Counting paths on the slit plane
Identifieur interne : 009F08 ( Main/Exploration ); précédent : 009F07; suivant : 009F09Counting paths on the slit plane
Auteurs : Mireille Bousquet-Mélou [France] ; Gilles Schaeffer [France]Source :
- Trends in Mathematics
Abstract
Abstract: We present a method, based on functional equations, to enumerate paths on the square lattice that avoid a horizontal half-line. The corresponding generating functions are algebraic, and sometimes remarkably simple: for instance, the number of paths of length 2n + 1 going from (0,0) to (1,0) and avoiding the nonpositive horizontal axis (except at their starting point) is C2n+1, the (2n + 1)th Catalan number More generally, we enumerate exactly all paths of length n starting from (0, 0) and avoiding the nonpositive horizontal axis. We then obtain limit laws for the coordinates of their endpoint: in particular, the average abscissa of their endpoint grows like $$ \sqrt {n}$$ (up to an explicit multiplicative constant), which shows that these paths are strongly repelled from the origin We derive from our results the distribution of the position where a random walk, starting from a given point, hits for the first time the horizontal half-line
Url:
DOI: 10.1007/978-3-0348-8405-1_9
Affiliations:
Links toward previous steps (curation, corpus...)
- to stream Istex, to step Corpus: 001A39
- to stream Istex, to step Curation: 001A19
- to stream Istex, to step Checkpoint: 002161
- to stream Main, to step Merge: 00A489
- to stream Main, to step Curation: 009F08
Le document en format XML
<record><TEI wicri:istexFullTextTei="biblStruct"><teiHeader><fileDesc><titleStmt><title xml:lang="en">Counting paths on the slit plane</title><author><name sortKey="Bousquet Melou, Mireille" sort="Bousquet Melou, Mireille" uniqKey="Bousquet Melou M" first="Mireille" last="Bousquet-Mélou">Mireille Bousquet-Mélou</name></author><author><name sortKey="Schaeffer, Gilles" sort="Schaeffer, Gilles" uniqKey="Schaeffer G" first="Gilles" last="Schaeffer">Gilles Schaeffer</name></author></titleStmt><publicationStmt><idno type="wicri:source">ISTEX</idno><idno type="RBID">ISTEX:722C0ACBA1EE44C8D61AD9BB0D305E5F716585A7</idno><date when="2000" year="2000">2000</date><idno type="doi">10.1007/978-3-0348-8405-1_9</idno><idno type="url">https://api.istex.fr/ark:/67375/HCB-2KC4TTK0-V/fulltext.pdf</idno><idno type="wicri:Area/Istex/Corpus">001A39</idno><idno type="wicri:explorRef" wicri:stream="Istex" wicri:step="Corpus" wicri:corpus="ISTEX">001A39</idno><idno type="wicri:Area/Istex/Curation">001A19</idno><idno type="wicri:Area/Istex/Checkpoint">002161</idno><idno type="wicri:explorRef" wicri:stream="Istex" wicri:step="Checkpoint">002161</idno><idno type="wicri:Area/Main/Merge">00A489</idno><idno type="wicri:Area/Main/Curation">009F08</idno><idno type="wicri:Area/Main/Exploration">009F08</idno></publicationStmt><sourceDesc><biblStruct><analytic><title level="a" type="main" xml:lang="en">Counting paths on the slit plane</title><author><name sortKey="Bousquet Melou, Mireille" sort="Bousquet Melou, Mireille" uniqKey="Bousquet Melou M" first="Mireille" last="Bousquet-Mélou">Mireille Bousquet-Mélou</name><affiliation wicri:level="3"><country xml:lang="fr">France</country><wicri:regionArea>CNRS, LaBRI, Université Bordeaux 1, 351 cours de la Libération, 33405, Talence Cedex</wicri:regionArea><placeName><region type="region" nuts="2">Nouvelle-Aquitaine</region><region type="old region" nuts="2">Aquitaine</region><settlement type="city">Talence</settlement></placeName></affiliation><affiliation wicri:level="1"><country wicri:rule="url">France</country></affiliation></author><author><name sortKey="Schaeffer, Gilles" sort="Schaeffer, Gilles" uniqKey="Schaeffer G" first="Gilles" last="Schaeffer">Gilles Schaeffer</name><affiliation wicri:level="3"><country xml:lang="fr">France</country><wicri:regionArea>CNRS, Loria, Campus Scientifique, 615 rue du Jardin Botanique - B.P. 101, 54602, Villers-lès-Nancy Cedex</wicri:regionArea><placeName><region type="region" nuts="2">Grand Est</region><region type="old region" nuts="2">Lorraine (région)</region><settlement type="city">Villers-lès-Nancy</settlement></placeName></affiliation><affiliation wicri:level="1"><country wicri:rule="url">France</country></affiliation></author></analytic><monogr></monogr><series><title level="s" type="main" xml:lang="en">Trends in Mathematics</title></series></biblStruct></sourceDesc></fileDesc><profileDesc><textClass></textClass></profileDesc></teiHeader><front><div type="abstract" xml:lang="en">Abstract: We present a method, based on functional equations, to enumerate paths on the square lattice that avoid a horizontal half-line. The corresponding generating functions are algebraic, and sometimes remarkably simple: for instance, the number of paths of length 2n + 1 going from (0,0) to (1,0) and avoiding the nonpositive horizontal axis (except at their starting point) is C2n+1, the (2n + 1)th Catalan number More generally, we enumerate exactly all paths of length n starting from (0, 0) and avoiding the nonpositive horizontal axis. We then obtain limit laws for the coordinates of their endpoint: in particular, the average abscissa of their endpoint grows like $$ \sqrt {n}$$ (up to an explicit multiplicative constant), which shows that these paths are strongly repelled from the origin We derive from our results the distribution of the position where a random walk, starting from a given point, hits for the first time the horizontal half-line</div></front></TEI><affiliations><list><country><li>France</li></country><region><li>Aquitaine</li><li>Grand Est</li><li>Lorraine (région)</li><li>Nouvelle-Aquitaine</li></region><settlement><li>Talence</li><li>Villers-lès-Nancy</li></settlement></list><tree><country name="France"><region name="Nouvelle-Aquitaine"><name sortKey="Bousquet Melou, Mireille" sort="Bousquet Melou, Mireille" uniqKey="Bousquet Melou M" first="Mireille" last="Bousquet-Mélou">Mireille Bousquet-Mélou</name></region><name sortKey="Bousquet Melou, Mireille" sort="Bousquet Melou, Mireille" uniqKey="Bousquet Melou M" first="Mireille" last="Bousquet-Mélou">Mireille Bousquet-Mélou</name><name sortKey="Schaeffer, Gilles" sort="Schaeffer, Gilles" uniqKey="Schaeffer G" first="Gilles" last="Schaeffer">Gilles Schaeffer</name><name sortKey="Schaeffer, Gilles" sort="Schaeffer, Gilles" uniqKey="Schaeffer G" first="Gilles" last="Schaeffer">Gilles Schaeffer</name></country></tree></affiliations></record>Pour manipuler ce document sous Unix (Dilib)
EXPLOR_STEP=$WICRI_ROOT/Wicri/Lorraine/explor/InforLorV4/Data/Main/Exploration
HfdSelect -h $EXPLOR_STEP/biblio.hfd -nk 009F08 | SxmlIndent | more
Ou
HfdSelect -h $EXPLOR_AREA/Data/Main/Exploration/biblio.hfd -nk 009F08 | SxmlIndent | more
Pour mettre un lien sur cette page dans le réseau Wicri
{{Explor lien
|wiki= Wicri/Lorraine
|area= InforLorV4
|flux= Main
|étape= Exploration
|type= RBID
|clé= ISTEX:722C0ACBA1EE44C8D61AD9BB0D305E5F716585A7
|texte= Counting paths on the slit plane
}}
| This area was generated with Dilib version V0.6.33. |  |