Topology of random real hypersurfaces

These are notes of the mini-course I gave during the CIMPA summer school at Villa de Leyva, Colombia, in July $2014$. The subject was my joint work with Damien Gayet on the topology of random real hypersurfaces, restricting myself to the case of projective spaces and focusing on our lower estimates. Namely, we estimate from (above and) below the mathematical expectation of all Betti numbers of degree $d$ random real projective hypersurfaces. For any closed connected hypersurface $\Sigma$ of $\mathbb{R}^n$, we actually estimate from below the mathematical expectation of the number of connected components of these degree $d$ random real projective hypersurfaces which are diffeomorphic to $\Sigma$.Comment: 18 pages, notes of the course I gave during the CIMPA summer school at Villa de Leyva, Colombia, in July 201

Linearly edge-reinforced random walks

We review results on linearly edge-reinforced random walks. On finite graphs, the process has the same distribution as a mixture of reversible Markov chains. This has applications in Bayesian statistics and it has been used in studying the random walk on infinite graphs. On trees, one has a representation as a random walk in an independent random environment. We review recent results for the random walk on ladders: recurrence, a representation as a random walk in a random environment, and estimates for the position of the random walker.Comment: Published at http://dx.doi.org/10.1214/074921706000000103 in the IMS Lecture Notes--Monograph Series (http://www.imstat.org/publications/lecnotes.htm) by the Institute of Mathematical Statistics (http://www.imstat.org

Notes on coherent backscattering from a random potential

We consider the quantum scattering from a random potential of strength $\lambda^{1/2}$ and with a support on the scale of the mean free path, which is of order $\lambda^{-1}$. On the basis of maximally crossed diagrams we provide a concise formula for the backscattering rate in terms of the Green's function for the kinetic Boltzmann equation. We briefly discuss the extension to wave scattering.Comment: 17 pages. 8 figure

Random processes via the combinatorial dimension: introductory notes

This is an informal discussion on one of the basic problems in the theory of empirical processes, addressed in our preprint "Combinatorics of random processes and sections of convex bodies", which is available at ArXiV and from our web pages.Comment: 4 page