Two-sided prudent walks: a solvable non-directed model of polymer adsorption

Beaton, Nicholas R.; Iliev, G K

doi:10.1088/1742-5468/2015/09/p09014

Cited by 4 publications

(11 citation statements)

References 26 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…However, the importance of Theorem 2.1 goes beyond IPSAW itself. The 2-sided prudent trajectories have indeed been studied already in the mathematical litterature, see e.g., Bousquet-Mélou (2010), Dethridge and Guttmann (2008) or Beaton and Iliev (2015). It was conjectured in Bousquet-Mélou (2010) or Dethridge and Guttmann (2008) that the exponential growth rate of the cardinality of 2-sided prudent paths (as a function of their length) equals that of generic prudent paths and this is precisely what Theorem 2.1 says at β = 0.…”

Section: Discussionmentioning

confidence: 86%

Collapse transition of the interacting prudent walk

Pétrélis

2018

Ann. Inst. Henri Poincaré Comb. Phys. Interact.

View full text Add to dashboard Cite

This article is dedicated to the study of the 2-dimensional interacting prudent self-avoiding walk (referred to by the acronym IPSAW) and in particular to its collapse transition. The interaction intensity is denoted by β > 0 and the set of trajectories consists of those self-avoiding paths respecting the prudent condition, which means that they do not take a step towards a previously visited lattice site. The IPSAW interpolates between the interacting partially directed self-avoiding walk (IPDSAW) that was analyzed in details in, e.g., Zwanzig and Lauritzen (1968), Brak et al. (1992), and Nguyen and Pétrélis (2013), and the interacting self-avoiding walk (ISAW) for which the collapse transition was conjectured in Saleur (1986).Three main theorems are proven. We show first that IPSAW undergoes a collapse transition at finite temperature and, up to our knowledge, there was so far no proof in the literature of the existence of a collapse transition for a non-directed model built with self-avoiding path. We also prove that the free energy of IPSAW is equal to that of a restricted version of IPSAW, i.e., the interacting two-sided prudent walk. Such free energy is computed by considering only those prudent path with a general north-east orientation. As a by-product of this result we obtain that the exponential growth rate of generic prudent paths equals that of two-sided prudent paths and this answers an open problem raised in e.g., Bousquet-Mélou (2010) or Dethridge and Guttmann (2008). Finally we show that, for every β > 0, the free energy of ISAW itself is always larger than β and this rules out a possible self-touching saturation of ISAW in its conjectured collapsed phase.2010 Mathematics Subject Classification. 82B26, 60K35, 82B41, 60K15.

show abstract

Section: Discussionmentioning

confidence: 86%

Collapse transition of the interacting prudent walk

Pétrélis

2018

Ann. Inst. Henri Poincaré Comb. Phys. Interact.

View full text Add to dashboard Cite

show abstract

“…We now state our first lemma about convergence of finite dimensional distributions. 1) , t (2) , . .…”

Section: Convergence To Brownian Motionmentioning

confidence: 99%

“…1) | 2 as n → ∞. (6.3) With an appropriate change in (5.20) we obtain (6.2) for N = 1 from Theorem 2.2.To advance the induction we assume that (6.2) holds when N is replaced by N − 1.…”

mentioning

confidence: 99%

Prudent walk in dimension six and higher

Heydenreich

Taggi

2022

Preprint

View full text Add to dashboard Cite

We study the high-dimensional uniform prudent self-avoiding walk, which assigns equal prob- ability to all nearest-neighbor self-avoiding paths of a fixed length that respect the prudent condi- tion, namely, the path cannot take any step in the direction of a previously visited site. We prove that the prudent self-avoiding walk converges to Brownian motion under diffusive scaling if the dimension is large enough. The same result is true for weakly prudent walk in dimension d > 5. A challenging property of the high-dimensional prudent walk is the presence of an infinite- range self-avoidance constraint. Interestingly, as a consequence of such a strong self-avoidance constraint, the upper critical dimension of the prudent walk is five, and thus greater than for the classical self-avoiding walk. Mathematics Subject Classification: 82B41 · 60G50

show abstract

“…The prudent walk has also been used in Beaton and Iliev (2015) to build and investigate a non-directed model of polymer adsorption.…”

Section: A Non Directed Model Of Isaw: the Ipsawmentioning

confidence: 99%

Interacting partially directed self-avoiding walk: a probabilistic perspective

Carmona¹,

Nguyen²,

Pétrélis³

2018

J. Phys. A: Math. Theor.

View full text Add to dashboard Cite

We review some recent results obtained in the framework of the 2-dimensional Interacting Self-Avoiding Walk (ISAW). After a brief presentation of the rigorous results that have been obtained so far for ISAW we focus on the Interacting Partially Directed Self-Avoiding Walk (IPDSAW), a model introduced in Zwanzig and Lauritzen (1968) to decrease the mathematical complexity of ISAW.In the first part of the paper, we discuss how a new probabilistic approach based on a random walk representation (see Nguyen and Pétrélis (2013)) allowed for a sharp determination of the asymptotics of the free energy close to criticality (see Carmona, Nguyen and Pétrélis (2016)). Some scaling limits of IPDSAW were conjectured in the physics literature (see e.g. Brak et al. (1993)). We discuss here the fact that all limits are now proven rigorously, i.e., for the extended regime in Carmona and Pétrélis (2016), for the collapsed regime in Carmona, Nguyen and Pétrélis (2016) and at criticality in Carmona and Pétrélis (2017a).The second part of the paper starts with the description of four open questions related to physically relevant extensions of IPDSAW. Among such extensions is the Interacting Prudent Self-Avoiding Walk (IPSAW) whose configurations are those of the 2-dimensional prudent walk. We discuss the main results obtained in Pétrélis and Torri (2016+) about IPSAW and in particular the fact that its collapse transition is proven to exist rigorously.MSC 2010 subject classifications: Primary 60K35; Secondary 82B26, 82B41.

show abstract

Two-sided prudent walks: a solvable non-directed model of polymer adsorption

Cited by 4 publications

References 26 publications

Collapse transition of the interacting prudent walk

Collapse transition of the interacting prudent walk

Prudent walk in dimension six and higher

Interacting partially directed self-avoiding walk: a probabilistic perspective

Contact Info

Product

Resources

About