Universality in survivor distributions: Characterising the winners of competitive dynamics

We investigate the survivor distributions of a spatially extended model of competitive dynamics in different geometries. The model consists of a deterministic dynamical system of individual agents at specified nodes, which might or might not survive the predatory dynamics: all stochasticity is brought in by the initial state. Every such initial state leads to a unique and extended pattern of survivors and non-survivors, which is known as an attractor of the dynamics. We show that the number of such attractors grows exponentially with system size, so that their exact characterisation is limited to only very small systems. Given this, we construct an analytical approach based on inhomogeneous mean-field theory to calculate survival probabilities for arbitrary networks. This powerful (albeit approximate) approach shows how universality arises in survivor distributions via a key concept -- the {\it dynamical fugacity}. Remarkably, in the large-mass limit, the survival probability of a node becomes independent of network geometry, and assumes a simple form which depends only on its mass and degree.Comment: 12 pages, 6 figures, 2 table

Slow synaptic dynamics in a network: from exponential to power-law forgetting

We investigate a mean-field model of interacting synapses on a directed neural network. Our interest lies in the slow adaptive dynamics of synapses, which are driven by the fast dynamics of the neurons they connect. Cooperation is modelled from the usual Hebbian perspective, while competition is modelled by an original polarity-driven rule. The emergence of a critical manifold culminating in a tricritical point is crucially dependent on the presence of synaptic competition. This leads to a universal $1/t$ power-law relaxation of the mean synaptic strength along the critical manifold and an equally universal $1/\sqrt{t}$ relaxation at the tricritical point, to be contrasted with the exponential relaxation that is otherwise generic. In turn, this leads to the natural emergence of long- and short-term memory from different parts of parameter space in a synaptic network, which is the most novel and important result of our present investigations.Comment: 12 pages, 8 figures. Phys. Rev. E (2014) to appea

Competition and cooperation:aspects of dynamics in sandpiles

In this article, we review some of our approaches to granular dynamics, now well known to consist of both fast and slow relaxational processes. In the first case, grains typically compete with each other, while in the second, they cooperate. A typical result of {\it cooperation} is the formation of stable bridges, signatures of spatiotemporal inhomogeneities; we review their geometrical characteristics and compare theoretical results with those of independent simulations. {\it Cooperative} excitations due to local density fluctuations are also responsible for relaxation at the angle of repose; the {\it competition} between these fluctuations and external driving forces, can, on the other hand, result in a (rare) collapse of the sandpile to the horizontal. Both these features are present in a theory reviewed here. An arena where the effects of cooperation versus competition are felt most keenly is granular compaction; we review here a random graph model, where three-spin interactions are used to model compaction under tapping. The compaction curve shows distinct regions where 'fast' and 'slow' dynamics apply, separated by what we have called the {\it single-particle relaxation threshold}. In the final section of this paper, we explore the effect of shape -- jagged vs. regular -- on the compaction of packings near their jamming limit. One of our major results is an entropic landscape that, while microscopically rough, manifests {\it Edwards' flatness} at a macroscopic level. Another major result is that of surface intermittency under low-intensity shaking.Comment: 36 pages, 23 figures, minor correction

Probability density of determinants of random matrices

In this brief paper the probability density of a random real, complex and quaternion determinant is rederived using singular values. The behaviour of suitably rescaled random determinants is studied in the limit of infinite order of the matrices

A two-species model of a two-dimensional sandpile surface: a case of asymptotic roughening

We present and analyze a model of an evolving sandpile surface in (2 + 1) dimensions where the dynamics of mobile grains ({\rho}(x, t)) and immobile clusters (h(x, t)) are coupled. Our coupling models the situation where the sandpile is flat on average, so that there is no bias due to gravity. We find anomalous scaling: the expected logarithmic smoothing at short length and time scales gives way to roughening in the asymptotic limit, where novel and non-trivial exponents are found.Comment: 7 Pages, 6 Figures; Granular Matter, 2012 (Online

Calculation of some determinants using the s-shifted factorial

Several determinants with gamma functions as elements are evaluated. This kind of determinants are encountered in the computation of the probability density of the determinant of random matrices. The s-shifted factorial is defined as a generalization for non-negative integers of the power function, the rising factorial (or Pochammer's symbol) and the falling factorial. It is a special case of polynomial sequence of the binomial type studied in combinatorics theory. In terms of the gamma function, an extension is defined for negative integers and even complex values. Properties, mainly composition laws and binomial formulae, are given. They are used to evaluate families of generalized Vandermonde determinants with s-shifted factorials as elements, instead of power functions.Comment: 25 pages; added section 5 for some examples of application

Perturbation approach to multifractal dimensions for certain critical random matrix ensembles

Fractal dimensions of eigenfunctions for various critical random matrix ensembles are investigated in perturbation series in the regimes of strong and weak multifractality. In both regimes we obtain expressions similar to those of the critical banded random matrix ensemble extensively discussed in the literature. For certain ensembles, the leading-order term for weak multifractality can be calculated within standard perturbation theory. For other models such a direct approach requires modifications which are briefly discussed. Our analytical formulas are in good agreement with numerical calculations.Comment: 28 pages, 7 figure