Faculty Agoston Pisztora, Associate Professor
Dr. Math., Swiss Federal Institute of Technology at Zurich
E-mail: pisztora@andrew.cmu.edu
Office: Wean Hall 7212
Phone: 412-268-8489

Research:

Probability theory and the mathematical aspects of statistical mechanics are my principal fields of interest. My research is focused on the probabilistic analysis of random fields, phase transitions and certain disordered systems. The basic program is to explain phenomena observed at a macroscopic level by a direct analysis which is based entirely on microscopic hypotheses on the underlying system. An often used tool in the analysis is large deviation theory. Its role can be explained by the fact that certain physical systems of interest can be modeled by a restricted set of possible configurations which can be seen as an unrestricted system (which is easier to understand) in a very unlikely state.

A typical example of this program is a recent project to obtain a mathematical derivation of the Wulff theory of coexisting phases in the context of the classical Ising-Potts model. The Wulff theory describes the equilibrium behavior of coexisting phases in a system "governed" by surface tension. In particular, the Wulff construction predicts the shape of a droplet (crystal) in a non-isotropic environment. The solution of this problem beautifully combines several branches of mathematics: the probabilistic theory of the Ising model, large deviations, minimal surfaces and calculus of variations, geometric measure theory.

Another recent project is aimed at understanding the precise mechanism of linking invasion percolation to regular percolation at (or near) criticality. Critical systems are known to be a challenging subject. As far as probability theory goes, the difficulties can partially be understood by recognising the lack of self-averaging of important characteristics of these systems. In particular there is no "ergodic" behavior or (abstract) laws of large numbers, rather, the system often remains self-similar on every (or many) scales. Such phenomenon has never occurred and therefore never been studied in classical probability theory.

Selected Publications:

• Scaling inequality for the length of shortest paths in regular and invasion percolation. To appear in Ann. Appl. Probab.
• On the Wulff crystal in the Ising model. Ann. Probab. 28 (2000), 947–1017 (with R. Cerf)
• Phase coexistence in Ising, Potts and percolation models. To appear in Ann. Inst. H. Poincare (with R. Cerf)
• Large deviation principle for random walk in a quenched random environment in the low speed regime. Ann. Probab. 27 (1999), 1389–1413 (with T. Povel)
• Surface order large deviations for Ising, Potts and percolation models. Probab. Theory Rel. Fields 104 (1996), 427–466