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Five systems belonging to same universality class (from left to right): selfavoiding walk, percolation cluster boundary, SLE(6), Brownian motion boudary and boundary of corroded part of metal.
The development of statistical mechanics started in the late 19th century, with the pioneering works of Boltzmann and Gibbs. It aims at describing the typical behavior of macroscopic systems, based on the knowledge of how their microscopic constituents behave and interact. Since then, it went on to become one of the cornerstones of modern physics, following massive expansion during the whole 20th century.
On the physics side, a host of powerful ideas and techniques was introduced, such as the renormalization group approach to critical phenomena. However, most of these farreaching methods are mathematically illdefined, and providing a rigorous understanding of these procedures constitutes for mathematicians both a major challenge and a source of many exciting questionings. The ongoing process of constructing a rigorous conceptual framework has revealed a lot of new methods and ideas, triggered a fruitful dialog between mathematics and physics, and led to a number of resounding successes.
This project lies at the interface between probability theory, combinatorics, analysis, and both theoretical and mathematical physics. Providing a firm mathematical framework for wellestablished physics methods is at the heart of NCCR SwissMAP. Hence, this project not only fits perfectly in the general perspective of NCCR SwissMAP, but it actually represents one of its central constituents.
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