Planar Markovian Holonomy Fields: A first step to the characterization of Markovian Holonomy Fields

Abstract : We study planar random holonomy fields which are processes indexed by paths on the plane which behave well under the concatenation and orientation-reversing operations on paths. We define the Planar Markovian Holonomy Fields as planar random holonomy fields which satisfy some independence and invariance by area-preserving homeomorphisms properties. We use the theory of braids in the framework of classical probabilities: for finite and infinite random sequences the notion of invariance by braids is defined and we prove a new version of the de-Finetti's Theorem. This allows us to construct a family of Planar Markovian Holonomy Fields, the Yang-Mills fields, and we prove that any regular Planar Markovian Holonomy Field is a planar Yang-Mills field. This family of planar Yang-Mills fields can be partitioned into three categories according to the degree of symmetry: we study some equivalent conditions in order to classify them. Finally, we recall the notion of Markovian Holonomy Fields and construct a bridge between the planar and non-planar theories. Using the results previously proved in the article, we compute, for any Markovian Holonomy Field, the "law" of any family of contractible loops drawn on a surface.
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Submitted on : Friday, October 28, 2016 - 5:15:15 PM
Last modification on : Wednesday, May 15, 2019 - 3:36:49 AM

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  • HAL Id : hal-01100388, version 2
  • ARXIV : 1501.05077

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Franck Gabriel. Planar Markovian Holonomy Fields: A first step to the characterization of Markovian Holonomy Fields. 2014. ⟨hal-01100388v2⟩

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