3 Search Results for "Raschel, Kilian"


Document
Polyharmonic Functions in the Quarter Plane

Authors: Andreas Nessmann

Published in: LIPIcs, Volume 225, 33rd International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2022)


Abstract
In this article, a novel method to compute all discrete polyharmonic functions in the quarter plane for models with small steps, zero drift and a finite group is proposed. A similar method is then introduced for continuous polyharmonic functions, and convergence between the discrete and continuous cases is shown.

Cite as

Andreas Nessmann. Polyharmonic Functions in the Quarter Plane. In 33rd International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 225, pp. 15:1-15:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


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@InProceedings{nessmann:LIPIcs.AofA.2022.15,
  author =	{Nessmann, Andreas},
  title =	{{Polyharmonic Functions in the Quarter Plane}},
  booktitle =	{33rd International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2022)},
  pages =	{15:1--15:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-230-3},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{225},
  editor =	{Ward, Mark Daniel},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.AofA.2022.15},
  URN =		{urn:nbn:de:0030-drops-161016},
  doi =		{10.4230/LIPIcs.AofA.2022.15},
  annote =	{Keywords: Polyharmonic functions, Functional equations, Lattice paths, Random walks, Brownian motion, Generating functions, Laplace transforms}
}
Document
Scaling and Local Limits of Baxter Permutations Through Coalescent-Walk Processes

Authors: Jacopo Borga and Mickaël Maazoun

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


Abstract
Baxter permutations, plane bipolar orientations, and a specific family of walks in the non-negative quadrant are well-known to be related to each other through several bijections. We introduce a further new family of discrete objects, called coalescent-walk processes, that are fundamental for our results. We relate these new objects with the other previously mentioned families introducing some new bijections. We prove joint Benjamini - Schramm convergence (both in the annealed and quenched sense) for uniform objects in the four families. Furthermore, we explicitly construct a new fractal random measure of the unit square, called the coalescent Baxter permuton and we show that it is the scaling limit (in the permuton sense) of uniform Baxter permutations. To prove the latter result, we study the scaling limit of the associated random coalescent-walk processes. We show that they converge in law to a continuous random coalescent-walk process encoded by a perturbed version of the Tanaka stochastic differential equation. This result has connections (to be explored in future projects) with the results of Gwynne, Holden, Sun (2016) on scaling limits (in the Peanosphere topology) of plane bipolar triangulations. We further prove some results that relate the limiting objects of the four families to each other, both in the local and scaling limit case.

Cite as

Jacopo Borga and Mickaël Maazoun. Scaling and Local Limits of Baxter Permutations Through Coalescent-Walk Processes. 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. 7:1-7:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


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@InProceedings{borga_et_al:LIPIcs.AofA.2020.7,
  author =	{Borga, Jacopo and Maazoun, Micka\"{e}l},
  title =	{{Scaling and Local Limits of Baxter Permutations Through Coalescent-Walk Processes}},
  booktitle =	{31st International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2020)},
  pages =	{7:1--7:18},
  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.2020.7},
  URN =		{urn:nbn:de:0030-drops-120370},
  doi =		{10.4230/LIPIcs.AofA.2020.7},
  annote =	{Keywords: Local and scaling limits, permutations, planar maps, random walks in cones}
}
Document
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)


Copy BibTex To Clipboard

@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}
}
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