Mathematical Physics, Analysis and Geometry最新文献

We introduce a kind of infinite-dimensional Lie algebra, it follows that a scheme for generating multi-component discrete integrable hierarchy of soliton equations is proposed. A multi-component discrete quadratic-form identity is presented which could be used to establish Hamiltonian structures of multi-component discrete integrable hierarchies. By considering the application, we obtain a coupled and a multi-component Volterra lattice hierarchies and their Liouville integrable Hamiltonian structures.

In this paper we compute the deformations of the Clarke–Oliveira instantons on the Bryant–Salamon Spin(7)-Manifold. The Bryant–Salamon Spin(7)-Manifold — the negative spinor bundle of (S^4) — is an asymptotically conical manifold where the link is the squashed 7-sphere. We use the deformation theory developed by the author in a previous paper to calculate the deformations of the Clarke–Oliveira instantons and calculate the virtual dimensions of the moduli spaces.

The Hirota–Miwa equation is one of the most celebrated fully discrete integrable systems. By introducing bilinear operators of trigonometric-type, we propose a novel variant of the Hirota–Miwa equation, which can be regarded as an integrable discretization of the Kadomtsev–Petviashvili-I (KPI) equation. It turns out that this new equation admits a number of physically significant solutions, including solitons, lumps, breathers, and periodic wave solutions. As far as we know, it is the first time that lump solutions have been reported in the context of fully discrete integrable systems. In addition, the numerical periodic wave solutions are computed by employing deep learning techniques. Finally, the reduction procedure is considered, which yields a trigonometric-type discrete Korteweg–de Vries (KdV) equation and a trigonometric-type discrete Boussinesq equation.

At the critical point, the probability density function of the Ising magnetization is believed to decay like (exp {(-x^{delta +1})}), where (delta ) is the Ising critical exponent that controls the decay to zero of the magnetization in a vanishing external field. In this paper, we discuss the presence of a power-law correction (x^{frac{delta -1}{2}}), which has been debated in the physics literature. We argue that whether such a correction is present or not is related to the asymptotic behavior of a function that measures the extent to which the average magnetization of a finite system with an external field is influenced by the boundary conditions. Our discussion is informed by a mixture of heuristic calculations and rigorous results. Along the way, we review some recent results on the critical Ising model and prove properties of the average magnetization in two dimensions which are of independent interest.

The present paper is a brief overview of random opinion dynamics on random graphs based on the Ising Lecture given by the author at the World Congress in Probability and Statistics, 12–16 August 2024, Bochum, Germany. The content is a snapshot of an interesting area of research that is developing rapidly.

We give a short non-technical introduction to the Ising model, and review some successes as well as challenges which have emerged from its study in probability and mathematical physics. This includes the infinite-volume theory of phase transitions, and ideas like scaling, renormalization group, universality, SLE, and random symmetry breaking in disordered systems and networks. This note is based on a talk given on 15 August 2024, as part of the Ising lecture during the 11th Bernoulli-IMS world congress, Bochum.

This paper investigates translation-invariant p-adic generalized Gibbs measures for the p-adic Ising model with a homogeneous external field on a Cayley tree of order three, assuming (p > 3). We demonstrate that if (p equiv 1 (operatorname {mod} {6})), then there exist four translation-invariant p-adic generalized Gibbs measures; if (p not equiv 1 (operatorname {mod} {6})), there exist exactly two. Additionally, for any prime (p > 3), we establish the occurrence of a phase transition in this model.

This paper provides an overview of the research on the metastable behaviour of the Ising model. We analyse the transition times from the set of metastable states to the set of the stable states by identifying the critical configurations that the system crosses with high probability during this transition and by computing the energy barrier that the system must overcome to reach the stable state starting from the metastable one. We describe the dynamical phase transition of the Ising model evolving under Glauber dynamics across various contexts, including different lattices, dimensions and anisotropic variants. The analysis is extended to related models, such as long-range Ising model, Blume-Capel and Potts models, as well as to dynamics like Kawasaki dynamics, providing insights into metastability across different systems.

We develop and prove new geometric and algebraic characterisations for locations of constituent skyrmions, as well as their signed multiplicity, using Sutcliffe’s JNR ansatz. Some low charge examples, and their similarity to BPS monopoles, are discussed. In addition, we provide Julia code for the further numerical study and visualisation of JNR skyrmions.

We study the spectral properties of the phase space localization operator (P_{R}), defined by the indicator function of a disk (D_{R}) of radius (R<1.) The localization is performed using a family of negative binomial states (NBS), labeled by points z in the unit disk (mathbb {D}) and parameterized by (nu > {frac{1}{2}}). These states are intrinsically connected to the 1D pseudo-harmonic oscillator (PHO) through superpositions of its eigenfunctions. The superposition coefficients form an orthonormal basis of a weighted Bergman space (mathcal {A}^{nu }left( mathbb {D}right) ), which also emerges as the eigenspace of a 2D Schrödinger operator with a magnetic field (proportional to (nu )) corresponding to the lowest hyperbolic Landau level. The eigenvalues (lambda _{j}^{nu ,R}) of (P_{R}) were obtained via a discrete spectral resolution within a shared eigenbasis for (P_{R}) and the PHO. By using these eigenvalues we obtain a closed-form expression for the variance of the particle count in (D_{R}) under the determinantal point process (DPP) defined by the weighted Bergman kernel. Beyond (D_{R}), the phase space content of (P_{R}) was estimated via the NBS photon-counting distribution, revealing a non-zero residual contribution. Using the coherent state transform associated with NBS, we mapped (P_{R}) to (mathcal {A}^{nu }left( mathbb {D}right) ) and we derive its explicit integral kernel (K_{nu ,R}left( z,wright) ), which converges to the Bergman kernel (K_{nu }left( z,wright) ) as (Rrightarrow 1).