Fredholm's Minors of Arbitrary Order: Their Representations as a Determinant of Resolvents and in Terms of Free Fermions and an Explicit Formula for Their Functional Derivative

Isomonodromic tau functions on a torus as Fredholm determinants, and charged partitions

We prove that the isomonodromic tau function on a torus with Fuchsian singularities and generic monodromies in $GL(N,\mathbb{C})$ can be written in terms of a Fredholm determinant of Cauchy-Plemelj operators. We further show that the minor expansion of this Fredholm determinant is described by a series labeled by charged partitions. As an example, we show that in the case of $SL(2,\mathbb{C})$ this combinatorial expression takes the form of a dual Nekrasov-Okounkov partition function, or equivalently of a free fermion conformal block on the torus. Based on these results, we also propose a defi…

The $τ$-function of the Ablowitz-Segur family of solutions to Painlevé II as a Widom constant

$τ$-functions of certain Painlevé equations (PVI,PV,PIII) can be expressed as a Fredholm determinant. Further, the minor expansion of these determinants provide an interesting connection to Random partitions. This paper is a step towards understanding whether the $τ$-function of Painlevé II has a Fredholm determinant representation. The Ablowitz-Segur family of solutions are special one parameter solutions of Painlevé II and the corresponding $τ$-function is known to be the Fredholm determinant of the Airy Kernel. We develop a formalism for open contour in parallel to the one formulated in \ci…

The tacnode kernel: equality of Riemann-Hilbert and Airy resolvent formulas

We study nonintersecting Brownian motions with two prescribed starting and ending positions, in the neighborhood of a tacnode in the time-space plane. Several expressions have been obtained in the literature for the critical correlation kernel $K\tac(x,y)$ that describes the microscopic behavior of the Brownian motions near the tacnode. One approach, due to Kuijlaars, Zhang and the author, expresses the kernel (in the single time case) in terms of a $4\times 4$ matrix valued Riemann-Hilbert problem. Another approach, due to Adler, Ferrari, Johansson, van Moerbeke and Vet\H o in a series of pap…

On an integral formula for Fredholm determinants related to pairs of spectral projections

We consider Fredholm determinants of the form identity minus product of spectral projections corresponding to isolated parts of the spectrum of a pair of self-adjoint operators. We show an identity relating such determinants to an integral over the spectral shift function in the case of a rank-one perturbation. More precisely, we prove $$ -\ln \left(\det \big(\mathbf{1} -\mathbf{1} _{I}(A) \mathbf{1}_{\mathbb R\backslash I}(B)\mathbf{1}_{I}(A)\big) \right) = \int_I \text{d} x \int_{\mathbb R\backslash I} \text{d} y\, \frac{ξ(x)ξ(y)}{(y-x)^2}, $$ where $\mathbf{1}_J (\cdot)$ denotes the spectra…

On the Witten index in terms of spectral shift functions

We study the model operator $\mathbf{D}_{\mathbf{A}} = (d/dt) + \mathbf{A}$ in $L^2(\mathbb{R};\mathcal{H})$ associated with the operator path $\{A(t)\}_{t=-\infty}^{\infty}$, where $(\mathbf{A} f)(t) = A(t) f(t)$ for a.e.\ $t\in\mathbb{R}$, and appropriate $f \in L^2(\mathbb{R};\mathcal{H})$ (with $\mathcal{H}$ a separable, complex Hilbert space). Denoting by $A_{\pm}$ the norm resolvent limits of $A(t)$ as $t \to \pm \infty$, our setup permits $A(t)$ in $\mathcal{H}$ to be an unbounded, relatively trace class perturbation of the unbounded self-adjoint operator $A_-$, and no discrete spectrum…

纯SU(2)规范场理论配分函数与广义贝塞尔核

Pure $SU(2)$ gauge theory partition function and generalized Bessel kernel

该研究通过广义贝塞尔核的Fredholm行列式和Painlevé III方程的tau函数，揭示了纯SU(2)规范场理论配分函数的双重性质，并关联了Nekrasov求和与Young图对的组合结构。

一般 q-Painlevé III₃ tau 函数的 Riemann-Hilbert 问题、Fredholm 行列式、显式组合展开与连接公式

Riemann-Hilbert problems, Fredholm determinants, explicit combinatorial expansions, and connection formulas for the general $q$-Painlevé III$_3$ tau functions

本文将 A_7^{(1)'} 型 q-Painlevé 方程对应的 q-差分线性系统化为圆周上的 Riemann-Hilbert 问题，证明其 Fredholm 行列式满足双线性关系并给出显式组合展开，进而解决连接问题以确定 tau 函数的全局行为。

The linear stability of weakly charged and slowly rotating Kerr-Newman family of charged black holes

In this paper, we prove the linear stability of weakly charged and slowly rotating Kerr-Newman black holes under coupled gravitational and electromagnetic perturbations. We show that the solutions to the linearized Einstein-Maxwell equations decay at an inverse polynomial rate to a linearized Kerr-Newman solution plus a pure gauge term. This work builds on the framework developed by Häfner-Hintz-Vasy for the study of the Einstein vacuum equations. We work in the generalized wave map and Lorenz gauge. The proof involves the analysis of the resolvent of the Fourier transformed linearized Einstei…

Dynamical correlation functions for an impenetrable Bose gas with Neumann or Dirichlet boundary conditions

We study the time and temperature dependent correlation functions for an impenetrable Bose gas with Neumann or Dirichlet boundary conditions $\langle ψ(x_1,0)ψ^\dagger(x_2,t)\rangle _{\pm,T}$. We derive the Fredholm determinant formulae for the correlation functions, by means of the Bethe Ansatz. For the special case $x_1=0$, we express correlation functions with Neumann boundary conditions $\langleψ(0,0)ψ^\dagger(x_2,t)\rangle _{+,T}$, in terms of solutions of nonlinear partial differential equations which were introduced in \cite{kojima:Sl} as a generalization of the nonlinear Schrödinger eq…

线性电磁场中的Wigner输运

Wigner transport in linear electromagnetic fields

针对电磁场中规范不变Wigner方程数值求解难度高的问题，研究通过线性电磁场与二维输运的简化假设降低复杂度，提出两种Monte Carlo算法求解物理量平均值或Wigner函数，并衍生出启发式量子输运粒子模型。