The mesoscopic geometry of sparse random maps

Autor: Nicolas Curien, Igor Kortchemski, Cyril Marzouk

We investigate the structure of large uniform random maps with $n$ edges, $\mathrm{f}_n$ faces, and with genus $\mathrm{g}_n$ in the so-called sparse case, where the ratio between the number vertices and edges tends to $1$. We focus on two regimes: t

Externí odkaz: https://explore.openaire.eu/search/publication?articleId=doi_dedup___::010daf23b2378524788e1b02a6b8fadc

Markovian explorations of random planar maps are roundish

Autor: Cyril Marzouk, Nicolas Curien

Publikováno v: Bulletin de la société mathématique de France
Bulletin de la société mathématique de France, Société Mathématique de France, 2020, 148 (4), pp.709-732. ⟨10.24033/bsmf.2821⟩
Bulletin de la société mathématique de France, 2020, 148 (4), pp.709-732. ⟨10.24033/bsmf.2821⟩
Bulletin de la Société mathématique de France

The infinite discrete stable Boltzmann maps are "heavy-tailed" generalisations of the well-known Uniform Infinite Planar Quadrangulation. Very efficient tools to study these objects are Markovian step-by-step explorations of the lattice called peelin

Externí odkaz: https://explore.openaire.eu/search/publication?articleId=doi_dedup___::5870fb587a06421726041669acce8ba9

How fast planar maps get swallowed by a peeling process

Autor: Nicolas Curien, Cyril Marzouk

Publikováno v: Electronic Communications in Probability
Electronic Communications in Probability, Institute of Mathematical Statistics (IMS), 2018, 23 (18), 11 p. ⟨10.1214/18-ECP123⟩
Electron. Commun. Probab.
Electronic Communications in Probability, 2018, 23 (18), 11 p. ⟨10.1214/18-ECP123⟩

The peeling process is an algorithmic procedure that discovers a random planar map step by step. In generic cases such as the UIPT or the UIPQ, it is known [Curien & Le Gall, Scaling limits for the peeling process on random maps, Ann. Inst. Henri Poi

Externí odkaz: https://explore.openaire.eu/search/publication?articleId=doi_dedup___::20eee8809978b17a2df972d8d71350d3
https://hal.archives-ouvertes.fr/hal-01719797/file/Sous_diff_UIPQ.pdf

Simply generated non-crossing partitions

Autor: Cyril Marzouk, Igor Kortchemski

Publikováno v: Combinatorics, Probability and Computing
Combinatorics, Probability and Computing, Cambridge University Press (CUP), 2017, 26 (4), pp.560-592. ⟨10.1017/S0963548317000050⟩
Combinatorics Probability and Computing

We introduce and study the model of simply generated non-crossing partitions, which are, roughly speaking, chosen at random according to a sequence of weights. This framework encompasses the particular case of uniform non-crossing partitions with con

Externí odkaz: https://explore.openaire.eu/search/publication?articleId=doi_dedup___::dc15a41b2e3cce8056ce5d9feedb6255
https://hal.archives-ouvertes.fr/hal-01199288/file/primes200.pdf

On the sizes of burnt and fireproof components for fires on a large Cayley tree

Autor: Cyril Marzouk

Publikováno v: Ann. Inst. H. Poincaré Probab. Statist. 52, no. 1 (2016), 355-375
Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques
Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques, 2016, 52 (1), pp.355-375. ⟨10.1214/14-AIHP640⟩
Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques, Institut Henri Poincaré (IHP), 2016, 52 (1), pp.355-375. ⟨10.1214/14-AIHP640⟩
Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques, Institut Henri Poincaré (IHP), 2016, 52 (1), pp.355-375
Annales de l'Institut Henri Poincaré

We continue the study initiated by Jean Bertoin in 2012 of a random dynamics on the edges of a uniform Cayley tree with $n$ vertices in which, successively, each edge is either set on fire with some fixed probability $p_n$ or fireproof with probabili

Externí odkaz: https://explore.openaire.eu/search/publication?articleId=doi_dedup___::a694d1fc43987d4e779d935a930eb1aa
http://projecteuclid.org/euclid.aihp/1452089272