability calculus to asymptotically large non-Hermitian random matrices. The main object in this generalization is a quaternionic extension of the R transform which is a generating function for planar (noncrossing) cumulants. We demonstrate that the quaternionic R transform generates all connected averages of all distinct powers of X and its Hermitian conjugate X†: 〈〈1/NTrX(a)X(†b)X(c)...〉〉 for N→∞. We show that the R transform for Gaussian elliptic laws is given by a simple linear quaternionic map R(z+wj)=x+σ(2)(µe(2iϕ)z+wj) where (z,w) is the Cayley-Dickson pair of complex numbers forming a quaternion q=(z,w)≡z+wj. This map has five real parameters Rex, Imx, ϕ, σ, and µ. We use the R transform to calculate the limiting eigenvalue densities of several products of Gaussian random matrices.

Dysonian dynamics of the Ginibre ensemble.

We study the time evolution of Ginibre matrices whose elements undergo Brownian motion. The non-Hermitian character of the Ginibre ensemble binds the dynamics of eigenvalues to the evolution of eigenvectors in a nontrivial way, leading to a system of coupled nonlinear equations resembling those for turbulent systems. We formulate a mathematical framework allowing simultaneous description of the flow of eigenvalues and eigenvectors, and we unravel a hidden dynamics as a function of a new complex variable, which in the standard description is treated as a regulator only. We solve the evolution equations for large matrices and demonstrate that the nonanalytic behavior of the Green's functions is associated with a shock wave stemming from a Burgers-like equation describing correlations of eigenvectors. We conjecture that the hidden dynamics that we observe for the Ginibre ensemble is a general feature of non-Hermitian random matrix models and is relevant to related physical applications.

Commutative law for products of infinitely large isotropic random matrices.

Ensembles of isotropic random matrices are defined by the invariance of the probability measure under the left (and right) multiplication by an arbitrary unitary matrix. We show that the multiplication of large isotropic random matrices is spectrally commutative and self-averaging in the limit of infinite matrix size N→∞. The notion of spectral commutativity means that the eigenvalue density of a product ABC... of such matrices is independent of the order of matrix multiplication, for example, the matrix ABCD has the same eigenvalue density as ADCB. In turn, the notion of self-averaging means that the product of n independent but identically distributed random matrices, which we symbolically denote by AAA..., has the same eigenvalue density as the corresponding power A(n) of a single matrix drawn from the underlying matrix ensemble. For example, the eigenvalue density of ABCCABC is the same as that of A(2)B(2)C(3). We also discuss the singular behavior of the eigenvalue and singular value densities of isotropic matrices and their products for small eigenvalues λ→0. We show that the singularities at the origin of the eigenvalue density and of the singular value density are in one-to-one correspondence in the limit N→∞: The eigenvalue density of an isotropic random matrix has a power-law singularity at the origin ~|λ|(-s) with a power sε(0,2) when and only when the density of its singular values has a power-law singularity ~λ(-σ) with a power σ=s/(4-s). These results are obtained analytically in the limit N→∞. We supplement these results with numerical simulations for large but finite N and discuss finite-size effects for the most common ensembles of isotropic random matrices.

Free random Lévy and Wigner-Lévy matrices.

We compare eigenvalue densities of Wigner random matrices whose elements are independent identically distributed random numbers with a Lévy distribution and maximally random matrices with a rotationally invariant measure exhibiting a power law spectrum given by stable laws of free random variables. We compute the eigenvalue density of Wigner-Lévy matrices using (and correcting) the method by Bouchaud and Cizeau, and of free random Lévy (FRL) rotationally invariant matrices by adapting results of free probability calculus. We compare the two types of eigenvalue spectra. Both ensembles are spectrally stable with respect to the matrix addition. The discussed ensemble of FRL matrices is maximally random in the sense that it maximizes Shannon's entropy. We find a perfect agreement between the numerically sampled spectra and the analytical results already for matrices of dimension N=100 . The numerical spectra show very weak dependence on the matrix size N as can be noticed by comparing spectra for N=400 . After a pertinent rescaling, spectra of Wigner-Lévy matrices and of symmetric FRL matrices have the same tail behavior. As we discuss towards the end of the paper the correlations of large eigenvalues in the two ensembles are, however, different. We illustrate the relation between the two types of stability and show that the addition of many randomly rotated Wigner-Lévy matrices leads by a matrix central limit theorem to FRL spectra, providing an explicit realization of the maximal randomness principle.

Spectral properties of empirical covariance matrices for data with power-law tails.

We present an analytic method for calculating spectral densities of empirical covariance matrices for correlated data. In this approach the data is represented as a rectangular random matrix whose columns correspond to sampled states of the system. The method is applicable to a class of random matrices with radial measures including those with heavy (power-law) tails in the probability distribution. As an example we apply it to a multivariate Student distribution.

Spectral moments of correlated Wishart matrices.

We present an analytic method to determine the spectral properties of the covariance matrices constructed of correlated Wishart random matrices. The method gives, in the limit of large matrices, exact analytic relations between the spectral moments and the eigenvalue densities of the covariance matrices and their estimators. The results can be used in practice to extract the information about genuine correlations from the given experimental realization of random matrices.

Free random Lévy matrices.

Using the theory of free random variables and the Coulomb gas analogy, we construct stable random matrix ensembles that are random matrix generalizations of the classical one-dimensional stable Lévy distributions. We show that the resolvents for the corresponding matrices obey transcendental equations in the large size limit. We solve these equations in a number of cases, and show that the eigenvalue distributions exhibit Lévy tails. For the analytically known Lévy measures we explicitly construct the density of states using the method of orthogonal polynomials. We show that the Lévy tail distributions are characterized by a different novel form of microscopic universality.