scholarly journals The critical surface fugacity for self-avoiding walks on a rotated honeycomb lattice

2013 ◽  
Vol DMTCS Proceedings vol. AS,... (Proceedings) ◽  
Author(s):  
Nicholas R. Beaton
Keyword(s):  
Phase Transition ◽  
Honeycomb Lattice ◽  
Critical Surface ◽  
Similar Model ◽  
Adsorption Phase ◽  
International Audience ◽  
Self Avoiding Walk ◽  
Self Avoiding Walks

International audience In a recent paper with Bousquet-Mélou, de Gier, Duminil-Copin and Guttmann (2012), we proved that a model of self-avoiding walks on the honeycomb lattice, interacting with an impenetrable surface, undergoes an adsorption phase transition when the surface fugacity is 1+√2. Our proof used a generalisation of an identity obtained by Duminil-Copin and Smirnov (2012), and confirmed a conjecture of Batchelor and Yung (1995). Here we consider a similar model of self-avoiding walk adsorption on the honeycomb lattice, but with the impenetrable surface placed at a right angle to the previous orientation. For this model there also exists a conjecture for the critical surface fugacity, made by Batchelor, Bennett-Wood and Owczarek (1998). We adapt the methods of the earlier paper to this setting in order to prove the critical surface fugacity, but have to deal with several subtle complications which arise. This article is an abbreviated version of a paper of the same title, currently being prepared for submission.

scholarly journals Families of prudent self-avoiding walks

2008 ◽  
Vol DMTCS Proceedings vol. AJ,... (Proceedings) ◽  
Author(s):  
Mireille Bousquet-Mélou
Keyword(s):  
Generating Function ◽  
Recurrence Relations ◽  
Square Lattice ◽  
Random Generation ◽  
Fourth Class ◽  
End Distance ◽  
International Audience ◽  
Self Avoiding Walk ◽  
Natural Classes ◽  
Self Avoiding Walks

International audience A self-avoiding walk on the square lattice is $\textit{prudent}$, if it never takes a step towards a vertex it has already visited. Préa was the first to address the enumeration of these walks, in 1997. For 4 natural classes of prudent walks, he wrote a system of recurrence relations, involving the length of the walks and some additional "catalytic'' parameters. The generating function of the first class is easily seen to be rational. The second class was proved to have an algebraic (quadratic) generating function by Duchi (FPSAC'05). Here, we solve exactly the third class, which turns out to be much more complex: its generating function is not algebraic, nor even $D$-finite. The fourth class ―- general prudent walks ―- still defeats us. However, we design an isotropic family of prudent walks on the triangular lattice, which we count exactly. Again, the generating function is proved to be non-$D$-finite. We also study the end-to-end distance of these walks and provide random generation procedures. Un chemin auto-évitant sur le réseau carré est $\textit{prudent}$, s'il ne fait jamais un pas en direction d'un point qu'il a déjà visité. Préa est le premier à avoir cherché à énumérer ces chemins, en 1997. Pour 4 classes naturelles de chemins prudents, il donne un système de relations de récurrence, impliquant la longueur des chemins et plusieurs paramètres "catalytiques'' supplémentaires. La première classe a une série génératrice simple, rationnelle. La deuxième a une série algébrique (quadratique) (Duchi, FPSAC'05). Nous comptons ici les chemins de la troisième classe, et observons un saut de complexité: la série obtenue n'est ni algébrique, ni même différentiellement finie. La quatrième classe, celle des chemins prudents généraux, résiste encore. Cependant, nous définissons un modèle isotrope de chemins prudents sur réseau triangulaire, que nous résolvons de nouveau, la série obtenue n'est pas différentiellement finie. Nous étudions aussi la vitesse d'éloignement de ces chemins, et proposons des algorithmes de génération aléatoire.

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