## Particle systems and statistical physics

For more than a hundred years, physicists have put great effort into trying to understand the connection between the microscopic and macroscopic laws of nature. How can the seemingly ordered structure of matter on the macroscopic level be consistent with the highly disordered structure on the atomic and molecular level? This subdiscipline of physics is known as statistical mechanics.

During the last few decades, probability theory has played an important role in statistical mechanics, through the construction and analysis of large (sometimes infinite) spatial stochastic particle system models that capture various dichotomies between the microscopic and macroscopic behaviors observed in nature.

Much of our work is concerned with such models. A famous example is the Ising model for spontaneous magnetization. Each integer point in 2- or 3-dimensional space is assigned the value +1 or -1 randomly, according to rules which favor agreement between nearest neighbors. How strongly favored, depends on the interaction parameter J. When J=0, different spins are independent, while as J increases, the tendency to take the same values becomes stronger and stronger. It turns out that there is a critical value for J, below which the macroscopic behavior is disordered and above which it is ordered; this is an example of an important phenomenon known as phase transition, which in this case explains the occurrence of spontaneous magnetization.

The Ising model and many of its relatives are so called Markov random fields, meaning that they exhibit a spatial variant of the property that characterizes Markov chains (see the Markov theory group). Such models are used extensively in many fields outside of physics, such as spatial statistics and image analysis.

We also study so called percolation models, which are models for the connectivity behavior of random media. These have a variety of applications, ranging from the spread of liquid through porous media and electrical conductivity, to forest fires and epidemics. An example is given in the figure below: disks of unit radius are spread out randomly in 2-dimensional space (according to a so called Poisson process) with an average of L disks per unit area. Again we see a phase transition phenomenon: there is a critical value for L, below which the set of disks exhibits short-range connectivity only, and above which there are long- (in fact, infinite-) range connections. The figure shows realizations below, near, and above the critical value.

A poetic name for this model is the lilypad model. Imagine a pond with lilypads, and a snail trying to cross the pond but being limited to going on the lilypads. How large must the "lilypad density" L be for this to be possible?

We also study clumping phenomena in such models. This is relevant e.g. for understanding strength of inhomogeneous materials (see the material fatigue group) and can also be viewed as an extreme value problem.

Rigorous mathmematical solutions to a large number of difficult problems in this area have been obtained in recent years, by us and others. However, many of the problems appear to be, at present, too difficult to admit such solutions. When that happens, it is common practice to revert to simulation studies, in which case the so called Markov chain Monte Carlo algorithms (developed e.g. in the Markov theory group) are of great use.

Senior researchers: Olle Häggström, Johan Jonasson, Marianne Månsson, Jeff Steif.
Ph.D. students: Erik Broman, Marcus Isaksson, Fredrik Lundin, Oskar Sandberg, Johan Tykesson, Marcus Warfheimer.

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