Mathematical Physics, Analysis and Geometry最新文献

We introduce a framework to identify Fluctuation Relations for vector-valued observables in physical systems evolving through a stochastic dynamics. These relations arise from the particular structure of a suitable entropic functional and are induced by transformations in trajectory space that are invertible but are not involutions, typical examples being spatial rotations and translations. In doing so, we recover as particular cases results known in the literature as isometric fluctuation relations or spatial fluctuation relations and moreover we provide a recipe to find new ones. We mainly discuss two case studies, namely stochastic processes described by a canonical path probability and non degenerate diffusion processes. In both cases we provide sufficient conditions for the fluctuation relations to hold, considering either finite time or asymptotically large times.

Existence of abelian BPS vortices on a manifold with boundary satisfying Neumann boundary conditions is proved. Numerical solutions are constructed on the Euclidean disk, and the (L^2)-metric of the moduli space of one vortex located at the interior of a rotationally symmetric disk is studied. The results presented extend previous work of Manton and Zhao on quotients of surfaces that admit a reflection.

In this paper, we apply the Fokas unified transform method to study the initial-boundary value problems for the coupled Sasa-Satsuma equation with a (5times 5) Lax pair on the half-line. The solution of the coupled Sasa-Satsuma equation is proved to be expressible in terms of the unique solution of a (5times 5) matrix Riemann-Hilbert problem in the complex k-plane. The relevant jump matrix is formulated using the matrix spectral functions S(k) and s(k), which are determined by the initial values and all boundary values at (x=0), respectively. While introducing the foundational Riemann-Hilbert formalism, we further investigate the corresponding generalized Dirichlet-Neumann mapping through the lens of the global relation. Moreover, by utilizing the perturbation expansion, we obtain an effective characterization of the unknown boundary values.

We study coined Random Quantum Walks on the hexagonal lattice, where the strength of disorder is monitored by the coin matrix. Each lattice site is equipped with an i.i.d. random variable that is uniformly distributed on the torus and acts as a random phase in every step of the QW. We show exponential decay of the fractional moments of the Green function in the regime of strong disorder, that is whenever the coin matrix is sufficiently close to the fully localized case, using a fractional moment criterion and a finite volume method. In the decorrelated case, we deduce dynamical localization. Moreover, we adapt a topological index to our model and thereby obtain transport for some coin matrices.

The random current representation of the Ising model, along with a related path expansion, has been a source of insight on the stochastic geometric underpinning of the ferromagnetic model’s phase structure and critical behavior in different dimensions. This representation is extended here to systems with a mild amount of frustration, such as generated by disorder operators and external field of mixed signs. Examples of the utility of such stochastic geometric representations are presented in the context of the deconfinement transition of the (mathbb {Z}_2) lattice gauge model – particularly in three dimensions– and in streamlined proofs of correlation inequalities with wide-ranging applications.

By blending Painlevé property with singularity confinement for a general arbitrary order Sawada-Kotera differential-difference equation, we find a proliferation of “tau-functions” (coming from confined patterns). However, only one of these function enters into the Hirota bilinear form (the others give multi-linear expressions) but it has specific relations with all others. We also discuss two modifications of the Sawada-Kotera equation. Fully discretizations and the express method for computing algebraic entropy are discussed.

We generalize double bracket vector fields, originally defined on semisimple Lie algebras, to Poisson manifolds equipped with a pseudo-Riemannian metric by utilizing a symmetric contravariant 2-tensor field. We extend the normal metric on an adjoint orbit of a compact semisimple Lie algebra to ensure that these vector fields become gradient vector fields on each symplectic leaf. Furthermore, we apply this construction to enhance the equilibria of Hamiltonian systems, specifically addressing the challenge of asymptotically stabilizing points that are already stable, through dissipation terms derived from generalized double bracket vector fields.

The transverse-field Ising model is widely studied as one of the simplest quantum spin systems. It is known that this model exhibits a phase transition at the critical inverse temperature (beta _textrm{c}), which is determined by the spin-spin couplings and the transverse field (qge 0). Björnberg Commun. Math. Phys. 323, 329–366 (2013) investigated the divergence rate of the susceptibility for the nearest-neighbor model as the critical point is approached by simultaneously changing the spin-spin coupling (Jge 0) and (q) in a proper manner, with fixed temperature. In this paper, we fix J and (q) and show that the susceptibility diverges as (({beta _textrm{c}}-beta )^{-1}) as (beta uparrow {beta _textrm{c}}) for (d>4) assuming an infrared bound on the space-time two-point function. One of the key elements is a stochastic-geometric representation in Björnberg & Grimmett J. Stat. Phys. 136, 231–273 (2009) and Crawford & Ioffe Commun. Math. Phys. 296, 447–474 (2010). As a byproduct, we derive a new lace expansion for the classical Ising model (i.e., (q=0)).

This paper calculates the fluctuations of eigenvalues of polynomials on large Haar unitaries cut by finite rank deterministic matrices. When the eigenvalues are all simple, we can give a complete algorithm for computing the fluctuations. When multiple eigenvalues are involved, we present several examples suggesting that a general algorithm would be much more complex.