Department of Mathematical Sciences
Colloquia and Seminars
Areas of Research
About the Department
, Associate Professor
Ph.D., ETH ZurichE-mail: firstname.lastname@example.orgOffice:
Wean Hall 7212Phone:
My research is concerned with probabilistic analysis of random fields, phase transitions and certain disordered systems. The basic program is to explain phenomena observed at a macroscopic level by probabilistic analysis based entirely on microscopic hypotheses on the underlying system. Random systems often behave deterministically on the macroscopic level due to some kind of ergodic behavior. In some cases however randomness might be retained or “created" on the macroscopic level as well (for instance turbulence or critical systems in statistical physics). To understand the mechanism responsible for either type of behavior poses challenging research problems. More specifically I worked on
- Large deviations (LD) of the magnetization in Ising-Potts models closely related to LD of the density of the “infinte" cluster in classical and Fortuin-Kasteleyn percolation. Developing a quite generally applicable block renormalization procedure opened up also the possibility of a far more refined analysis of these models which lead to a justification (proof) of the classical Wulff construction predicting the shape of the macroscopic phase boundary in certain types of materials.
- A geometric characteristics of statistically homogenous random medium is the length of shortest connections between points and its scaling behavior with respect to the euclidean distance between the points. In supercritical percolation (a simple model for porous medium) the shortest scales linearly with the distance, and the proportionality factor F should diverge as the medium is thinned out to reach critical density. I studied LD behavior of the shortest distance in the supercritical case and its critical behavior in terms of the correlation length.
- Random Walk in Random Environment (RWRE): we established full LD principles (leading order asymptotics) for random walk with a drift in random media, in both the annealed (the environment and the walk is randomly picked for each trial) and quenched (only the walk is updated for every trial, the environment is random but frozen afterwards) cases.
- Conditional correlation inequalities for regular and FK percolation. Established an FKG type inequality with respect to the conditional measure where the conditioning is on certain connectivity events.
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