A short proof of the Kac-Ward formula

The critical Ising model via Kac-Ward matrices

The Kac-Ward formula allows to compute the Ising partition function on any finite graph G from the determinant of 2^{2g} matrices, where g is the genus of a surface in which G embeds. We show that in the case of isoradially embedded graphs with critical weights, these determinants have quite remarkable properties. First of all, they satisfy some generalized Kramers-Wannier duality: there is an explicit equality relating the determinants associated to a graph and to its dual graph. Also, they are proportional to the determinants of the discrete critical Laplacians on the graph G, exactly when t…

Combinatorial formulation of Ising model revisited

In 1952, Kac and Ward developed a combinatorial formulation for the two dimensional Ising model which is another method of obtaining Onsager's formula for the free energy per site in the thermodynamic limit of the model. Feynman gave an important contribution to this formulation conjecturing a crucial mathematical relation which completed Kac and Ward ideas. In this paper, the method of Kac, Ward and Feynman for the free field Ising model in two dimensions is reviewed in a selfcontained way.

通过平坦SU(2)联络研究平面伊辛模型的零点

Zeros of planar Ising models via flat SU(2) connections

本文严格证明了Livine-Bonzom关于平面伊辛模型配分函数零点的几何公式，核心方法是构造Kac-Ward矩阵的零特征向量，并揭示其转移矩阵给出的SU(2)联络平坦性与零点存在性的等价关系。

Kac-Wakimoto character formula for the general linear Lie superalgebra

Character formulas for Lie superalgebras have been shown to have important applications to number theory and combinatorics. We prove the Kac-Wakimoto character formula for the general linear Lie superalgebra gl(m|n). This formula specializes to the well-known Kac-Weyl character formula when the modules are typical and to the Weyl denominator identity when the module is trivial. We also prove a determinantal character formula for KW-modules using the Kac-Wakimoto character formula.

Feynman-Kac formula for Levy processes and semiclassical (Euclidean) momentum representation

We prove a version of the Feynman-Kac formula for Levy processes and integro-differential operators, with application to the momentum representation of suitable quantum (Euclidean) systems whose Hamiltonians involve Lévy-type potentials. Large deviation techniques are used to obtain the limiting behavior of the systems as the Planck constant approaches zero. It turns out that the limiting behavior coincides with fresh aspects of the semiclassical limit of (Euclidean) quantum mechanics. Non-trivial examples of Levy processes are considered as illustrations and precise asymptotics are given for …

Spectral rigidity of random Schrödinger operators via Feynman-Kac formulas

We develop a technique for proving number rigidity (in the sense of Ghosh-Peres) of the spectrum of general random Schrödinger operators (RSOs). Our method makes use of Feynman-Kac formulas to estimate the variance of exponential linear statistics of the spectrum in terms of self-intersection local times. Inspired by recent results concerning Feynman-Kac formulas for RSOs with multiplicative white noise by Gorin, Shkolnikov and the first-named author, we use this method to prove number rigidity for a class of one-dimensional continuous RSOs of the form $-\frac12Δ+V+ξ$, where $V$ is a determini…

A Feynman-Kac Formula for Unbounded Semigroups

We prove a Feynman-Kac formula for Schrodinger operators with potentials V(x) that obey (for all ε> 0): V(x) \geq - ε|x|^2 - C_ε. Even though e^{-tH} is an unbounded operator, any ϕ, ψ\in L^2 with compact support lie in D(e^{-tH}) and <ϕ, e^{-tH}ψ> is given by a Feynman-Kac formula.

A Feynman-Kac-Itô Formula for magnetic Schrödinger operators on graphs

In this paper we prove a Feynman-Kac-Itô formula for magnetic Schrödinger operators on arbitrary weighted graphs. To do so, we have to provide a natural and general framework both on the operator theoretic and the probabilistic side of the equation. On the operator side we identify a very general class of potentials that allows the definition of magnetic Schrödinger operators. On the probabilistic side, we introduce an appropriate notion of stochastic line integrals with respect to magnetic potentials. Apart from linking the world of discrete magnetic operators with the probabilistic world thr…

Magnetic energies and Feynman-Kac-Itô formulas for symmetric Markov processes

Given a (conservative) symmetric Markov process on a metric space we consider related bilinear forms that generalize the energy form for a particle in an electromagnetic field. We obtain one bilinear form by semigroup approximation and another, closed one, by using a Feynman-Kac-Itô formula. If the given process is Feller, its energy measures have densities and its jump measure has a kernel, then the two forms agree on a core and the second is a closed extension of the first. In this case we provide the explicit form of the associated Hamiltonian.

伊藤过程下鞅可观察性的平滑性及广义费曼-科茨公式

Smoothness of martingale observables and generalized Feynman-Kac formulas

本文证明了伊藤过程中满足霍尔曼德准则时所有鞅可观察量的平滑性，并推导出允许退化扩散的广义费曼-科茨公式，同时应用于施拉姆-洛威演化的相关问题。

A Feynman-Kac formula for magnetic monopoles

We consider the quantum mechanics of a charged particle in the presence of Dirac's magnetic monopole. Wave functions are sections of a complex line bundle and the magnetic potential is a connection on the bundle. We use a continuum eigenfunction expansion to find an invariant domain of essential self-adjointness for the Hamiltonian. This leads to a proof of the a Feynman-Kac formula expressing solutions of the imaginary time Schrodinger equation as stochastic integrals.

An analog of the Feynman-Kac formula for Dirac's electron in electromagnetic field and the correspondence principle

The article describes a relation between the fundamental solutions of Dirac's equations for free electron and electron in given electromagnetic field viewed as functionals on bump functions. We explain how classical relativistic mechanics of a charged particle in given electromagnetic field arises from quantum mechanics of Dirac's electron in that field.

Feynman-Kac formulas for Dirichlet-Pauli-Fierz operators with singular coefficients

We derive Feynman-Kac formulas for Dirichlet realizations of Pauli-Fierz operators generating the dynamics of nonrelativistic quantum mechanical matter particles, which are minimally coupled to both classical and quantized radiation fields and confined to an arbitrary open subset of the Euclidean space. Thanks to a suitable interpretation of the involved Stratonovich integrals, we are able to retain familiar formulas for the Feynman-Kac integrands merely assuming local square-integrability of the classical vector potential and the coupling function in the quantized vector potential. Allowing f…

亚临界紫外正规化自旋玻色模型的费曼-科茨公式

A Feynman-Kac Formula for the Subcritical Ultraviolet-Renormalized Spin Boson Model

本文证明了亚临界紫外正规化自旋玻色模型的费曼-科茨公式，通过将自能正规化的哈密顿量作为两态量子系统与玻色量子场的线性耦合，将自旋视为跳跃过程，同时在通常的玻色福克空间上处理场，证明了红外正则模型的基态存在性在去除紫外截止后仍成立。

二维相对论Nelson模型中纤维哈密顿量的费曼-卡茨公式

Feynman-Kac formula for fiber Hamiltonians in the relativistic Nelson model in two spatial dimensions

本文研究了二维相对论Nelson模型中纤维哈密顿量的费曼-卡茨公式，通过能量重整化定义哈密顿量，并推导了新的费曼-卡茨公式，提供了另一种全译不变哈密顿量的推导方法。

磁场中及一般区域上的多极化子哈密顿量所生成半群的Feynman-Kac公式

Feynman-Kac formulas for semigroups generated by multi-polaron Hamiltonians in magnetic fields and on general domains

本文为包含Fröhlich哈密顿量的自伴算子半群证明Feynman-Kac公式，处理多极化子模型，涵盖奇异磁场、静电势及受限系统，通过新方法克服声子色散常数带来的困难。

形变量子化中的费曼-卡茨公式

The Feynman-Kac formula in deformation quantization

本文将费曼-卡茨公式引入形变量子化框架，证明经维克旋转后可由哈密顿算符星指数的相空间积分渐近极限求得基态能量，并给出高效计算实例。

Feynman-Kac formulas for the ultra-violet renormalized Nelson model

We derive Feynman-Kac formulas for the ultra-violet renormalized Nelson Hamiltonian with a Kato decomposable external potential and for corresponding fiber Hamiltonians in the translation invariant case. We simultaneously treat massive and massless bosons. Furthermore, we present a non-perturbative construction of a renormalized Nelson Hamiltonian in the non-Fock representation defined as the generator of a corresponding Feynman-Kac semi-group. Our novel analysis of the vacuum expectation of the Feynman-Kac integrands shows that, if the external potential and the Pauli-principle are dropped, t…

Feynman-Kac formula for perturbations of order $\leq 1$ and noncommutative geometry

Let $Q$ be a differential operator of order $\leq 1$ on a complex metric vector bundle $\mathscr{E}\to \mathscr{M}$ with metric connection $\nabla$ over a possibly noncompact Riemannian manifold $\mathscr{M}$. Under very mild regularity assumptions on $Q$ that guarantee that $\nabla^{\dagger}\nabla/2+Q$ generates a holomorphic semigroup $\mathrm{e}^{-zH^{\nabla}_{Q}}$ in $Γ_{L^2}(\mathscr{M},\mathscr{E})$ (where $z$ runs through a complex sector which contains $[0,\infty)$), we prove an explicit Feynman-Kac type formula for $\mathrm{e}^{-tH^{\nabla}_{Q}}$, $t>0$, generalizing the standard self…

Brownian-Time Processes: The PDE Connection II and the Corresponding Feynman-Kac Formula

We delve deeper into our study of the connection of Brownian-time processes (BTPs) to fourth order parabolic PDEs, which we introduced in a recent joint article with W. Zheng. Probabilistically, BTPs and their cousins BTPs with excursions form a unifying class of interesting stochastic processes that includes the celebrated IBM of Burdzy and other new intriguing processes, and is also connected to the Markov snake of Le Gall. BTPs also offer a new connection of probability to PDEs that is fundamentally different from the Markovian one. They solve fourth order PDEs in which the initial function…

Rigidity of Kac-Schwarz operators

In his work on the mathematical formulation of 2d quantum gravity Schwarz established a rigidity result for Kac-Schwarz operators for the n-KdV hierarchies. Later on, Adler and van Moerbeke as well as Fastré obtained different proofs of this result. We give yet another proof of the rigidity, one that in fact holds for all Drinfeld-Sokolov hierarchies.

星指数与费米系统及费曼-凯克公式

Star-exponential for Fermi systems and the Feynman-Kac formula

本文提出费米系统星指数的计算方法，并推导出费米版本的费曼-凯克公式，为费米系统研究提供新的计算工具。