Master Operators Govern Multifractality in Percolation

Using renormalization group methods we study multifractality in percolation at the instance of noisy random resistor networks. We introduce the concept of master operators. The multifractal moments of the current distribution (which are proportional to the noise cumulants $C_R^{(l)} (x, x^\prime)$ of the resistance between two sites x and $x^\prime$ located on the same cluster) are related to such master operators. The scaling behavior of the multifractal moments is governed exclusively by the master operators, even though a myriad of servant operators is involved in the renormalization procedure. We calculate the family of multifractal exponents ${\psi_l}$ for the scaling behavior of the noise cumulants, $C_R^{(l)} (x, x^\prime) \sim | x - x^\prime |^{\psi_l /\nu}$, where $\nu$ is the correlation length exponent for percolation, to two-loop order.Comment: 6 page

Scale Invariance and Self-averaging in disordered systems

In a previous paper we found that in the random field Ising model at zero temperature in three dimensions the correlation length is not self-averaging near the critical point and that the violation of self-averaging is maximal. This is due to the formation of bound states in the underlying field theory. We present a similar study for the case of disordered Potts and Ising ferromagnets in two dimensions near the critical temperature. In the random Potts model the correlation length is not self-averaging near the critical temperature but the violation of self-averaging is weaker than in the random field case. In the random Ising model we find still weaker violations of self-averaging and we cannot rule out the possibility of the restoration of self-averaging in the infinite volume limit.Comment: 7 pages, 4 ps figure

Rational families of vector bundles on curves, I

Let C be a smooth complex projective curve of genus at least 2 and let M be the moduli space of rank 2, stable vector bundles on C, with fixed determinant of degree 1. For any k>1, we find two irreducible components of the space of rational curves of degree k on M. One component, which we call the nice component has the property that the general element is a very free curve if k is sufficiently large. The other component has the general element a free curve. Both components have the expected dimension and their maximal rationally connected fibration is the Jacobian of the curve C.Comment: 23 page

Current fluctuations in stochastic systems with long-range memory

We propose a method to calculate the large deviations of current fluctuations in a class of stochastic particle systems with history-dependent rates. Long-range temporal correlations are seen to alter the speed of the large deviation function in analogy with long-range spatial correlations in equilibrium systems. We give some illuminating examples and discuss the applicability of the Gallavotti-Cohen fluctuation theorem.Comment: 10 pages, 1 figure. v2: Minor alterations. v3: Very minor alterations for consistency with published version appearing at http://stacks.iop.org/1751-8121/42/34200

Radial basis function classifier construction using particle swarm optimisation aided orthogonal forward regression

We develop a particle swarm optimisation (PSO) aided orthogonal forward regression (OFR) approach for constructing radial basis function (RBF) classifiers with tunable nodes. At each stage of the OFR construction process, the centre vector and diagonal covariance matrix of one RBF node is determined efficiently by minimising the leave-one-out (LOO) misclassification rate (MR) using a PSO algorithm. Compared with the state-of-the-art regularisation assisted orthogonal least square algorithm based on the LOO MR for selecting fixednode RBF classifiers, the proposed PSO aided OFR algorithm for constructing tunable-node RBF classifiers offers significant advantages in terms of better generalisation performance and smaller model size as well as imposes lower computational complexity in classifier construction process. Moreover, the proposed algorithm does not have any hyperparameter that requires costly tuning based on cross validation

Detecting binocular 3-D motion in static 3-D noise: No effect of viewing distance.

Relative binocular disparity cannot tell us the absolute 3-D shape of an object, nor its 3-D trajectory if it is moving, unless the visual system has independent access to how far away the object is at any moment. Indeed, as the viewing distance is changed, the same disparate retinal motions will correspond to very different real 3-D trajectories. In this paper we were interested in whether binocular 3-D motion detection is affected by viewing distance. We used a visual search task in which the observer is asked to detect a target dot, moving in 3-D, amidst 3-D stationary distractor dots. We found that distance does not affect detection performance. Motion-in-depth is consistently harder to detect than the equivalent lateral motion, for all viewing distances. For a constant retinal motion with both lateral and motion-in-depth components, detection performance is constant despite variations in viewing distance that produce large changes in the direction of the 3-D trajectory. We conclude that binocular 3-D motion detection relies on retinal, not absolute visual signals