Table of contents

The transition between Hermitian and non-Hermitian matrices of the Gaussian unitary ensemble is revisited. An expression for the kernel of the rescaled Hermite polynomials is derived which expresses the sum in terms of the highest order polynomials. From this Christoffel–Darboux-like formula some results are derived including an extension to the complex plane of the Airy kernel.

We study the probability distribution function P^(β)_n(w) of the Schmidt-like random variable w = x²₁/(∑_{j = 1}ⁿx²_j/n), where x_j, (j = 1, 2, ..., n), are unordered eigenvalues of a given n × n β-Gaussian random matrix, β being the Dyson symmetry index. This variable, by definition, can be considered as a measure of how any individual (randomly chosen) eigenvalue deviates from the arithmetic mean value of all eigenvalues of a given random matrix, and its distribution is calculated with respect to the ensemble of such β-Gaussian random matrices. We show that in the asymptotic limit n → ∞ and for arbitrary β the distribution P^(β)_n(w) converges to the Marčenko–Pastur form, i.e. is defined as $P_{n}^{( \beta )}(w) \sim \sqrt{(4 - w)/w}$ for w ∈ [0, 4] and equals zero outside of the support, despite the fact that formally w is defined on the interval [0, n]. Furthermore, for Gaussian unitary ensembles (β = 2) we present exact explicit expressions for P^{(β = 2)}_n(w) which are valid for arbitrary n and analyse their behaviour.

Conformal field theories do not only classify two dimensional (2D) classical critical behaviour, but they also govern a certain class of 2D quantum critical behaviour. In this latter case, it is the ground state wave functional of the quantum theory that is conformally invariant, rather than the classical action. We show that the superconducting-insulating (SI) quantum phase transition in 2D Josephson junction arrays (JJAs) is a (doubled) c = 1 Gaussian conformal quantum critical point. The quantum action describing this system is a doubled Maxwell–Chern–Simons model in the strong coupling limit. We also argue that the SI quantum transitions in frustrated JJAs realize the other possible universality classes of conformal quantum critical behaviour, corresponding to the unitary minimal models at central charge c = 1 − 6/m(m + 1) for m ⩾ 3.

Using exact enumerations of self-avoiding walks (SAWs) we compute the inhomogeneous pressure exerted by a two-dimensional end-grafted polymer on the grafting line which limits a semi-infinite square lattice. The results for SAWs show that the asymptotic decay of the pressure as a function of the distance to the grafting point follows a power law with an exponent similar to that of Gaussian chains and is, in this sense, independent of excluded volume effects.

We give an explicit and simple derivation of the propagator for fractional Lévy motion using path integral methods. Some known forms of propagators, like for Brownian motion, are recovered in limiting cases. We also discuss the associated kinetic equations, as well as some extensions.

In this work, we perform a systematic analysis of various structural parameters that have an influence on the thermal rectification effect and the negative differential thermal resistance present in a one-dimensional anharmonic lattice with a mass gradient. After determining the optimal number of thermostated oscillators to enhance the rectification effect we determined that, for a random perturbation to the mass profile, rectification is enhanced, whereas negative differential thermal resistance is diminished. When a cubic term is included in the interaction potential thermal rectification is reduced, an effect more evident for large mass gradients. All results are obtained for larger system sizes than those so far considered for the present problem, thus revealing a complex interplay between the magnitude of the mass gradient and system size that is far from trivial and remains to be fully understood.

In recent times, attention has been directed to the problem of solving the Poisson equation, either in engineering scenarios (computational) or in regard to crystal structure (theoretical). Herein we study a class of lattice sums that amount to Poisson solutions, namely the n-dimensional forms

$$\begin{eqnarray*} &&\phi _n(r_1, \dots ,r_n) = \frac{1}{\pi ^2} \sum _{m_1, \dots , m_n {\rm \ odd}} \frac{{\rm e}^{{\rm i} \pi ( m_1 r_1 + \cdots + m_n r_n)}}{m_1^2 + \cdots + m_n^2}. \end{eqnarray*}$$

$\frac{1}{\pi } \log A$

A directed path in the vicinity of a hard wall exerts pressure on the wall because of loss of entropy. The pressure at a particular point may be estimated by estimating the loss of entropy if the point is excluded from the path. In this paper we determine asymptotic expressions for the pressure on the X-axis in models of adsorbing directed paths in the first quadrant. Our models show that the pressure vanishes in the limit of long paths in the desorbed phase, but there is a non-zero pressure in the adsorbed phase. We determine asymptotic approximations of the pressure for finite length Dyck paths and directed paths, as well as for a model of adsorbing staircase polygons with both ends grafted to the X-axis.

In this paper, we propose the study of the Boiti–Leon–Manna–Pempinelli equation from two points of view: the classical and the supersymmetric. In the classical case, we construct new solutions of this equation from Wronskian formalism and the Hirota method. Then, we introduce an $\mathcal {N}=1$ supersymmetric extension of the Boiti–Leon–Manna–Pempinelli equation. We thus produce a bilinear form and give multisolitons and superpartner solutions. As an application, we produce Bäcklund transformations.

We study bound states of a three-particle system in $\mathbb {R}^3$ described by the Hamiltonian H(λ_n) = H₀ + v₁₂ + λ_n(v₁₃ + v₂₃), where the particle pair {1, 2} has a zero-energy resonance and no bound states, while other particle pairs have neither bound states nor zero-energy resonances. It is assumed that for a converging sequence of coupling constants λ_n → λ_cr the Hamiltonian H(λ_n) has a sequence of levels with negative energies E_n and wavefunctions ψ_n, where the sequence ψ_n totally spreads in the sense that lim_{n → ∞}∫_{|ζ| ⩽ R}|ψ_n(ζ)|²dζ = 0 for all R > 0. We prove that for large n the angular probability distribution of three particles determined by ψ_n approaches the universal analytical expression, which does not depend on pair-interactions. The result has applications in Efimov physics and in the physics of halo nuclei.

The multi-indexed Jacobi polynomials are the main part of the eigenfunctions of exactly solvable quantum mechanical systems obtained by certain deformations of the Pöschl–Teller potential (Odake–Sasaki). By fine-tuning the parameter(s) of the Pöschl–Teller potential, we obtain several families of explicit and global solutions of certain second-order Fuchsian differential equations with an apparent singularity of characteristic exponents −2 and −1. They form orthogonal polynomials over x ∈ ( − 1, 1) with weight functions of the form (1 − x)^α(1 + x)^β/{(ax + b)⁴q(x)²}, in which q(x) is a polynomial in x.

In this work the Casimir effect is studied for scalar fields in the presence of boundaries and under the influence of arbitrary smooth potentials of compact support. In this setting, piston configurations are analyzed in which the piston is modeled by a potential. For these configurations, analytic results for the Casimir energy and force are obtained by employing the zeta function regularization method. Also, explicit numerical results for the Casimir force are provided for pistons modeled by a class of compactly supported potentials that are realizable as delta-sequences. These results are then generalized to higher dimensional pistons by considering additional Kaluza–Klein dimensions.

We evaluate the exact one-loop partition function for fundamental strings whose world surface ends on a cusp at the boundary of AdS₄ and has a 'jump' in $\mathbb {C}{ {\rm P}}^3$. This allows us to extract the stringy prediction for the ABJM-generalized cusp anomalous dimension Γ^ABJM_cusp(ϕ, θ) up to NLO in sigma-model perturbation theory. With a similar analysis, we present the exact partition functions for folded closed string solutions moving in the AdS₃ parts of ${ {\rm AdS}}_4\times \mathbb {C}{ {\rm P}}^3$ and AdS₃ × S³ × S³ × S¹ backgrounds. Results are obtained applying to the string solutions relevant for the AdS₄/CFT₃ and AdS₃/CFT₂ correspondence the tools previously developed for their AdS₅ × S⁵ counterparts.

The Von Mises second-order quasi-linear partial differential equation describes the dynamics of an irrotational, compressible and barotropic classical perfect fluid through a scalar function only, i.e. the velocity potential. It is shown here how to derive it in the case of a polytropic equation of state starting from a least action principle. The Lagrangian density is found to coincide with pressure. Once re-expressed in terms of the velocity potential, the action integral presents some similarities with other classical and quantum field theories. Aided by the Legendre transformation tool, we show that the nonlinear equation is completely integrable in the case of a non-steady parallel flow dynamics. A numerical solution of the equation in a critical shock wave forming scenario allows one to analyze then such a particular dynamics by using the so-called analogue gravity formalism which impressively links ordinary perfect fluids to curved spacetimes. Finally, implications for fluid dynamics and field theories are discussed.

This paper presents a numerical study on the dynamics of point vortex pairs moving on convex closed surfaces (ovaloids) approximated by triangular icosahedral meshes. Firstly, both a discretized conformal mapping between the ovaloid and the unit sphere $\mathbb {S}^2$ and a least-squares fitting for this approximated conformal map in terms of a spherical harmonic expansion are obtained. Then, an approximation of the conformal spherical metric of the ovaloid is derived by obtaining a spherical harmonic expansion of the corresponding conformal factor and its gradient on $\mathbb {S}^2$. The equations of motion for a point vortex pair on $\mathbb {S}^2$ with this quasi-conformal metric are integrated by using the classical Gaussian collocation method for some particular models.