AustralieFrV1, Main, Exploration, bibRecord, 00A450

Self-avoiding walks crossing a square

Identifieur interne : 00A450 ( Main/Exploration ); précédent : 00A449; suivant : 00A451

Self-avoiding walks crossing a square

Auteurs : M. Bousquet-Meelou [France] ; A. J. Guttmann [Australie] ; I. Jensen [Australie]

Source :

Journal of physics A : mathematical and general [ 0305-4470 ] ; 2005.

RBID : Pascal:05-0463180

Descripteurs français

Pascal (Inist)
- Réseau carré, Borne supérieure, Borne inférieure, Modèle, Transition phase, Vertex, Hamiltonien.

English descriptors

KwdEn :
- Hamiltonians, Lower bound, Models, Phase transitions, Square lattices, Upper bound, Vertex.

Abstract

We study a restricted class of self-avoiding walks (SAWs) which start at the origin (0, 0), end at (L, L), and are entirely contained in the square [0, L] x [0, L] on the square lattice Z². The number of distinct walks is known to grow as λ^L2+o(L2). We estimate λ = 1.744550 ± 0.000005 as well as obtaining strict upper and lower bounds, 1.628 < λ < 1.782. We give exact results for the number of SAWs of length 2L + 2K for K = 0, 1, 2 and asymptotic results for K = o(L^1/3). We also consider the model in which a weight orfugacity x is associated with each step of the walk. This gives rise to a canonical model of a phase transition. For x < 1/μ the average length of a SAW grows as L, while for x > 1/μ it grows as L². Here μ is the growth constant of unconstrained SAWs in Z². For x = 1/μ we provide numerical evidence, but no proof, that the average walk length grows as L^4/3. Another problem we study is that of SAWs, as described above, that pass through the central vertex of the square. We estimate the proportion of such walks as a fraction of the total, and find it to be just below 80% of the total number of SAWs. We also consider Hamiltonian walks under the same restriction. They are known to grow as τ^L2+o(L2) on the same L x L lattice. We give precise estimates for τ as well as upper and lower bounds, and prove that T < λ.

Affiliations:

Links toward previous steps (curation, corpus...)

to stream PascalFrancis, to step Corpus: 004708
to stream PascalFrancis, to step Curation: 001995
to stream PascalFrancis, to step Checkpoint: 004393
to stream Main, to step Merge: 00B043
to stream Main, to step Curation: 00A450

Le document en format XML

<record><TEI><teiHeader><fileDesc><titleStmt><title xml:lang="en" level="a">Self-avoiding walks crossing a square</title>
<author><name sortKey="Bousquet Meelou, M" sort="Bousquet Meelou, M" uniqKey="Bousquet Meelou M" first="M." last="Bousquet-Meelou">M. Bousquet-Meelou</name>
<affiliation wicri:level="4"><inist:fA14 i1="01"><s1>CNRS, LaBRI, Université Bordeaux I, 351 cours de la Libération</s1>
<s2>33405 Talence</s2>
<s3>FRA</s3>
<sZ>1 aut.</sZ>
</inist:fA14>
<country>France</country>
<placeName><region type="region" nuts="2">Nouvelle-Aquitaine</region>
<region type="old region" nuts="2">Aquitaine</region>
<settlement type="city">Talence</settlement>
<settlement type="city">Bordeaux</settlement>
</placeName>
<orgName type="university">Université Bordeaux I</orgName>
</affiliation>
</author>
<author><name sortKey="Guttmann, A J" sort="Guttmann, A J" uniqKey="Guttmann A" first="A. J." last="Guttmann">A. J. Guttmann</name>
<affiliation wicri:level="4"><inist:fA14 i1="02"><s1>ARC Centre of Excellence for Mathematics and Statistics of Complex Systems, Department of Mathematics and Statistics, The University of Melbourne</s1>
<s2>Victoria 3010</s2>
<s3>AUS</s3>
<sZ>2 aut.</sZ>
<sZ>3 aut.</sZ>
</inist:fA14>
<country>Australie</country>
<placeName><settlement type="city">Melbourne</settlement>
<region type="état">Victoria (État)</region>
</placeName>
<orgName type="university">Université de Melbourne</orgName>
</affiliation>
</author>
<author><name sortKey="Jensen, I" sort="Jensen, I" uniqKey="Jensen I" first="I." last="Jensen">I. Jensen</name>
<affiliation wicri:level="4"><inist:fA14 i1="02"><s1>ARC Centre of Excellence for Mathematics and Statistics of Complex Systems, Department of Mathematics and Statistics, The University of Melbourne</s1>
<s2>Victoria 3010</s2>
<s3>AUS</s3>
<sZ>2 aut.</sZ>
<sZ>3 aut.</sZ>
</inist:fA14>
<country>Australie</country>
<placeName><settlement type="city">Melbourne</settlement>
<region type="état">Victoria (État)</region>
</placeName>
<orgName type="university">Université de Melbourne</orgName>
</affiliation>
</author>
</titleStmt>
<publicationStmt><idno type="wicri:source">INIST</idno>
<idno type="inist">05-0463180</idno>
<date when="2005">2005</date>
<idno type="stanalyst">PASCAL 05-0463180 INIST</idno>
<idno type="RBID">Pascal:05-0463180</idno>
<idno type="wicri:Area/PascalFrancis/Corpus">004708</idno>
<idno type="wicri:Area/PascalFrancis/Curation">001995</idno>
<idno type="wicri:Area/PascalFrancis/Checkpoint">004393</idno>
<idno type="wicri:explorRef" wicri:stream="PascalFrancis" wicri:step="Checkpoint">004393</idno>
<idno type="wicri:doubleKey">0305-4470:2005:Bousquet Meelou M:self:avoiding:walks</idno>
<idno type="wicri:Area/Main/Merge">00B043</idno>
<idno type="wicri:Area/Main/Curation">00A450</idno>
<idno type="wicri:Area/Main/Exploration">00A450</idno>
</publicationStmt>
<sourceDesc><biblStruct><analytic><title xml:lang="en" level="a">Self-avoiding walks crossing a square</title>
<author><name sortKey="Bousquet Meelou, M" sort="Bousquet Meelou, M" uniqKey="Bousquet Meelou M" first="M." last="Bousquet-Meelou">M. Bousquet-Meelou</name>
<affiliation wicri:level="4"><inist:fA14 i1="01"><s1>CNRS, LaBRI, Université Bordeaux I, 351 cours de la Libération</s1>
<s2>33405 Talence</s2>
<s3>FRA</s3>
<sZ>1 aut.</sZ>
</inist:fA14>
<country>France</country>
<placeName><region type="region" nuts="2">Nouvelle-Aquitaine</region>
<region type="old region" nuts="2">Aquitaine</region>
<settlement type="city">Talence</settlement>
<settlement type="city">Bordeaux</settlement>
</placeName>
<orgName type="university">Université Bordeaux I</orgName>
</affiliation>
</author>
<author><name sortKey="Guttmann, A J" sort="Guttmann, A J" uniqKey="Guttmann A" first="A. J." last="Guttmann">A. J. Guttmann</name>
<affiliation wicri:level="4"><inist:fA14 i1="02"><s1>ARC Centre of Excellence for Mathematics and Statistics of Complex Systems, Department of Mathematics and Statistics, The University of Melbourne</s1>
<s2>Victoria 3010</s2>
<s3>AUS</s3>
<sZ>2 aut.</sZ>
<sZ>3 aut.</sZ>
</inist:fA14>
<country>Australie</country>
<placeName><settlement type="city">Melbourne</settlement>
<region type="état">Victoria (État)</region>
</placeName>
<orgName type="university">Université de Melbourne</orgName>
</affiliation>
</author>
<author><name sortKey="Jensen, I" sort="Jensen, I" uniqKey="Jensen I" first="I." last="Jensen">I. Jensen</name>
<affiliation wicri:level="4"><inist:fA14 i1="02"><s1>ARC Centre of Excellence for Mathematics and Statistics of Complex Systems, Department of Mathematics and Statistics, The University of Melbourne</s1>
<s2>Victoria 3010</s2>
<s3>AUS</s3>
<sZ>2 aut.</sZ>
<sZ>3 aut.</sZ>
</inist:fA14>
<country>Australie</country>
<placeName><settlement type="city">Melbourne</settlement>
<region type="état">Victoria (État)</region>
</placeName>
<orgName type="university">Université de Melbourne</orgName>
</affiliation>
</author>
</analytic>
<series><title level="j" type="main">Journal of physics A : mathematical and general</title>
<title level="j" type="abbreviated">J. phys. A : math. gen.</title>
<idno type="ISSN">0305-4470</idno>
<imprint><date when="2005">2005</date>
</imprint>
</series>
</biblStruct>
</sourceDesc>
<seriesStmt><title level="j" type="main">Journal of physics A : mathematical and general</title>
<title level="j" type="abbreviated">J. phys. A : math. gen.</title>
<idno type="ISSN">0305-4470</idno>
</seriesStmt>
</fileDesc>
<profileDesc><textClass><keywords scheme="KwdEn" xml:lang="en"><term>Hamiltonians</term>
<term>Lower bound</term>
<term>Models</term>
<term>Phase transitions</term>
<term>Square lattices</term>
<term>Upper bound</term>
<term>Vertex</term>
</keywords>
<keywords scheme="Pascal" xml:lang="fr"><term>Réseau carré</term>
<term>Borne supérieure</term>
<term>Borne inférieure</term>
<term>Modèle</term>
<term>Transition phase</term>
<term>Vertex</term>
<term>Hamiltonien</term>
</keywords>
</textClass>
</profileDesc>
</teiHeader>
<front><div type="abstract" xml:lang="en">We study a restricted class of self-avoiding walks (SAWs) which start at the origin (0, 0), end at (L, L), and are entirely contained in the square [0, L] x [0, L] on the square lattice Z<sup>2</sup>
. The number of distinct walks is known to grow as λ<sup>L2+o(L2)</sup>
. We estimate λ = 1.744550 ± 0.000005 as well as obtaining strict upper and lower bounds, 1.628 < λ < 1.782. We give exact results for the number of SAWs of length 2L + 2K for K = 0, 1, 2 and asymptotic results for K = o(L<sup>1/3</sup>
). We also consider the model in which a weight orfugacity x is associated with each step of the walk. This gives rise to a canonical model of a phase transition. For x < 1/μ the average length of a SAW grows as L, while for x > 1/μ it grows as L<sup>2</sup>
. Here μ is the growth constant of unconstrained SAWs in Z<sup>2</sup>
. For x = 1/μ we provide numerical evidence, but no proof, that the average walk length grows as L<sup>4/3</sup>
. Another problem we study is that of SAWs, as described above, that pass through the central vertex of the square. We estimate the proportion of such walks as a fraction of the total, and find it to be just below 80% of the total number of SAWs. We also consider Hamiltonian walks under the same restriction. They are known to grow as τ<sup>L2+o(L2)</sup>
 on the same L x L lattice. We give precise estimates for τ as well as upper and lower bounds, and prove that T < λ.</div>
</front>
</TEI>
<affiliations><list><country><li>Australie</li>
<li>France</li>
</country>
<region><li>Aquitaine</li>
<li>Nouvelle-Aquitaine</li>
<li>Victoria (État)</li>
</region>
<settlement><li>Bordeaux</li>
<li>Melbourne</li>
<li>Talence</li>
</settlement>
<orgName><li>Université Bordeaux I</li>
<li>Université de Melbourne</li>
</orgName>
</list>
<tree><country name="France"><region name="Nouvelle-Aquitaine"><name sortKey="Bousquet Meelou, M" sort="Bousquet Meelou, M" uniqKey="Bousquet Meelou M" first="M." last="Bousquet-Meelou">M. Bousquet-Meelou</name>
</region>
</country>
<country name="Australie"><region name="Victoria (État)"><name sortKey="Guttmann, A J" sort="Guttmann, A J" uniqKey="Guttmann A" first="A. J." last="Guttmann">A. J. Guttmann</name>
</region>
<name sortKey="Jensen, I" sort="Jensen, I" uniqKey="Jensen I" first="I." last="Jensen">I. Jensen</name>
</country>
</tree>
</affiliations>
</record>

Pour manipuler ce document sous Unix (Dilib)

EXPLOR_STEP=$WICRI_ROOT/Wicri/Asie/explor/AustralieFrV1/Data/Main/Exploration

HfdSelect -h $EXPLOR_STEP/biblio.hfd -nk 00A450 | SxmlIndent | more

HfdSelect -h $EXPLOR_AREA/Data/Main/Exploration/biblio.hfd -nk 00A450 | SxmlIndent | more

Pour mettre un lien sur cette page dans le réseau Wicri

{{Explor lien
   |wiki=    Wicri/Asie
   |area=    AustralieFrV1
   |flux=    Main
   |étape=   Exploration
   |type=    RBID
   |clé=     Pascal:05-0463180
   |texte=   Self-avoiding walks crossing a square
}}

This area was generated with Dilib version V0.6.33.
Data generation: Tue Dec 5 10:43:12 2017. Site generation: Tue Mar 5 14:07:20 2024

	Serveur d'exploration sur les relations entre la France et l'Australie
	Attention, ce site est en cours de développement ! Attention, site généré par des moyens informatiques à partir de corpus bruts. Les informations ne sont donc pas validées.

Serveur d'exploration sur les relations entre la France et l'Australie

Self-avoiding walks crossing a square

Self-avoiding walks crossing a square

Source :

Descripteurs français

English descriptors

Abstract

Links toward previous steps (curation, corpus...)

Le document en format XML

Pour manipuler ce document sous Unix (Dilib)

Pour mettre un lien sur cette page dans le réseau Wicri