Mathematical Physics, Analysis and Geometry最新文献

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.

Consider a non-relativistic quantum particle with wave function inside a region (Omega subset mathbb {R}^3), and suppose that detectors are placed along the boundary (partial Omega ). The question how to compute the probability distribution of the time at which the detector surface registers the particle boils down to finding a reasonable mathematical definition of an ideal detecting surface; a particularly convincing definition, called the absorbing boundary rule, involves a time evolution for the particle’s wave function (psi ) expressed by a Schrödinger equation in (Omega ) together with an “absorbing” boundary condition on (partial Omega ) first considered by Werner in 1987, viz., (partial psi /partial n=ikappa psi ) with (kappa >0) and (partial /partial n) the normal derivative. We provide here a discussion of the rigorous mathematical foundation of this rule. First, for the viability of the rule it plays a crucial role that these two equations together uniquely define the time evolution of (psi ); we point out here how, under some technical assumptions on the regularity (i.e., smoothness) of the detecting surface, the Lumer-Phillips theorem implies that the time evolution is well defined and given by a contraction semigroup. Second, we show that the collapse required for the N-particle version of the problem is well defined. We also prove that the joint distribution of the detection times and places, according to the absorbing boundary rule, is governed by a positive-operator-valued measure.

Fermionic time-reversal-invariant insulators in two dimension – class AII in the Kitaev table – come in two different topological phases. These are characterized by a (mathbb {Z}_2)-invariant: the Fu–Kane–Mele index. We prove that if two such insulators with different indices occupy regions containing arbitrarily large balls, then the spectrum of the resulting operator fills the bulk spectral gap. Our argument follows a proof by contradiction developed in [16] for quantum Hall systems. It boils down to showing that the (mathbb {Z}_2)-index can be computed only from bulk information in sufficiently large balls. This is achieved via a result of independent interest: a local trace formula for the (mathbb {Z}_2)-index.

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.