Free probability for probabilists

This is an introduction to some of the most probabilistic aspects of free probability theory

Brownian motion ensembles and parametric correlations of the transmission eigenvalues: Application to coupled quantum billiards and to disordered wires

The parametric correlations of the transmission eigenvalues $T_i$ of a $N$-channel quantum scatterer are calculated assuming two different Brownian motion ensembles. The first one is the original ensemble introduced by Dyson and assumes an isotropic diffusion for the $S$-matrix. The second Brownian motion ensemble assumes for the transfer matrix $M$ an isotropic diffusion yielded by a multiplicative combination law. We review the qualitative differences between transmission through two weakly coupled quantum dots and through a disordered line and we discuss the mathematical analogies between the Fokker-Planck equations of the two Brownian motion models.Comment: 33 pages, 7 postscript figures, the presented abstract is shortened in comparison to the abstract of the pape

Complex Gaussian multiplicative chaos

In this article, we study complex Gaussian multiplicative chaos. More precisely, we study the renormalization theory and the limit of the exponential of a complex log-correlated Gaussian field in all dimensions (including Gaussian Free Fields in dimension 2). Our main working assumption is that the real part and the imaginary part are independent. We also discuss applications in 2D string theory; in particular we give a rigorous mathematical definition of the so-called Tachyon fields, the conformally invariant operators in critical Liouville Quantum Gravity with a c=1 central charge, and derive the original KPZ formula for these fields.Comment: 66 pages, 5 figures, contains open problems and application in 2 dimensional string theory. The new version contains the KPZ formula for the Tachyon fields and further discussions about application