Vladas Sidoravicius (IMPA)

Title: Structure of near critical clusters and continuity of the phase transition for Bernoulli percolation and Ising models in dimensions 2 and 3.

Abstract: During my tutorial lectures I will address to several different (but somehow strongly interrelated) questions of near-critical and critical behavior of Bernoulli percolation model in dimensions 2 and 3, and the Ising model in dimension 3.

The first set of questions is related to the so called model of self-destructive percolation. In 1992, B. Drossel and F. Schwabl introduced a paradigm model of the forest fire model in 2D, which received a lot of attention in theoretical and numerical physics literature, especially due to its relation to "self-organized" criticality. However, more than ten years later Rob van den Berg et al. (2004) raised doubts about existence of such model, in the same spirit as the non-existence of D. Aldous frozen percolation model in 2D, and they related it to the question of the critical behavior of the model of self-destructive percolation. It took another ten years of intense research to show that van den Berg et al. were right and the Drossel-Schwabl model indeed does not exist in 2D. I will discuss technical details of the proof of this remarkable fact. (Based of joint work with D. Kiss and I. Manolescu).

The second question is related to the continuity of phase transition for Bernoulli percolation in dimensions d > 2. I will give a brief overview of the current state of the problem, present known techniques and methods, and discuss in detail new ideas and the proof of the continuity for the two dimensional slabs. (Based on joint work with H. Duminil-Copin and V. Tassion).

The final, third part of my lectures will concern the proof of continuity of the phase transition for the Ising model in d > 2. This question has a very long history: L. Onsager in 1944 proved it for d=2 and M. Aizenman and R. Fernandez in 1986 proved it for d > 3. The case d=3 remained open for decades. In the last part of my lectures I will present a proof for all dimensions, and will explain the basic techniques, such as random current representation, reflection positivity, etc, which are used in the proof. (Based on joint work with M. Aizenman and H. Duminil-Copin).