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Documents authored by Fusy, Éric

Document
DOI: 10.4230/LIPIcs.AofA.2024.6
Phase Transition for Tree-Rooted Maps

Authors: Marie Albenque, Éric Fusy, and Zéphyr Salvy

Published in: LIPIcs, Volume 302, 35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024)

Abstract
We introduce a model of tree-rooted planar maps weighted by their number of 2-connected blocks. We study its enumerative properties and prove that it undergoes a phase transition. We give the distribution of the size of the largest 2-connected blocks in the three regimes (subcritical, critical and supercritical) and further establish that the scaling limit is the Brownian Continuum Random Tree in the critical and supercritical regimes, with respective rescalings √{n/log(n)} and √n.
Cite as

Marie Albenque, Éric Fusy, and Zéphyr Salvy. Phase Transition for Tree-Rooted Maps. In 35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 302, pp. 6:1-6:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{albenque_et_al:LIPIcs.AofA.2024.6,
  author =	{Albenque, Marie and Fusy, \'{E}ric and Salvy, Z\'{e}phyr},
  title =	{{Phase Transition for Tree-Rooted Maps}},
  booktitle =	{35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024)},
  pages =	{6:1--6:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-329-4},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{302},
  editor =	{Mailler, C\'{e}cile and Wild, Sebastian},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.AofA.2024.6},
  URN =		{urn:nbn:de:0030-drops-204413},
  doi =		{10.4230/LIPIcs.AofA.2024.6},
  annote =	{Keywords: Asymptotic Enumeration, Planar maps, Random trees, Phase transition}
}
Document
DOI: 10.4230/LIPIcs.AofA.2020.9
Polyharmonic Functions And Random Processes in Cones

Authors: François Chapon, Éric Fusy, and Kilian Raschel

Published in: LIPIcs, Volume 159, 31st International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2020)

Abstract
We investigate polyharmonic functions associated to Brownian motions and random walks in cones. These are functions which cancel some power of the usual Laplacian in the continuous setting and of the discrete Laplacian in the discrete setting. We show that polyharmonic functions naturally appear while considering asymptotic expansions of the heat kernel in the Brownian case and in lattice walk enumeration problems. We provide a method to construct general polyharmonic functions through Laplace transforms and generating functions in the continuous and discrete cases, respectively. This is done by using a functional equation approach.
Cite as

François Chapon, Éric Fusy, and Kilian Raschel. Polyharmonic Functions And Random Processes in Cones. In 31st International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 159, pp. 9:1-9:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{chapon_et_al:LIPIcs.AofA.2020.9,
  author =	{Chapon, Fran\c{c}ois and Fusy, \'{E}ric and Raschel, Kilian},
  title =	{{Polyharmonic Functions And Random Processes in Cones}},
  booktitle =	{31st International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2020)},
  pages =	{9:1--9:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-147-4},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{159},
  editor =	{Drmota, Michael and Heuberger, Clemens},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.AofA.2020.9},
  URN =		{urn:nbn:de:0030-drops-120390},
  doi =		{10.4230/LIPIcs.AofA.2020.9},
  annote =	{Keywords: Brownian motion in cones, Heat kernel, Random walks in cones, Harmonic functions, Polyharmonic functions, Complete asymptotic expansions, Functional equations}
}
Document
DOI: 10.4230/LIPIcs.SEA.2018.24
Fast Spherical Drawing of Triangulations: An Experimental Study of Graph Drawing Tools

Authors: Luca Castelli Aleardi, Gaspard Denis, and Éric Fusy

Published in: LIPIcs, Volume 103, 17th International Symposium on Experimental Algorithms (SEA 2018)

Abstract
We consider the problem of computing a spherical crossing-free geodesic drawing of a planar graph: this problem, as well as the closely related spherical parameterization problem, has attracted a lot of attention in the last two decades both in theory and in practice, motivated by a number of applications ranging from texture mapping to mesh remeshing and morphing. Our main concern is to design and implement a linear time algorithm for the computation of spherical drawings provided with theoretical guarantees. While not being aesthetically pleasing, our method is extremely fast and can be used as initial placer for spherical iterative methods and spring embedders. We provide experimental comparison with initial placers based on planar Tutte parameterization. Finally we explore the use of spherical drawings as initial layouts for (Euclidean) spring embedders: experimental evidence shows that this greatly helps to untangle the layout and to reach better local minima.
Cite as

Luca Castelli Aleardi, Gaspard Denis, and Éric Fusy. Fast Spherical Drawing of Triangulations: An Experimental Study of Graph Drawing Tools. In 17th International Symposium on Experimental Algorithms (SEA 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 103, pp. 24:1-24:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{aleardi_et_al:LIPIcs.SEA.2018.24,
  author =	{Aleardi, Luca Castelli and Denis, Gaspard and Fusy, \'{E}ric},
  title =	{{Fast Spherical Drawing of Triangulations: An Experimental Study of Graph Drawing Tools}},
  booktitle =	{17th International Symposium on Experimental Algorithms (SEA 2018)},
  pages =	{24:1--24:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-070-5},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{103},
  editor =	{D'Angelo, Gianlorenzo},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SEA.2018.24},
  URN =		{urn:nbn:de:0030-drops-89597},
  doi =		{10.4230/LIPIcs.SEA.2018.24},
  annote =	{Keywords: Graph drawing, planar triangulations, spherical parameterizations}
}
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