Stochastic dynamics of random surfaces
Today, there is a revival of interest in the mathematical theory of lattice dimers, because these statistical mechanics models are in some sense solvable. Furthermore, dimer models play an important role in the study of high-temperature superconductivity and 2D quantum gravity, where the surfaces are of a more mathematical nature. The EU-funded project DMCP (Dimers, Markov chains and critical phenomena) did not focus on these applications. Instead, the properties of conformal invariance and Gaussian free-field-type fluctuations attracted the interest of scientists. An aspect of relevance to the project goals was the associated Markov dynamics and in particular, the speed of convergence. Working closely with experts from around the world, the DMCP team developed a new model for 2D stochastic growth that appears as irreversible random dynamics of discrete surfaces. If positive interface gradients are identified with 'particles' and negative gradients with 'holes', this model can be used to describe an interacting particle system. Breakthrough results were also obtained on the universality of height fluctuations for non-integrable models of dimers and their convergence to a massless Gaussian free field. Working on 2D square lattices, scientists extended the results to more general lattice models and studied the effects of non-periodic boundary conditions. DMCP work was communicated to the scientific community through several seminars in Europe and USA as well as publications in high-impact peer-reviewed journals or the preprint archive arXiv. The numerous results of the project indicate the commitment of the EU towards supporting progress in pure science and applying these findings.
Keywords
Random surfaces, statistical mechanics models, dimer models, DMCP, square lattices, boundary conditions
