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CRiSM Seminar - Tomasz Schreiber

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Location: A1.01

Professor Tomasz Schreiber (Nicolaus Copernicus University)
Polygonal Markov fields in the plane
Abstract: Polygonal Markov fields (PMFs), originally introduced by Arak, Clifford and Surgailis, can be regarded as random ensembles of non-intersecting planar polygonal contours with interaction determined by a rather flexible class of potentials. Not unexpectedly, such models share a number of important properties with the two-dimensional Ising model, including Ising-like phase transitions, spontaneous magnetisation and low temperature phase separation (Wulff construction). On the other hand, the polygonal fields exhibit remarkable features of their own, such as consistency property and explicit expressions for many crucial numeric and functional characteristics (free energy, correlation functions, integral geometric characteristics). Arguably the most important property of polygonal fields is that they admit a rich class of graphical constructions, all yielding the same field and often used as a crucial tool in theoretical developments on PMFs. In this talk we take the algorithmic graphical constructions as the starting point for defining the polygonal Markov fields, rather than the usual Gibbsian formalism. This point of view is compatible with applications of the PMFs for Bayesian image segmentation which we shall present (joint work with M.N.M. van Lieshout, R. Kluszczynski and M. Matuszak). Further, we shall also discuss our latest theoretical developments made possible by this approach, examples including the evaluation of higher order correlation functions, factorisation theorems and duality theory, where the dual object - the polygonal web - arises as the union of interacting critical branching polygonal walks in the plane. We shall conclude the talk indicating existing open problems and conjectures on the PMFs.

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