48th Probability Summer School
Saint-Flour (France), 08 - 20 July 2018
IHES, Institut des hautes études scientifiques (France)
Graphical representations of the Ising model.
In these lectures, we will describe certain aspects of the classical Ising model on the hypercubic lattice using graphical representations. More specifically, we will introduce the low/high temperature expansions, the random current representation and the random-cluster model. We will explain how these models complement each other and enable us to prove very precise results on the Ising model. Among others, we will prove the following results: sharpness and continuity of the phase transition, classification of the invariant Gibbs states, triviality in high dimension, conformal invariance in 2 dimensions.
Tel Aviv University (Israel)
Planar maps, random walks and the circle packing theorem.
Koebe's circle packing theorem (1936) states that any simple, finite planar graph can be drawn in the plane by mapping the set of vertices to a set of circles with disjoint interiors so that two circles are tangent if and only if the corresponding vertices form an edge in the graph. This classical geometric theorem has deep applications in various fields of math such as complex analysis, spectral geometry, theory of computer science, and more recently, probability.
A guiding principle, pioneered by the late Oded Schramm, is that circle packing endows a planar graph with a geometry that, for many probabilistic purposes, is much better than the usual graph metric. For graphs with bounded degrees, a rich theory has been established connecting the geometry of the circle packing and the behavior of the random walk allowing to determine, among other properties, the recurrence or transience of the walk. However, most of this classical theory collapses without the assumption of bounded vertex degrees and hence cannot be applied to random planar maps.
In this course we will study this classical theory and develop a parallel theory for random planar maps. We will show various recent results within this theme: the almost sure recurrence for the uniform infinite planar triangulation (UIPT), the convergence to the geometric boundary of random walks on random hyperbolic maps, and the connectivity of the uniform spanning forest on planar maps.
University of Bristol (England) and Rényi Institute Budapest (Hungary)
Scaling limits for random walks and diffusion with long memory.
I will survey problems and results related to diffusivity and super-diffusivity of tracer particle motion in various models of random environment and interacting particles. The typical examples considered and treated in detail will be (maybe, in increasing order of complexity):
(a) random walks in random environment
(b) random walks and diffusions with self-interaction (typically, self-repellence)
(c) tagged particle diffusion in interacting particle systems of physical nature
In (a) and (b) I will analyse diffusive cases (where the central limit theorem/invariance principle holds) in three and more dimensions and anomalously fast diffusive (or, superdiffusive) cases in dimensions 1 and 2. In (c) I plan to present some selection of physically somewhat more relevant problems and few results.
On a technical level, I will (at least partially) cover topics like martingale approximation and Kipnis-Varadhan theory, Nash inequality and Nash moment bounds (with some extensions), Hilbert space and resolvent methods, and of course, ad hoc methods designed for the particular problems.