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Some questions on the Brownian Map motivated by its higher-genus analogues
Multi angle
Authors : ... (Author of the conference)
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05A15 - Exact enumeration problems, generating functions
05A16 - Asymptotic enumeration
05C80 - Random graphs
60J80 - Branching processes (Galton-Watson, birth-and-death, etc.)
60J85 - Applications of branching processes
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Abstract :
Several operations of combinatorial surgery can be used to relate maps of a given genus g to maps of genus g' is smaller than g. One of them is the Tutte/Lehman-Walsh decomposition, but more advanced constructions exist in the combinatorial toolbox, based on the Marcus-Schaeffer/ Miermont or the trisection bijections.
At the asymptotic level, these constructions lead to surprising relations between the enumeration of maps of genus g, and the genus 0 Brownian map. I will talk about some fascinating identities and (open) problems resulting from these connections, related to Voronoi diagrams, 'W-functionals', and properties of the ISE measure. Although the motivation comes from 'higher genus', these questions deal with the usual Brownian map as everyone likes it.
This is not very new material, and a (mostly French) part of the audience may have heard these stories one million times. But still I hope it will be interesting to advertise them here. In particular, I do not know if recent 'Liouville-based' approaches have anything to say about all this.
Keywords : random maps; high-genus maps; Brownian maps; Voronoï diagrams; bijections; trisections
MSC Codes : At the asymptotic level, these constructions lead to surprising relations between the enumeration of maps of genus g, and the genus 0 Brownian map. I will talk about some fascinating identities and (open) problems resulting from these connections, related to Voronoi diagrams, 'W-functionals', and properties of the ISE measure. Although the motivation comes from 'higher genus', these questions deal with the usual Brownian map as everyone likes it.
This is not very new material, and a (mostly French) part of the audience may have heard these stories one million times. But still I hope it will be interesting to advertise them here. In particular, I do not know if recent 'Liouville-based' approaches have anything to say about all this.
Keywords : random maps; high-genus maps; Brownian maps; Voronoï diagrams; bijections; trisections
05A15 - Exact enumeration problems, generating functions
05A16 - Asymptotic enumeration
05C80 - Random graphs
60J80 - Branching processes (Galton-Watson, birth-and-death, etc.)
60J85 - Applications of branching processes
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Information on the Video
Language : EnglishAvailable date : 04/02/2022 Conference Date : 20/01/2022 Subseries : Research talks arXiv category : Probability ; Combinatorics Mathematical Area(s) : Combinatorics ; Probability & Statistics Format : MP4 (.mp4) - HD Video Time : 00:52:16 Targeted Audience : Researchers ; Graduate Students Download : https://videos.cirm-math.fr/2022-01-20_Chapuis.mp4 
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Information on the Event
Event Title : Random Geometry / Géométrie aléatoireDates : 17/01/2022 - 21/01/2022 Event Year : 2022 Event URL : https://conferences.cirm-math.fr/2528.html
Citation Data
DOI : 10.24350/CIRM.V.19884603Cite this video as: (2022). Some questions on the Brownian Map motivated by its higher-genus analogues. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.19884603 URI : http://dx.doi.org/10.24350/CIRM.V.19884603 |
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Bibliography
- CHAPUY, Guillaume. The structure of unicellular maps, and a connection between maps of positive genus and planar labelled trees. Probability Theory and Related Fields, 2010, vol. 147, no 3, p. 415-447. - http://dx.doi.org/10.1007/s00440-009-0211-0
- CHAPUY, Guillaume. On tessellations of random maps and the $$ t_g $$ tg-recurrence. Probability Theory and Related Fields, 2019, vol. 174, no 1, p. 477-500. - http://dx.doi.org/10.1007/s00440-018-0865-6
- ADDARIO-BERRY, Louigi, ANGEL, Omer, CHAPUY, Guillaume, et al. Voronoi tessellations in the CRT and continuum random maps of finite excess. In : Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms. Society for Industrial and Applied Mathematics, 2018. p. 933-946. - https://doi.org/10.1137/1.9781611975031.60
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