Statistical Divergence and Paths Thereof to Socioeconomic Inequality and to Renewal Processes.

This paper establishes a general framework for measuring statistical divergence. Namely, with regard to a pair of random variables that share a common range of values: quantifying the distance of the statistical distribution of one random variable from that of the other. The general framework is then applied to the topics of socioeconomic inequality and renewal processes.

Gumbel central limit theorem for max-min and min-max.

The max-min and min-max of matrices arise prevalently in science and engineering. However, in many real-world situations the computation of the max-min and min-max is challenging as matrices are large and full information about their entries is lacking. Here we take a statistical-physics approach and establish limit laws-akin to the central limit theorem-for the max-min and min-max of large random matrices.

Poisson-process limit laws yield Gumbel max-min and min-max.

"A chain is only as strong as its weakest link" says the proverb. But what about a collection of statistically identical chains: How long till all chains fail? The answer to this question is given by the max-min of a matrix whose (i,j) entry is the failure time of link j of chain i: take the minimum of each row, and then the maximum of the rows' minima. The corresponding min-max is obtained by taking the maximum of each column, and then the minimum of the columns' maxima.

First Passage under Restart with Branching.

First passage under restart with branching is proposed as a generalization of first passage under restart. Strong motivation to study this generalization comes from the observation that restart with branching can expedite the completion of processes that cannot be expedited with simple restart; yet a sharp and quantitative formulation of this statement is still lacking. We develop a comprehensive theory of first passage under restart with branching.

Occupation probabilities and fluctuations in the asymmetric simple inclusion process.

The asymmetric simple inclusion process (ASIP), a lattice-gas model of unidirectional transport and aggregation, was recently proposed as an "inclusion" counterpart of the asymmetric simple exclusion process. In this paper we present an exact closed-form expression for the probability that a given number of particles occupies a given set of consecutive lattice sites. Our results are expressed in terms of the entries of Catalan's trapezoids-number arrays which generalize Catalan's numbers and Catalan's triangle.

Rank distributions: a panoramic macroscopic outlook.

This paper presents a panoramic macroscopic outlook of rank distributions. We establish a general framework for the analysis of rank distributions, which classifies them into five macroscopic "socioeconomic" states: monarchy, oligarchy-feudalism, criticality, socialism-capitalism, and communism. Oligarchy-feudalism is shown to be characterized by discrete macroscopic rank distributions, and socialism-capitalism is shown to be characterized by continuous macroscopic size distributions.

Topography of chance.

We present a model of multiplicative Langevin dynamics that is based on two foundations: the Langevin equation and the notion of multiplicative evolution. The model is a nonlinear mechanism transforming a white-noise input to a dynamic-equilibrium output, using a single control: an underlying convex U-shaped potential function. The output is quantified by a stationary density which can attain a given number of shapes and a given number of randomness categories.

Poissonian renormalizations, exponentials, and power laws.

This paper presents a comprehensive "renormalization study" of Poisson processes governed by exponential and power-law intensities. These Poisson processes are of fundamental importance, as they constitute the very bedrock of the universal extreme-value laws of Gumbel, Fréchet, and Weibull. Applying the method of Poissonian renormalization we analyze the emergence of these Poisson processes, unveil their intrinsic dynamical structures, determine their domains of attraction, and characterize their structural phase transitions.

Anomalous statistics of random relaxations in random environments.

We comprehensively analyze the emergence of anomalous statistics in the context of the random relaxation (RARE) model [Eliazar and Metzler, J. Chem. Phys.

Assessing the inherent uncertainty of one-dimensional diffusions.

In this paper we assess the inherent uncertainty of one-dimensional diffusion processes via a stochasticity classification which provides an à la Mandelbrot categorization into five states of uncertainty: infra-mild, mild, borderline, wild, and ultra-wild. Two settings are considered. (i) Stopped diffusions: the diffusion initiates from a high level and is stopped once it first reaches a low level; in this setting we analyze the inherent uncertainty of the diffusion's maximal exceedance above its initial high level.

Limit laws for the asymmetric inclusion process.

The Asymmetric Inclusion Process (ASIP) is a unidirectional lattice-gas flow model which was recently introduced as an exactly solvable 'Bosonic' counterpart of the 'Fermionic' asymmetric exclusion process. An iterative algorithm that allows the computation of the probability generating function (PGF) of the ASIP's steady state exists but practical considerations limit its applicability to small ASIP lattices. Large lattices, on the other hand, have been studied primarily via Monte Carlo simulations and were shown to display a wide spectrum of intriguing statistical phenomena.

The RARE model: A generalized approach to random relaxation processes in disordered systems.

This paper introduces and analyses a general statistical model, termed the RAndom RElaxations (RARE) model, of random relaxation processes in disordered systems. The model considers excitations that are randomly scattered around a reaction center in a general embedding space. The model's input quantities are the spatial scattering statistics of the excitations around the reaction center, and the chemical reaction rates between the excitations and the reaction center as a function of their mutual distance.

Poissonian steady states: from stationary densities to stationary intensities.

Markov dynamics are the most elemental and omnipresent form of stochastic dynamics in the sciences, with applications ranging from physics to chemistry, from biology to evolution, and from economics to finance. Markov dynamics can be either stationary or nonstationary. Stationary Markov dynamics represent statistical steady states and are quantified by stationary densities.

Geometric theory for Weibull's distribution.

Weibull's distribution is the principal phenomenological law of relaxation in the physical sciences and spans three different relaxation regimes: subexponential ("stretched exponential"), exponential, and superexponential. The probabilistic theory of extreme-value statistics asserts that the linear scaling limits of minima of ensembles of positive-valued random variables, which are independent and identically distributed, are universally governed by Weibull's distribution. However, this probabilistic theory does not take into account spatial geometry, which often plays a key role in the physical sciences.

Asymmetric inclusion process as a showcase of complexity.

The asymmetric inclusion process is a lattice-gas model which replaces the "fermionic" exclusion interactions of the asymmetric exclusion process by "bosonic" inclusion interactions. Combining together probabilistic and Monte Carlo analyses, we showcase the model's rich statistical complexity-which ranges from "mild" to "wild" displays of randomness: gaussian load and draining, Rayleigh outflow with linear aging, inverse-gaussian coalescence, intrinsic power-law scalings and power-law fluctuations and condensation.

Langevin dynamics, entropic crowding, and stochastic cloaking.

We consider a pack of independent probes--within a spatially inhomogeneous thermal bath consisting of a vast number of randomly moving particles--which are subjected to an external force. The stochastic dynamics of the probes are governed by Langevin's equation. The probes attain a steady state distribution which, in general, is different than the concentration of the particles in the spatially inhomogeneous thermal bath.

Asymmetric inclusion process.

We introduce and explore the asymmetric inclusion process (ASIP), an exactly solvable bosonic counterpart of the fermionic asymmetric exclusion process (ASEP). In both processes, random events cause particles to propagate unidirectionally along a one-dimensional lattice of n sites. In the ASEP, particles are subject to exclusion interactions, whereas in the ASIP, particles are subject to inclusion interactions that coalesce them into inseparable clusters.

Randomized central limit theorems: A unified theory.

The central limit theorems (CLTs) characterize the macroscopic statistical behavior of large ensembles of independent and identically distributed random variables. The CLTs assert that the universal probability laws governing ensembles' aggregate statistics are either Gaussian or Lévy, and that the universal probability laws governing ensembles' extreme statistics are Fréchet, Weibull, or Gumbel. The scaling schemes underlying the CLTs are deterministic-scaling all ensemble components by a common deterministic scale.

Universal generation of 1/f noises.

This paper establishes a universal mechanism for the generation of 1/f noises. The mechanism is based on a signal-superposition model, which superimposes signals transmitted from independent sources. All sources transmit a statistically common-yet arbitrary-stochastic signal pattern; each source has its own random transmission parameters-amplitude, frequency, and initiation epoch.

Universal self-similarity of propagating populations.

This paper explores the universal self-similarity of propagating populations. The following general propagation model is considered: particles are randomly emitted from the origin of a d-dimensional Euclidean space and propagate randomly and independently of each other in space; all particles share a statistically common--yet arbitrary--motion pattern; each particle has its own random propagation parameters--emission epoch, motion frequency, and motion amplitude. The universally self-similar statistics of the particles' displacements and first passage times (FPTs) are analyzed: statistics which are invariant with respect to the details of the displacement and FPT measurements and with respect to the particles' underlying motion pattern.

Diversity of Poissonian populations.

Populations represented by collections of points scattered randomly on the real line are ubiquitous in science and engineering. The statistical modeling of such populations leads naturally to Poissonian populations-Poisson processes on the real line with a distinguished maximal point. Poissonian populations are infinite objects underlying key issues in statistical physics, probability theory, and random fractals.

Record events in growing populations: universality, correlation, and aging.

This paper studies the occurrence of record events in score populations which grow stochastically in time. In Rényi's basic record model, a population of independent and identically distributed (i.i.

Universal generation of statistical self-similarity: a randomized central limit theorem.

A universal mechanism for the generation of statistical self-similarity-i.e., fractality in the context of random processes-is established.

A unified and universal explanation for Lévy laws and 1/f noises.

Lévy laws and 1/f noises are shown to emerge uniquely and universally from a general model of systems which superimpose the transmissions of many independent stochastic signals. The signals are considered to follow, statistically, a common--yet arbitrary--generic signal pattern which may be either stationary or dissipative. Each signal is considered to have its own random transmission amplitude and frequency.

From Ornstein-Uhlenbeck dynamics to long-memory processes and fractional Brownian motion.

This article establishes a natural physical path leading from "regular" Ornstein-Uhlenbeck dynamics to "anomalous" long-memory processes and, thereafter, to fractional Brownian motion. Considering a system composed of n different parts-each part conducting its own Ornstein-Uhlenbeck dynamics, and all parts being perturbed by a common external Lévy noise-we show that the collective system-dynamics, in the limit n-->infinity , converges to a temporal moving-average of the driving noise. The limiting moving-average process, in turn, can possess a long memory-in which case, when observed over large time scales, further yields fractional Brownian motion.