Scaling models of random N x N hermitian matrices and passing to the limit N -» oo leads to integral operators whose Fredholm determinants describe the statistics of the spacing of the eigenvalues of hermitian matrices of large order. For the Gaussian Unitary Ensemble, and for many others' as well, the kernel one obtains by scaling in the "bulk" of the spectrum is the "sine kernel" — — . Rescal...

We compare various topologies on the space of (possibly unbounded) Fredholm selfadjoint operators and explain their K-theoretic relevance.∗ Introduction The work of Atiyah and Singer on the index of elliptic operators on manifolds has singled out the role of the space of bounded Fredholm operators in topology. It is a classifying space for a very useful functor, the topological K-theory. This m...

In this introductory review, we study Hankel and Toeplitz opera- tors considering them as acting on certain spaces of analytic functions, namely Hardy compare their spectral properties such compactness criteria. contrast to operators, the symbol a operator is not uniquely determined by operator. We also connect operators with Fredholm give some most beautiful essential spectrum continuous index...

The random point field which describes the position distribution of the system of ideal boson gas in a state of Bose-Einstein condensation is obtained through the thermodynamic limit. The resulting point field is given by convolution of two independent point fields: the so called boson process whose generating functional is represented by inverse of the Fredholm determinant for an operator rela...

The six-vertex model on an N × N square lattice with domain wall boundary conditions is considered. A Fredholm determinant representation for the partition function of the model is given. The kernel of the corresponding integral operator is of the so-called integrable type, and involves classical orthogonal polynomials. From this representation, a " reconstruction " formula is proposed, which e...

The six-vertex model on an N × N square lattice with domain wall boundary conditions is considered. A Fredholm determinant representation for the partition function of the model is given. The kernel of the corresponding integral operator is of the so-called integrable type, and involves classical orthogonal polynomials. From this representation, a " reconstruction " formula is proposed, which e...

We study the absolute continuity of the image measure of the canonical Poisson probability measure under nonlinear shifts. The Radon-Nykodim density function is expressed using a Carleman-Fredholm determinant and a divergence operator. Results are obtained for non-necessarily invertible transformations, under almost-sure differentiability hypothesis.