Table of contents

We consider the one-dimensional Kardar–Parisi–Zhang (KPZ) equation with the half Brownian motion initial condition, studied previously through the weakly asymmetric simple exclusion process. We employ the replica Bethe ansatz and show that the generating function of the exponential moments of the height is expressed as a Fredholm determinant. From this, the height distribution and its asymptotics are studied. Furthermore, using the replica method we also discuss the multi-point height distribution. We find that some good properties of the deformed Airy functions play an important role in the analysis.

Making use of the complete calculation [1] of the chiral six-point correlation function

with the four ϕ_{1, 2} operators at the corners of an arbitrary rectangle and the point z = x + i y in the interior, for arbitrary central charge (equivalently, SLE parameter κ > 0), we calculate various quantities of interest for percolation (κ = 6) and many other two-dimensional critical points. In particular, we use C to specify the density at z of critical clusters conditioned to touch either or both vertical sides of the rectangle, with these sides 'wired', i.e. constrained to be in a single cluster, and the horizontal sides free. These quantities probe the structure of various cluster configurations, including those that contribute to the crossing probability. We first examine the effects of boundary conditions on C for the critical O(n) loop models in both high- and low-density phases and for both Fortuin–Kasteleyn (FK) and spin clusters in the critical Q-state Potts models. A Coulomb gas analysis then allows us to calculate the cluster densities with various conditionings in terms of the conformal blocks calculated in [1]. Explicit formulas generalizing Cardy's horizontal crossing probability to these models (using previously known results) are also presented. These solutions are employed to generalize previous results demonstrating factorization of higher order correlation functions to the critical systems mentioned. An explicit formula for the density of critical percolation clusters that cross a rectangle horizontally with free boundary conditions is also given. Simplifications of the hypergeometric functions in our solutions for various models are presented. High-precision simulations verify these predictions for percolation and for the Q = 2- and 3-state Potts models, including both FK and spin clusters. Our formula for the density of crossing clusters in percolation with free boundary conditions is also verified.

Interactions between a quantum system and its environment at low temperatures can lead to apparent violations of thermal laws for the system. The source of these violations is the coupling between the system and environment, which prevents the system from entering into a thermal state, a state of Gibb's form. On the other hand, for two-state systems, we show that one can define an effective temperature, placing the system into a 'pseudo-thermal' state where effective thermal laws can be applied. We then numerically explore these assertions for an n-state environment inspired by the spin-boson environment.

Lévy stable noise is often used to describe impulsive noise bursting in communication systems. This paper investigates the effects of small time delay on a bistable system driven by an aperiodic bipolar pulse signal and Lévy stable noise. We obtain the dynamical probability density of the system response by solving the approximated time-delayed fractional Fokker–Planck equation (FFPE) via an implicit finite difference method. A new approach to evaluate the system response time is presented. The bit error rate (BER) is employed to measure the performance of the bistable system in detecting binary signals. The theoretical BER is validated by the Monte–Carlo simulation. We find that the existence of time delay can change both the drift term and the diffusion coefficient in time-delayed FFPE. For small noise intensity, the time delay extends the system response time and thus reduces the detection performance. However, effects of this kind will fade away with the increase of noise intensity.

We present integrable lattice equations on a two-dimensional square lattice with coupled vertex and bond variables. In some of the models, the vertex dynamics is independent of the evolution of the bond variables, and one can write the equations as non-autonomous 'Yang–Baxter maps'. We also present a model in which the vertex and bond variables are fully coupled. Integrability is tested with algebraic entropy as well as multidimensional consistency.

The general features and characteristics of Kapteyn series, which are a special type of series involving the Bessel function, are investigated. For many applications in physics, astrophysics and mathematics, it is crucial to have closed-form expressions in order to determine their functional structure and parametric behavior. The closed-form expressions of Kapteyn series have mostly been limited to special cases, even though there are often similarities in the approaches used to reduce the series to analytically tractable forms. The goal of this paper is to review the previous work in the area and to show that Kapteyn series can be expressed as trigonometric or gamma function series, which can be evaluated in a closed form for specific parameters. Two examples with a similar structure are given, showing the complexity of Kapteyn series.

The Lagrangian–Hamiltonian unified formalism of Skinner and Rusk was originally stated for autonomous dynamical systems in classical mechanics. It has been generalized for non-autonomous first-order mechanical systems, as well as for first-order and higher order field theories. However, a complete generalization to higher order mechanical systems is yet to be described. In this work, after reviewing the natural geometrical setting and the Lagrangian and Hamiltonian formalisms for higher order autonomous mechanical systems, we develop a complete generalization of the Lagrangian–Hamiltonian unified formalism for these kinds of systems, and we use it to analyze some physical models from this new point of view.

The object of this paper is to investigate, classically and quantum mechanically, the relation existing between the position-dependent effective mass and damping–antidamping dynamics. The quantization of the equations of motion is carried out using the geometric interpretation of the motion, and we compare it with the one based on the ordering ambiguity scheme. Furthermore, we apply the obtained results to a Fermi gas of damped–antidamped particles, and solve the Schrödinger equation for an exponentially increasing (decreasing) mass in the presence of the Morse potential.

We study closed bosonic strings propagating both in a flat background with constant H-flux and in its T-dual configurations. We define a conformal field theory capturing linear effects in the flux and compute scattering amplitudes of tachyons, where the Rogers dilogarithm plays a prominent role. For the scattering of four tachyons, a fluxed version of the Virasoro–Shapiro amplitude is derived and its pole structure is analysed. In the case of an R-flux background obtained after three T-dualities, we find indications for a nonassociative target-space structure which can be described in terms of a deformed tri-product. Remarkably, this product is compatible with crossing symmetry of conformal correlation functions. We finally argue that the R-flux background flows to an asymmetric CFT.

A general regularization/renormalization scheme based on intrinsic properties of quantum fields as operator-valued distributions with adequate test functions is presented. The paracompactness property of the Minkowskian or Euclidean manifolds permits a unique definition of fields through integrals over the manifold based on test functions which are partition of unity (PU). These test functions turn out to provide a direct Lorentz-invariant scheme to the extension procedure of singular distributions and possess the unique property of being equal to their Taylor remainder of any order. When expressed through Lagrange's formulae, this remainder leads to specific procedures of extension in the UV and IR domains. These results, directly obtainable at the physical dimension D = 4, are found to depend on an arbitrary scale present in the definition of any PU test functions and relevant to the final RG-analysis of physical amplitudes. The method is of general character in that it comprises the well-known symmetry-preserving procedures of Bogoliubov–Parasiuk–Hepp–Zimmermann, Pauli–Villars subtractions at the level of propagators and dispersion relations. Symmetry preservation is indeed verified explicitly in simple QED/QCD gauge-boson contributions and in a covariant light-front dynamics treatment of the Yukawa model.