Exchange relations, Dyck paths and copolymer adsorption

Rechnitzer, Andrew; Rensburg, E J Janse van

doi:10.1016/j.dam.2003.08.008

Cited by 12 publications

(27 citation statements)

References 31 publications

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“…Directed versions of lattice models of absorbing polymers are mathematically more tractable, while they also retain some of the rich combinatorial content so evident in more general models. The most well-known directed model of polymer adsorption is a model of Dyck paths [15,16,23] and this model has also been considered in directed models of copolymer adsorption [16,26].…”

Section: Introductionmentioning

confidence: 99%

“…However, not all Dyck path problems have been solved; no similar explicit general expressions are known for coloured Dyck paths (these are models of adsorbing copolymers), except in the simplest of cases [16]. Further investigation of these models [26] suggests instead that a general solution would be unlikely, and only asymptotic expressions are known for critical points in these models. In particular, the asymptotic expression up to decaying terms for the critical point in a {AB p−1 } * A-copolymer 1 model of Dyck paths adsorbing in the main diagonal is…”

Section: Introductionmentioning

confidence: 99%

“…This model can be turned into a model of copolymer adsorption by colouring the vertices with two colours, say A and B, and where only colour A will interact with the adsorbing line. The problem is unsolved for general colourings [26], and in this paper we only focus on the colouring {AB p−1 } * A with period p. Even in this case the model is unsolved -and we focus only on finding an asymptotic expression for the critical adsorption point in terms of p, similar to equation (1). The starting point is an exchange symmetry for Motzkin paths, analogous to the exchange symmetry for Dyck paths discussed in [26].…”

Section: Introductionmentioning

confidence: 99%

“…In these cases, we obtain generating functions M(z, v) and M(z, w) instead, and the key property of these will be their radii of convergence z c (v) and z c (w). In particular, these curves have a non-analytic point v c (and w c respectively) which corresponds to an adsorption transition in this model [2,16,19,26]. We are interested in the numerical values of these critical points; in particular how the position of the critical point depends upon the colouring of the path.…”

Section: Introductionmentioning

confidence: 99%

“…In Section 2 we first consider a Motzkin path model with vertex-visits. We show that the generating function of this model satisfies an exchange relation [26] which can be used to determine an asymptotic expression for the adsorption critical point a c (p) in a Motzkin path model of adsorbing directed copolymers whose vertices are coloured {AB p−1 } * A with a the generating variable of A-vertex-visits:…”

Section: Introductionmentioning

confidence: 99%

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Exchange Symmetries in Motzkin Path and Bargraph Models of Copolymer Adsorption

Rensburg¹,

Rechnitzer²

2002

Electron. J. Combin.

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In a previous work [26], by considering paths that are partially weighted, the generating function of Dyck paths was shown to possess a type of symmetry, called an exchange relation, derived from the exchange of a portion of the path between weighted and unweighted halves. This relation is particularly useful in solving for the generating functions of certain models of vertex-coloured Dyck paths; this is a directed model of copolymer adsorption, and in a particular case it is possible to find an asymptotic expression for the adsorption critical point of the model as a function of the colouring. In this paper we examine Motzkin path and partially directed walk models of the same adsorbing directed copolymer problem. These problems are an interesting generalisation of previous results since the colouring can be of either the edges, or the vertices, of the paths. We are able to find asymptotic expressions for the adsorption critical point in the Motzkin path model for both edge and vertex colourings, and for the partially directed walk only for edge colourings. The vertex colouring problem in partially directed walks seems to be beyond the scope of the methods of this paper, and remains an open question. In both these cases we first find exchange relations for the generating functions, and use those to find the asymptotic expression for the adsorption critical point.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Exchange Symmetries in Motzkin Path and Bargraph Models of Copolymer Adsorption

Rensburg¹,

Rechnitzer²

2002

Electron. J. Combin.

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Trajectories of directed lattice paths

Rensburg

2023

Phys. Scr.

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The distribution of monomers along a linear polymer grafted on a hard wall is modelled by determining the probability distribution of occupied vertices of Dyck and ballot path models of adsorbing linear polymers. For example, the probability that a Dyck path passes through the lattice site with coordinates (⌊εn⌋,⌊δ√n⌋) in the square lattice, for 0 < ε < 1 and δ ≥ 0, is determined asymptotically as n →∞ and this uncovers the probability density of vertices along Dyck path in the limit as the length of the path n approaches inﬁnity: Pr(ε,δ) = 4δ^2 exp(-δ^2/ε(1-ε)) / √(π ε3(1-ε)3). The properties of a polymer coating of a hard wall and the density or distribution of monomers in the coating is relevant in applications such as the stabilisation of a colloid dispersion by a polymer or in a drug delivery system such as a drug-eluding stent covered by a grafted polymer.

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Trajectories of square lattice staircase polygons

Janse van Rensburg

2023

Phys. Scr.

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The distribution of monomers in a coating of grafted and adsorbing polymers is modelled using a grafted staircase polygon in the square lattice. The adsorbing staircase polygon consists of a bottom and a top lattice path (branches) and the asymptotic probability density that vertices in the lattice are occupied by these paths is determined. This is a model consisting of two linear polymers grafted to a hard wall in a coating. The probability that either the bottom path, or the top path, of a staircase polygon passes through a lattice site with coordinates ( ⌊ ϵ n ⌋ , ⌊ δ n ⌋ ) , for 0 < ϵ < 1 and δ ≥ 0, is determined asymptotically as n → ∞ . This gives the probability density of vertices in the staircase polygon in the scaling limit (as n → ∞ ): P r f , ( both ) ( ϵ , δ ) = δ 2 ( 15 ϵ 2 1 − ϵ 2 − 12 δ 2 ϵ ( 1 − ϵ ) + 4 δ 4 ) 3 π ϵ 7 1 − ϵ 7 e − δ 2 / ϵ ( 1 − ϵ ) . The densities of other cases, grafted and adsorbed, and at the adsorption critical point (or special point), are also determined. In these cases the most likely and mean trajectories of the paths are determined. The results show that low density regions in the distribution induce entropic forces on test particles which confine them next to the hard wall, or between branches of the staircase polygon. These results give qualitative mechanisms for the stabilisation of a drug particle confined to a polymer coating in a drug delivery system such as a drug-eluding stent covered by a grafted polymer.

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Exchange relations, Dyck paths and copolymer adsorption

Cited by 12 publications

References 31 publications

Exchange Symmetries in Motzkin Path and Bargraph Models of Copolymer Adsorption

Exchange Symmetries in Motzkin Path and Bargraph Models of Copolymer Adsorption

Trajectories of directed lattice paths

Trajectories of square lattice staircase polygons

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