Table of contents

We show an exact equivalence of the free energy of the q-state Potts antiferromagnet on a lattice for the full temperature interval and the free energy of the q-state Potts model on the dual lattice for a semi-infinite interval of complex temperatures (CTs). This implies the existence of two quite different types of CT singularities: the generic kind, which does not obey universality or various scaling relations, and a special kind which does obey such properties and encodes information of direct physical relevance. We apply this observation to characterize CT properties of the Potts model on several lattices, to rule out two existing conjectures, and to determine the critical value of q above which the Potts antiferromagnet on the diced lattice has no phase transition.

We report a new, and relatively simple, procedure for finding new integrable differential-difference equations. The procedure starts from a rather general differential-difference equation taken in bilinear form and then searches for appropriate Bäcklund transformations for it. In this way three particular new integrable differential-difference systems are found and their corresponding Bäcklund transformations presented.

The basic features of the zeros of the Husimi phase-space density of a quantum eigenstate are discussed. It is demonstrated that some special properties of the harmonic oscillator related to the number of zeros and their location within the classical energy shell are not valid for anharmonic potentials, as illustrated by numerical examples for the Morse oscillator. Also discussed is the distribution of zeros in quantum Poincaré sections for systems with two degrees of freedom.

The steady-state particle-size distribution is examined, resulting from a breakage process with a maximum stable size. If the latter is much smaller than the characteristic size of the initial distribution, the steady-state distribution for continuous breakage kernels is independent of the breakage frequency and of initial conditions and is shown to be a simple function of the breakage kernel. For discontinuous kernels, the steady-state size distribution is always dependent on the initial conditions.

We consider self-avoiding polygons on the simple cubic lattice with a torsion fugacity. We use Monte Carlo methods to generate large samples as a function of the torsion fugacity and the number of edges in the polygon. Using these data we investigate the shapes of the polygons at large torsion fugacity and find evidence that the polygons have substantial helical character. In addition, we show that these polygons have induced writhe for any non-zero torsion fugacity, and that torsion and writhe are positively correlated.

A class of highest-weight irreducible representations of the quantum algebra is constructed, which is considerably larger than the currently known representations (Levendorskii S and Soibelman Y 1991 Commun. Math. Phys. 140 399). Within each module a basis is introduced and the transformation relations of the basis under the action of the Chevalley generators are explicitly written.

A Lorentz lattice gas with a fraction of the scatterers being pure backscatterers, the remaining being stochastic (right and left) rotators, is considered. The problem at hand is the evaluation of the probability x that the moving particle returns to its original site, in the limit where the density of sites occupied by scatterers is small and the lattice becomes a Cayley tree. In the special case of deterministic collision rules (Gunn - Ortuño model on a Cayley tree), an exact cubic equation for the Laplace transform of the distribution function of first-return times is derived and its asymptotic form near the threshold value (beyond which x = 1) is obtained. In the general case of stochastic models, a mean-field approximation is proposed for x and the mean return time . The approximation reduces to the exact result in the case of the deterministic model, as well as in the stochastic model without backscatterers. Comparison with Monte Carlo simulations shows a reasonable agreement in the dependence of x and on . The relevance of the results to the development of approximate analytic expressions for the diffusion coefficient is discussed.

We consider curvatures of all orders, as defined by the generalized Frenet - Serret formulae, along the trajectories of a classical Hamiltonian system with N degrees of freedom. In the spirit of previous experiments on the first two of them, time averages are numerically computed for the curvatures up to fifth order and for the microcanonical density in a typical anharmonic system (the FPU quartic chain), with checks in other models. Neat breakdowns of harmonic-like behaviour define thresholds to anharmonicity for every at distinct values of the order parameter (the energy density u). The threshold at fixed order i is independent of the total N, and it rapidly decreases as i grows. However, all curvatures are simultaneously sensitive or not to the initial conditions, for or respectively, confirming the previous identification of as an efficient indicator of the strong stochasticity transition. This phenomenology, which is discussed within the weak/strong stochasticity problem, gives a new insight into the progressive enforcement of a harmonic-like structure as u decreases.

We study the geometrical structure of the states in the low-temperature phase of a mean-field model for generalized spin glasses, the p-spin spherical model. This structure cannot be revealed by the standard methods, mainly due to the presence of an exponentially high number of states, each one having a vanishing weight in the thermodynamic limit. Performing a purely entropic computation, based on the TAP equations for this model, we define a constrained complexity which gives the overlap distribution of the states. We find that this distribution is continuous, non-random and highly dependent on the energy range of the considered states. Furthermore, we show which is the geometrical shape of the threshold landscape, giving some insight into the role played by threshold states in the dynamical behaviour of the system.

We present a field-theoretic analysis of high-precision Monte Carlo data for the Domb - Joyce model on the sc lattice. We vary the repulsion between two segments at the same point from zero (random walk) to infinity (self-avoiding walk). Eventually, we even include a repulsion between segments at neighbour points to increase the excluded volume beyond that of self-avoiding walks. The data for the end-to-end distance, the radius of gyration and the partition function clearly show the existence of two branches of universal behaviour. These two branches can be identified with the weak- and strong-coupling branch of the renormalization group, respectively. A quantitative analysis shows the ability of the standard field theoretic approach to describe the data, including the data for strong coupling, i.e. renormalized coupling u greater than its fixed point value . We conclude, in contrast with some claims in the literature, that the standard formalism of the renormalized field theory can be used even for (strong-coupling branch). In addition, exploiting the fast approach to asymptotic behaviour at the transition between weak and strong coupling, we obtain very precise estimates for the critical exponents of self-avoiding walks.

We use the concept of block variables to obtain a measure of order/disorder for some one-dimensional deterministic aperiodic sequences. For the Thue - Morse sequence, the Rudin - Shapiro sequence and the period-doubling sequence it is possible to obtain analytical expressions in the limit of infinite sequences. For the Fibonacci sequence, we present some analytical results which can be supported by numerical arguments. It turns out that the block variables show a wide range of different behaviour, some of them indicating that some of the considered sequences are more `random' than other. However, the method does not give any definite answer to the question of which sequence is more disordered than the other and, in this sense, the results obtained are negative. We compare this with some other ways of measuring the amount of order/disorder in such systems, and there seems to be no direct correspondence between the measures.

Many dynamical systems are thought to exhibit windows of attracting periodic behaviour for arbitrarily small perturbations from parameter values yielding chaotic attractors. This structural instability of chaos is particularly well documented and understood for the case of the one-dimensional quadratic map. In this paper we attempt to numerically characterize the global parameter-space structure of the dense set of periodic `windows' occurring in the chaotic regime of the quadratic map. In particular, we use scaling techniques to extract information on the probability distribution of window parameter widths as a function of period and location of the window in parameter space. We also use this information to obtain the uncertainty exponent which is a quantity that globally characterizes the ability to identify chaos in the presence of small parameter uncertainties.

We consider the sub-dominant eigenstates of the transfer matrix for the square - triangle random tiling model on an infinite strip of width L. A numerical algorithm for generation of the corresponding solution of the Bethe ansatz is developed. Numerical finite-size scaling analysis of the associated eigenvalues reveals the presence of both integer and non-integer critical exponents. The analytical value of one of the non-integer exponents is found. It is also shown numerically that, along with the leading correction to the free-energy density, for some excitations there is a term proportional to .

We revisit the hard-spheres lattice gas model in the spherical approximation proposed by Lebowitz and Percus. Although no disorder is present in the model, we find that the short-range dynamical restrictions in the model induce glassy behaviour. We examine the off-equilibrium Langevin dynamics of this model and study the relaxation of the density as well as the correlation, response and overlap two-time functions. We find that the relaxation proceeds in two steps as well as absence of anomaly in the response function. By studying the violation of the fluctuation - dissipation ratio we conclude that the glassy scenario of this model corresponds to the dynamics of domain growth in phase-ordering kinetics.

We have studied numerically the remanent magnetization in the six- and eight-dimensional Ising spin glass and we have compared it with the behaviour observed in the SK model, that we have also computed analytically. We also report the value of the dynamical critical exponent z in six dimensions measured in three different ways: from the behaviour of the energy and the susceptibility as a function of the Monte Carlo time and by studying the overlap - overlap correlation function as a function of space and time. These three results are in very good agreement with the mean field prediction z = 4. Finally we have checked numerically the analytical prediction, obtained by assuming spontaneously broken replica symmetry, for the most singular part of the propagator in the spin-glass phase. This last result supports the existence of spontaneously broken replica symmetry in finite-dimensional spin glasses.

The paper presents a method for finding the absolute best basis out of the library of bases offered by the wavelet packet decomposition of a discrete signal. Data-adaptive optimality is achieved with respect to an objective function, e.g. minimizing entropy, and concerns the choice of the Heisenberg rectangles tiling the time - frequency domain over which the energy of the signal is distributed. It is also shown how optimizing a concave objective function is equivalent to concentrating maximal energy into a few basis elements. Signal-adaptive basis selection algorithms currently in use do not generally find the absolute best basis, and moreover have an asymmetric time - frequency adaptivity - although a complete wavepacket decomposition comprises a symmetric set of tilings with respect to time and frequency. The higher adaptivity in frequency than in time can lead to ignoring frequencies that exist over short time intervals (short as compared to the length of the whole signal, not to the period corresponding to these frequencies). Revealing short-lived frequencies to the investigator can bring up important features of the studied process, such as the presence of coherent (`persistent') structures in a time series.

We work out examples of tensor products for distinct q-generalizations of Euclidean, oscillator and type superalgebras in cases where the method of highest-weight vectors will not apply. In particular, we use the three-term recurrence relations for Askey - Wilson polynomials to decompose the tensor product of representations from the positive discrete series and representations from the negative discrete series. We show that various q-analogues of the exponential function can be used to mimic the exponential mapping from a Lie algebra to its Lie group and we compute the corresponding matrix elements of the `group operators' on these representation spaces. We show that the matrix elements themselves transform irreducibly under the action of the quantum superalgebra. The most important q-series identities derived here are interpreted as the expansion of the matrix elements of a `group operator' (via the exponential mapping) in a tensor product basis in terms of the matrix elements in a reduced basis. They involve q-hypergeometric series with base .

We study an example of a perturbed Floquet Hamiltonian depending on a coupling constant . The spectrum is pure point and dense. We pick up an eigenvalue, namely , and show the existence of a function defined on such that for all , 0 is a point of density for the set I, and the Rayleigh - Schrödinger perturbation series represents an asymptotic series for the function . All ideas are developed and demonstrated when treating the explicit example, but some of them are expected to have an essentially wider range of application.

We accurately calculate the strong coupling expansion for anharmonic oscillators by means of a robust and stable numerical algorithm. The method applies to any state and to any anharmonicity degree. By means of the perturbation coefficients, we estimate the location of the branch points that determine the convergence radius of the strong coupling expansion.

We show that Floquet states can be constructed as strong-field limits of cavity-dressed states. The interaction between laser beams propagating outside the cavity with atoms or molecules are described by Floquet states, constructed from dressed states of average photon number quantized in a cavity of volume V, by taking the limit , , while keeping the photon density constant. Thus, Floquet theory can be seen as a fully quantum-mechanical model in the sense that it describes the photon exchanges between matter and radiation containing a large amount of photons. We discuss in this context adiabatic Floquet theory to treat a slow time dependence of the laser amplitude, to describe pulses, and of its frequency (chirping).

We show that appropriate q-analogues of the Schur polynomials provide rational solutions of a q-deformation of the Nth KdV hierarchy. This allows us to construct explicit examples of bispectral commutative rings of q-difference operators.

We describe a method for obtaining an analytic form for an inverse of a class of symmetric semi-infinite banded matrices, which are, apart from a finite number of terms, of the Toeplitz type. The results are applied to the determination of the spectrum of two-magnon excitations in Heisenberg spin chains with next-nearest-neighbour interactions.

The group theoretical analysis of Coulomb scattering based on the SO(3,1) group is revisited. Using matrix-valued differential operators, modifying the angular momentum and the Runge - Lenz vector used hitherto for the realization of the so(3,1) (Lorentz) algebra, we obtain a three-dimensional solvable two-channel scattering problem. The interaction term besides the Coulomb potential contains a non-local potential of LS-type. Using the momentum representation the S-matrix can be calculated analytically. By employing a canonical transformation, another solvable three-dimensional scattering problem is found, in agreement with the expectations of algebraic scattering theory. The potential in this case is of Pöschl - Teller type with an LS term. It is also pointed out that our matrix-valued realization of the so(3,1) algebra can be cast to an instructive form with the help of su(2) gauge fields. An interesting connection between gauge transformations and supersymmetry transformations of supersymmetric quantum mechanics is also observed. These results enable us to construct other solvable scattering problems by using su(2) gauge transformations.

Using some different Miura-type transformations, a C-integrable ordinary differential equation, the Riccati equation, is deformed to some different S-integrable models such as the (1 + 1)-dimensional and (2 + 1)-dimensional sinh - Gordon equations and Mikhailov - Dodd - Bullough equations.

We consider the general problem of motion of a rigid body about a fixed point under the action of an axisymmetric combination of potential and gyroscopic forces. We introduce six cases of this problem which are completely integrable for arbitrary initial conditions. The new cases generalize by several parameters all, but one, of the known results in the subject of rigid body dynamics. Namely, we generalize all the results due to Euler, Lagrange, Clebsch, Kovalevskaya, Brun and Lyapunov and also their subsequent generalizations by Rubanovsky and the present author.