Table of contents

We investigate the percolation thresholds of both random and invasion percolation in two and three dimensions on elongated lattices; lattices with a geometry of L^d-1 × nL in d dimensions, where n denotes the aspect ratio of the lattice. Scaling laws for the threshold and spanning cluster density for random percolation are derived and simulation confirms the behaviour. A direct relationship between thresholds obtained for random percolation and invasion percolation is given and verified numerically.

We calculated some of the critical exponents of the directed percolation universality class through exact numerical diagonalizations of the master operator of the one-dimensional basic contact process. Perusal of the power method together with finite-size scaling allowed us to achieve a high degree of accuracy in our estimates with relatively little computational effort. A simple reasoning leading to the appropriate choice of the microscopic time scale for time-dependent simulations of Markov chains within the so-called quantum chain formulation is discussed. Our approach is applicable to any stochastic process with a finite number of absorbing states.

Dual material space-time with defect field is presented in the language of differential forms: one is the strain space-time whose basic equation is the continuity equation for the dislocation 2-form; the other is the stress space-time whose basic equation is the continuity equation for the couple-stress and angular momentum 2-form. Continuity and kinematic equations in each space can be derived by the transformation from p-form to (p + 1)-form. Moreover, several constitutive equations can be recognized as the transformation between the p-form of the strain space-time and the (4-p)-form of the stress space-time. These kinematic, continuity and constitutive equations can be interpreted geometrically as Cartan structure equations, Bianchi identities and Hodge duality transformations, respectively.

In this work we present a theoretical and numerical study of the behaviour of the maximum Lyapunov exponent in products of random tridiagonal matrices in the limit of small coupling and small fluctuations. Such a problem is directly motivated by the investigation of coupled-map lattices in a regime where the chaotic properties are quite robust and yet a complete understanding has still not been reached. We derive some approximate analytic expressions by introducing a suitable continuous-time formulation of the evolution equation. As a first result, we show that the perturbation of the Lyapunov exponent due to the coupling depends only on a single scaling parameter which, in the case of strictly positive multipliers, is the ratio of the coupling strength with the variance of local multipliers. An explicit expression for the Lyapunov exponent is obtained by mapping the original problem onto a chain of nonlinear Langevin equations, which are eventually reduced to a single stochastic equation. The probability distribution of this dynamical equation provides an excellent description for the behaviour of the Lyapunov exponent. Finally, multipliers with random signs are also considered, finding that the Lyapunov exponent again depends on a single scaling parameter, which, however, has a different expression.

We show the integrability of spin-½ XXZ Heisenberg chain with two arbitrary spin boundary impurities. By using the fusion method, we generalize it to the spin-1 XXZ chain. Then the eigenvalues of Hamiltonians of these models are obtained by the means of Bethe ansatz method.

We develop a practical method of computing the stationary drift velocity V and the diffusion coefficient D of a particle (or a few particles) in a periodic system with arbitrary transition rates. We solve this problem in a physically relevant continuous-time approach as well as for models with discrete-time kinetics, which are often used in computer simulations. We show that both approaches yield the same value of the drift, but the difference between the diffusion coefficients obtained in each of them equals ½V². Generalization to spaces of arbitrary dimension and several applications of the method are also presented.

We study the relaxation properties of the voter model with i.i.d. random bias. We prove under mild conditions that the disorder-averaged relaxation of this biased random voter model is faster than a stretched exponential with exponent d/(d+), where 0< 2 depends on the transition rates of the non-biased voter model. Under an additional assumption, we show that the above upper bound is optimal. The main ingredient of our proof is a result of Donsker and Varadhan.

We study the mean survival probability (n) at time n on a random one-dimensional chain with perfect absorbers at 0 and L. The transition probabilities g_i at the lattice sites i, are independent identically distributed random variables having the distribution p(g_i) = 1 for 0g_i1. We prove the asymptotic inequality, C₁(n)n²/(logn)^L-3C₂ where C₁ and C₂ are finite positive constants which depend on the lattice size L, but not on n. We confirm this result by simulations for lattice sizes up to L = 17.

The distribution of masses of clusters smaller than the infinite cluster is evaluated at the percolation threshold. The clusters are ranked according to their masses and the distribution P(M/L^D,r) of the scaled masses M for any rank r shows a universal behaviour for different lattice sizes L (D is the fractal dimension). For different ranks however, there is a universal distribution function only in the large-rank limit, i.e. P(M/L^D,r)r^-y~g(Mr^y/L^D) (y and are defined in the text), where the universal scaling function g is found to be Gaussian in nature.

We investigate the influence of diffraction on the statistics of energy levels in quantum systems with a chaotic classical limit. By applying the geometrical theory of diffraction we show that diffraction on singularities of the potential can lead to modifications in semiclassical approximations for spectral statistics that persist in the semiclassical limit 0. This result is obtained by deriving a classical sum rule for trajectories that connect two points in coordinate space.

We propose a new approach to calculate perturbatively the effects of a particular deformed Heisenberg algebra on an energy spectrum. We use this method to calculate the harmonic oscillator spectrum and find that the corrections are in agreement with a previous calculation. Then, we apply this approach to obtain the hydrogen atom spectrum and we find that splittings of degenerate energy levels appear. Comparison with experimental data yields an interesting upper bound for the deformation parameter of the Heisenberg algebra.

A new reaction kernel, K(j,k) = 2-q^j-q^k with 0<q<1, is introduced, for which the Smoluchowski equations of aggregation can be solved. The time evolution of the concentrations c_j(t) and of their moments is analysed. The c_j(t) decay at large times as t^-(2-qj) in striking contrast to the behaviour of the constant kernel K(j,k) = 2, for which c_j(t) behaves as t^-2 at large times. On the other hand, the moments behave in leading order at large times exactly like the moments of the constant kernel, though differences appear at higher orders.

In a previous paper, a new reaction kernel for the Smoluchowski equations of aggregation was solved exactly. This kernel, K(j,k) = 2-q^j-q^k, for 0<q<1 a real positive quantity, interpolates between two well understood exactly solved cases, namely that of K(j,k) = 2 and that of K(j,k) = j+k. This new model, however, shows a number of unexpected features, not found in either of the two limiting cases. It is shown that this model has a remarkable behaviour with respect to the commonly accepted scaling theory. On the one hand, it satisfies a rigorous form of the scaling hypothesis, but, on the other hand, it clearly violates some relations which are ordinarily assumed to follow from it. These issues are discussed, as well as the nature of the singular limit in which q is very close to one, for which our kernel becomes close to the sum kernel mentioned above. In particular, the form of the crossover between two kernels with different degrees of homogeneity can be discussed here in an exact way.

We study a system of equations which models the formation of clusters by coagulation, with particles of unit size being injected at a time-dependent rate. We observe that the criteria under which gelation occurs are the same as for the constant mass and constant monomer cases, which have been studied previously. We identify a variety of types of behaviour in the large-time limit, depending on the coagulation kernel and on the rate at which monomer is introduced into the system. The results are obtained by means of exact (generating function) techniques, matched asymptotic expansions and numerical simulations.

We found Hermitian realizations of the position vector , the angular momentum and the linear momentum , all behaving like vectors under the su_q(2) algebra, generated by L₀ and L_±. They are used to introduce a q-deformed Schrödinger equation. Its solutions for the particular cases of the Coulomb and the harmonic oscillator potentials are given and briefly discussed.

We present efficient algorithms for simplifying tensor expressions that obey generic symmetries. We define the canonical form of a single tensor and we show that the problem of finding the canonical form of a generic tensor expression reduces to finding the canonical form of single tensors. Special symmetries are considered in order to push the efficiency further. We also present algorithms to address the cyclic symmetry of the Riemann tensor. With these algorithms it is possible to simplify generic Riemann tensor polynomials.

We introduce vector phase states for multipath quantum interferometry and construct the vector phase positive operator-valued measure. We calculate SU(3) phase distributions for three-path quantum interferometry and discuss measurement limits.

We study the classical motion in bidimensional polygonal billiards on the sphere. In particular, we investigate the dynamics in tiling and generic rational and irrational equilateral triangles. Unlike the plane or the negative curvature cases we obtain a complex but regular dynamics.