{"title":"Reversible Poisson-Kirchhoff Systems","authors":"Alexandre Boyer, Jérôme Casse, Nathanaël Enriquez, Arvind Singh","doi":"10.24033/bsmf.2863","DOIUrl":null,"url":null,"abstract":"We define a general class of random systems of horizontal and vertical weighted broken lines on the quarter plane whose distribution are proved to be translation invariant. This invariance stems from a reversibility property of the model. This class of systems generalizes several classical processes of the same kind, such as Hammersley's broken line processes involved in Last Passage Percolation theory or such as the six-vertex model for some special sets of parameters. The novelty comes here from the introduction of a weight associated with each line. The lines are initially generated by spatially homogeneous weighted Poisson Point Process and their evolution (turn, split, crossing) are ruled by a Markovian dynamics which preserves Kirchhoff's node law for the line weights at each intersection. Among others, we derive some new explicit invariant measures for some bullet models as well as new reversible properties for some six-vertex models with an external electromagnetic field.","PeriodicalId":55332,"journal":{"name":"Bulletin De La Societe Mathematique De France","volume":null,"pages":null},"PeriodicalIF":0.5000,"publicationDate":"2023-05-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Bulletin De La Societe Mathematique De France","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.24033/bsmf.2863","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
We define a general class of random systems of horizontal and vertical weighted broken lines on the quarter plane whose distribution are proved to be translation invariant. This invariance stems from a reversibility property of the model. This class of systems generalizes several classical processes of the same kind, such as Hammersley's broken line processes involved in Last Passage Percolation theory or such as the six-vertex model for some special sets of parameters. The novelty comes here from the introduction of a weight associated with each line. The lines are initially generated by spatially homogeneous weighted Poisson Point Process and their evolution (turn, split, crossing) are ruled by a Markovian dynamics which preserves Kirchhoff's node law for the line weights at each intersection. Among others, we derive some new explicit invariant measures for some bullet models as well as new reversible properties for some six-vertex models with an external electromagnetic field.
期刊介绍:
The Bulletin de la Société Mathématique de France was founded in 1873, and it has published works by some of the most prestigious mathematicians, including for example H. Poincaré, E. Borel, E. Cartan, A. Grothendieck and J. Leray. It continues to be a journal of the highest mathematical quality, using a rigorous refereeing process, as well as a discerning selection procedure. Its editorial board members have diverse specializations in mathematics, ensuring that articles in all areas of mathematics are considered. Promising work by young authors is encouraged.