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pro vyhledávání: '"Marzouk, Cyril"'
Autor:
Kortchemski, Igor, Marzouk, Cyril
What is the analogue of L\'evy processes for random surfaces? Motivated by scaling limits of random planar maps in random geometry, we introduce and study L\'evy looptrees and L\'evy maps. They are defined using excursions of general L\'evy processes
Externí odkaz:
http://arxiv.org/abs/2402.04098
Autor:
Marzouk, Cyril
We first rephrase and unify known bijections between bipartite plane maps and labelled trees with the formalism of looptrees, which we argue to be both more relevant and technically simpler since the geometry of a looptree is explicitly encoded by th
Externí odkaz:
http://arxiv.org/abs/2202.08666
Publikováno v:
Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques, Tome 9 (2022), pp. 1305-1345
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:
http://arxiv.org/abs/2112.10719
Autor:
Kortchemski, Igor, Marzouk, Cyril
Publikováno v:
Ann. Appl. Probab. 33(5): 3755-3802 (October 2023)
We first establish new local limit estimates for the probability that a nondecreasing integer-valued random walk lies at time $n$ at an arbitrary value, encompassing in particular large deviation regimes. This enables us to derive scaling limits of s
Externí odkaz:
http://arxiv.org/abs/2101.01682
Autor:
Curien, Nicolas, Marzouk, Cyril
Publikováno v:
Prob. Math. Phys. 2 (2021) 1-26
The infinite discrete stable Boltzmann maps are generalisations of the well-known Uniform Infinite Planar Quadrangulation in the case where large degree faces are allowed. We show that the simple random walk on these random lattices is always subdiff
Externí odkaz:
http://arxiv.org/abs/1910.09623
Autor:
Marzouk, Cyril
Publikováno v:
Ann. Henri Lebesgue, Vol. 5 (2022), pp. 317-386
We study a configuration model on bipartite planar maps in which, given $n$ even integers, one samples a planar map with $n$ faces uniformly at random with these face degrees. We prove that when suitably rescaled, such maps always admit nontrivial su
Externí odkaz:
http://arxiv.org/abs/1903.06138
Autor:
Curien, Nicolas, Marzouk, Cyril
Publikováno v:
Bull. Soc. Math. France. 148 (4), 709-732, 2020
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:
http://arxiv.org/abs/1902.10624
Autor:
Marzouk, Cyril
For non-negative integers $(d_n(k))_{k \ge 1}$ such that $\sum_{k \ge 1} d_n(k) = n$, we sample a bipartite planar map with $n$ faces uniformly at random amongst those which have $d_n(k)$ faces of degree $2k$ for every $k \ge 1$ and we study its asym
Externí odkaz:
http://arxiv.org/abs/1902.04539
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Autor:
Marzouk, Cyril
Publikováno v:
ALEA, Lat. Am. J. Probab. Math. Stat. 15 , 1089-1122 (2018)
We discuss the asymptotic behaviour of random critical Boltzmann planar maps in which the degree of a typical face belongs to the domain of attraction of a stable law with index $\alpha \in (1,2]$. We prove that when conditioning such maps to have $n
Externí odkaz:
http://arxiv.org/abs/1803.07899