Markov loops and renormalization. (English) Zbl 1197.60075

From the introduction: The purpose of this paper is to explore some simple relations between Markov path and loop measures, spanning trees, determinants and Markov fields such as the free field. The main emphasis is put on the study of occupation fields defined by Poissonian ensembles of Markov loops. These were defined by G. F. Lawler and W. Werner [Probab. Theory Relat. Fields 128, No. 4, 565–588 (2004; Zbl 1049.60072)] for planar Brownian motion. We first present the results in the elementary framework of symmetric Markov chains on a finite space, proving also in passing several interesting results such as the relation between loop ensembles and spanning trees. We can show that the renormalized powers of the occupation field (i. e. the self intersection local times of the loop ensemble) converge in the two dimensional case and that they can be identified with higher even Wick powers of the free field when the intensity parameter is a half integer.


60J27 Continuous-time Markov processes on discrete state spaces
60K35 Interacting random processes; statistical mechanics type models; percolation theory
60J45 Probabilistic potential theory


Zbl 1049.60072
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