Mathematical Physics, Analysis and Geometry最新文献

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).

We propose a systematic scheme for computing the variation of rearrangement multilinear functionals arising in the recently developed spectral geometry on noncommutative tori and (theta )-deformed Riemannian manifolds. It can be summarized as a category whose objects consist of spectral functions of the rearrangement multilinear functionals and morphisms are generated by transformations associated with basic operations of the variational calculus. The generators of the morphisms fulfill most, but not all of the relations in Connes’s cyclic category. Compare-and-contrast with, cyclic theory and Hopf cyclic theory have also been discussed.

Woronowicz proved the existence of the Haar state for compact quantum groups under a separability assumption later removed by Van Daele in a new existence proof. A minor adaptation of Van Daele’s proof yields an idempotent state in any non-empty weak-* compact convolution-closed convex subset of the state space. Such subsets, and their associated idempotent states, are studied in the case of quantum permutation groups.

A variety of local index formulas is constructed for gapped quantum Hamiltonians with periodic boundary conditions. All dimensions of physical space as well as many symmetry constraints are covered, notably one-dimensional systems in Class DIII as well as two- and three-dimensional systems in Class AII. The constructions are based on several periodic variations of the spectral localizer and are rooted in the existence of underlying fuzzy tori. For these latter, a general invariant theory is developed.

We give two characterizations for the essential self-adjointness of the weighted Laplacian on birth–death chains. The first involves the edge weights and vertex measure and is classically known; however, we give another proof using stability results, limit point-limit circle theory and the connection between essential self-adjointness and harmonic functions. The second characterization involves a new notion of capacity. Furthermore, we also analyze the essential self-adjointness of Schrödinger operators, use the characterizations for birth–death chains and stability results to characterize essential self-adjointness for star-like graphs, and give some connections to the (ell ^2)-Liouville property.

We proved that the normalized Ricci flow does not preserve the positivity of the Ricci curvature of invariant Riemannian metrics on every generalized Wallach space with (a_1+a_2+a_3le 1/2), in particular, such a property takes place on the homogeneous spaces (operatorname {SU}(k+l+m)/operatorname {S}(operatorname {U}(k)times operatorname {U}(l) times operatorname {U}(m))) and (operatorname {Sp}(k+l+m)/operatorname {Sp}(k)times operatorname {Sp}(l) times operatorname {Sp}(m)) independently on their parameters k, l and m. We proved that the positivity of the Ricci curvature is preserved under the normalized Ricci flow on generalized Wallach spaces with (a_1+a_2+a_3> 1/2) if the conditions (4left( a_j+a_kright) ^2ge (1-2a_i)(1+2a_i)^{-1}) are satisfied for all ({i,j,k}={1,2,3}). We also established that the homogeneous spaces (operatorname {SO}(k+l+m)/operatorname {SO}(k)times operatorname {SO}(l)times operatorname {SO}(m)) satisfy the above conditions if (max {k,l,m}le 11), moreover, additional conditions were found to keep (operatorname {Ric}>0) in cases when (max {k,l,m}le 11) is violated. Answers have also been found to similar questions about maintaining or non-maintaining the positivity of the Ricci curvature on all other generalized Wallach spaces given in the classification of Yu. G. Nikonorov.

This paper introduces a new (1,1/2) mixed spin Ising model (shortly, (1,1/2)-MSIM) having (J_1) and (J_2) competing interactions on (m, k)-ary trees. By constructing splitting Gibbs measures, we establish the presence of multiple Gibbs measures, which implies the occurrence of the phase transition for the (1, 1/2)-MSIM on the (m, k)-ary trees. Furthermore, the extremality of the two translation-invariant Gibbs measures is demonstrated for (1, k)-ary trees. Moreover, the extremality condition for the disordered phases is found and its non-extremality regimes are examined as well. It is well known that, to investigate lattice models on tree-like structures, conducting a stability analysis of the dynamical systems that represent the model and examining the behavior of the fixed points of these dynamical systems can provide significant insights into the model. We conducted a stability analysis around fixed points to investigate the behavior of the MSIM on the (m, k)-ary trees, also referred to as k-ary trees.