There will be two main courses lasting for the entire school, and three mini-courses.
We will discuss diverse properties of random matrices, including the convergence of their spectral measure and its fluctuations, universality properties of the local behaviour of their eigenvalues. We will also study topological expansions and their applications in combinatorics. Considering several random matrices together, we will visit free probability theory and the combinatorics of colored graphs. This course is a graduate course which will require no particular prerequisite, but some basics in linear algebra and probability theory.X
Percolation is one of the simplest models displaying a phase transition. On a base graph, make bonds open with probability p and closed with probability 1-p, independently across the edges. Then, for p small, there is no infinite connected component of open bonds, while for p close to 1, an infinite connected component appears. The appearance of the infinite component occurs at a sharply defined critical threshold pc. Percolation displays fascinating scaling behavior close to the critical threshold pc. In this course, we investigate such phenomena on high-dimensional base graphs.
This year, it is 25 years ago that Hara and Slade used the lace expansion to analyze the high-dimensional percolation two-point function. Using differential inequalities by Aizenman, Barsky and Newman, this allowed them to prove mean-field behavior for percolation in high-dimensions. Mean-field behavior for percolation can be interpreted as the statement that the percolation critical exponents, that describe the behavior at or close to criticality, agree with those of percolation on an infinite tree, embedded into space. In particular, these results imply that there is no infinite component at the critical value. Intuitively, mean-field behavior is due to the space in high-dimensions being so vast that faraway parts of connected component become independent, as on a tree.
Recently, our understanding of high-dimensional percolation has been significantly improved. The precise scaling of the two-point function has been derived in work with Hara and Slade. Kozma and Nachmias have derived the scaling of arm probabilities using clever recursive inequalities that can be viewed as discrete analogues of the Aizenman-Barsky differential inequalities. The so-called incipient infinite cluster, the infinite connected component that is on the verge of appearing at the critical value, has been proved to exist and the anomalous behavior of random walk on it has been proved.
Further, the strong relations between finite-size effects for percolation on high-dimensional tori on the one hand, and random graphs, a lively research area in combinatorics and modern probability theory, on the other, have been uncovered. At criticality, the largest connected components have sizes of order n2/3, where n denotes the size of the base graph, similarly as for the Erdős-Rényi random graph or percolation on the complete graph. While the picture of the super-critical phase is yet far from complete, the sub-and critical phases of percolation on high-dimensional tori behave similarly as on the complete graph.
In this course, we summarizes the state of the art in high-dimensional percolation and its connections to random graphs, and address many open problems that could form inspiration for future research. The course requires no prior knowledge about percolation.X