I am searching for a reference book on Fredholm determinants. I am mainly interested in applications to probability theory, where cumulative distribution functions of limit laws are expressed in terms of Fredholm determinants. I would like to answer questions like :

• How to express a Fredholm determinant on $L^2(\mathcal{C})$, where $\mathcal{C}$ is a contour in $\mathbb{C}$ and the kernel takes a parameter $x$, as a deteminant on $L^2(x, +\infty)$ ; and vice versa.

• Which types of kernels give which distributions. For example, in which cases we get the cumulative distribution function of the gaussian distribution ?

These questions are quite vague, but I mostly need to be more familiar with the theory and the classical tricks in $\mathbb{C}$.

I found the book "Trace ideals ans their applications", of Simon Barry, but I wonder if an other reference exists, ideally with applications to probability theory.

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