Dyson's constant in the asymptotics of the Fredholm determinant of the sine kernel

Dyson's constants in the asymptotics of the determinants of Wiener-Hopf-Hankel operators with the sine kernel

In this paper we are going to prove two asymptotic formulas for determinants det(I-K_s), as s goes to infinity, where K_s are the Wiener-Hopf-Hankel operators acting on L^2[0,s] with the kernels K(x-y)+K(x+y) and K(x-y)-K(x+y), respectively, and K(t):=sin(t)/(π*t). These formulas were conjectured by Dyson. The identification of the constant term in the asymptotics was an open problem for a long time.

Traces of creation-annihilation operators and Fredholm's formulas

We prove the formula for the traces of certain class of operators in bosonic and fermionic Fock spaces. Vertex operators belong to this class. Traces of vertex operators can be used for calculation of correlation functions and formfactors of integrable models (XXZ, Sine-Gordon, etc.), that is why we are interested in this problem. Also we show that Fredholm's minor and determinant can be expressed by such traces. We obtain a short proof of the Fredholm's formula for the solution of an integral equation. Analogue of Fredholm's formula in terms of permanents is obtained.

Riemann--Hilbert approach to the time-dependent generalized sine kernel

We derive the leading asymptotic behavior and build a new series representation for the Fredholm determinant of integrable integral operators appearing in the representation of the time and distance dependent correlation functions of integrable models described by a six-vertex R-matrix. This series representation opens a systematic way for the computation of the long-time, long-distance asymptotic expansion for the correlation functions of the aforementioned integrable models away from their free fermion point. Our method builds on a Riemann--Hilbert based analysis.

Dyson Brownian Motion 的曲率界

Curvature bound of Dyson Brownian Motion

本文在 sine β 系综构型空间上构造满足 BE(0,∞) 曲率界的对称 Dirichlet 型，证明局部泛函不等式与半群正则性，并刻画对偶半群为熵意义下的唯一梯度流，同时推广到一维 (β,s)-circular Riesz 气体。

Gap probability in the spectrum of random matrices and asymptotics of polynomials orthogonal on an arc of the unit circle

We obtain uniform asymptotics for polynomials orthogonal on a fixed and varying arc of the unit circle with a positive analytic weight function. We also complete the proof of the large $s$ asymptotic expansion for the Fredholm determinant with the kernel $\sin z/(πz)$ on the interval $[0,s]$, verifying a conjecture of Dyson for the constant term in the expansion. In the Gaussian Unitary Ensemble of random matrices, this determinant describes the probability for an interval of length $s$ in the bulk scaling limit to be free from the eigenvalues.

Scattering, determinants, hyperfunctions in relation to Gamma(1-s)/Gamma(s)

The method of realizing certain self-reciprocal transforms as (absolute) scattering, previously presented in summarized form in the case of the Fourier cosine and sine transforms, is here applied to the self-reciprocal transform f(y)-> H(f)(x) = \int_0^\infty J_0(2\sqrt{xy})f(y) dy, which is isometrically equivalent to the Hankel transform of order zero and is related to the functional equations of the Dedekind zeta functions of imaginary quadratic fields. This also allows to re-prove and to extend theorems of de Branges and V. Rovnyak regarding square integrable functions which are self-or-sk…