Esaias J Janse van Rensburg

York University · Department of Mathematics and Statistics

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...

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...

A linear polymer grafted to a hard wall and underneath an AFM tip can be modelled in a lattice as a grafted lattice polymer (or self-avoiding walk) compressed underneath a piston approaching the wall. As the piston approaches the wall the increasingly confined polymer escapes from the confined region to explore conformations beside the piston. This...

The surface entropic exponents of half-space lattice stars grafted at their central nodes in a hard wall are estimated numerically using the PERM algorithm. In the square half-lattice the exact values of the exponents are verified, including Barber's scaling relation and a generalization for 2-stars with one and two surface loops respectively. This...

Two ring polymers close to each other in space may be either in a segregated phase if there is a strong repulsion between monomers in the polymers, or intermingle in a mixed phase if there is a strong attractive force between the monomers. These phases are separated by a critical point which has a θ-point character. The metric and topological prope...

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 $(\lfloor \epsilon n \rfloor,\lfloor \delt...

Two ring polymers close to each other in space may be either in a segregated phase if there is a strong repulsion between monomers in the polymers, or intermingle in a mixed phase if there is a strong attractive force between the monomers. These phases are separated by a critical point which has a $\theta$-point character. The metric and topologica...

The surface entropic exponents of half-space lattice stars grafted at their central nodes in a hard wall are estimated numerically using the PERM algorithm. In the square half-lattice the exact values of the exponents are verified, including Barber's scaling relation and a generalisation for $2$-stars with one and two surface loops respectively. Th...

We consider a model of star copolymers, based on self-avoiding walks, where the arms of the star can be chemically distinct. The copolymeric star is attached to an impenetrable surface at the end of an arm and the different monomers constituting the star have different interaction strengths with the surface. When the star is adsorbed at the surface...

We consider a model of star copolymers, based on self-avoiding walks, where the arms of the star can be chemically distinct. The copolymeric star is attached to an impenetrable surface at a vertex of unit degree and the different monomers constituting the star have different interaction strengths with the surface. When the star is adsorbed at the s...

The phase diagrams of two models of two confined and dense two-dimensional ring polymers are examined numerically. The ring polymers are modeled by square lattice polygons in a square cavity and are placed to be either unlinked or linked in the plane. The phase diagrams of the two models are found to be a function of the placement of the ring polym...

Numerical values of lattice star entropic exponents $\gamma_f$, and star vertex exponents $\sigma_f$, are estimated using parallel implementations of the PERM and Wang-Landau algorithms. Our results show that the numerical estimates of the vertex exponents deviate from predictions of the $\eps$-expansion and confirms and improves on estimates in th...

Numerical values of lattice star entropic exponents $\gamma_f$, and star vertex exponents $\sigma_f$, are estimated using parallel implementations of the PERM and Wang-Landau algorithms. Our results show that the numerical estimates of the vertex exponents deviate from predictions of the $\epsilon$-expansion and confirms and improves on estimates i...

We study by Monte Carlo simulations and scaling analysis two models of pairs of confined and dense ring polymers in two dimensions. The pair of ring polymers are modelled by squared lattice polygons confined within a square cavity and they are placed in relation to each other to be either unlinked or linked in the plane. The observed rich phase dia...

We implement parallel versions of the generalized atmospheric Rosenbluth methods and Wang-Landau algorithms for stars and for acyclic uniform branched networks in the square lattice. These are models of monodispersed branched polymers, and we estimate the star vertex exponents σf for f stars, and the entropic exponent γG for networks with comb and...

Fekete’s lemma shows the existence of limits in subadditive sequences. This lemma, and generalisations of it, also have been used to prove the existence of thermodynamic limits in statistical mechanics. In this paper it is shown that the two variable supermultiplicative relation p n 1 ( m 1 ) p n 2 ( m 2 ) ⩽ p n 1 + n 2 ( m 1 + m 2 ) together with...

We investigate self-avoiding walk models of linear block copolymers adsorbed at a surface and desorbed by the action of a force. We rigorously establish the dependence of the free energy on the adsorption and force parameters, and the form of the phase diagram for several cases, including AB -diblock copolymers and ABA -triblock copolymers, pulled...

We investigate self-avoiding walk models of linear block copolymers adsorbed at a surface and desorbed by the action of a force. We rigorously establish the dependence of the free energy on the adsorption and force parameters, and the form of the phase diagram for several cases, including $AB$-diblock copolymers and $ABA$-triblock copolymers, pulle...

Numerical values of lattice star vertex exponents are estimated using parallel implementations of the GARM and Wang-Landau algorithms in the square and cubic lattices. In the square lattice the results are consistent with exact values of the exponents, but in the cubic lattice there are deviations from the predictions of the $\epsilon$-expansion. I...

We develop and implement a parallel flatPERM algorithm (Grassberger 1997 Phys. Rev. E 56 3682–3693, Prellberg and Krawczyk 2004 Phys. Rev. Lett. 92 120602) with mutually interacting parallel flatPERM sequences and use it to sample self-avoiding walks in two and three dimensions. Our data show that the parallel implementation accelerates the converg...

We develop and implement a parallel implementation of the flatPERM algorithm \cite{G97,PK04}. Our data show that this accelerates the convergence of the algorithm. Moreover, increasing the number of interacting flatPERM sequences (chains) rather than running longer simulations improves the rate of convergence, and supports the notion that the effic...

The free energy of a model of uniformly weighted lattice knots of length n and knot type K confined to a lattice cube of side length L−1 is given by FL(ϕ)=−1Vlogpn,L(K), where V=L3 and where ϕ=n/V is the concentration of monomers of the lattice knot in the confining cube. The limiting free energy of the model is F∞(ϕ)=limL→∞FL(ϕ) and the limiting o...

A compressed knotted ring polymer in a confining cavity is modelled by a knotted lattice polygon confined in a cube in . The GAS algorithm (Janse van Rensburg and Rechnitzer 2011 J. Knot Theol. Raman 20 1145–71) is used to sample lattice polygons of fixed knot type in a confining cube and to estimate the free energy of confined lattice knots. Latti...

IOP Publishing has withdrawn this article upon the author's request due to several issues with the mathematical content. The osmotic pressure of monomers in a knotted ring polymer in a confining cavity is modelled by a lattice polygon confined in a cube in ${\mathbb Z}^3$. These polygons can be knotted and are called lattice knots. In this paper th...

We consider self-avoiding walks terminally attached to a surface at which they can adsorb. A force is applied, normal to the surface, to desorb the walk and we investigate how the behaviour depends on the vertex of the walk at which the force is applied. We use rigorous arguments to map out some features of the phase diagram, including bounds on th...

A numerical simulation shows that the osmotic pressure of compressed lattice knots is a function of knot type, and so of entanglements. The osmotic pressure for the unknot goes through a negative minimum at low concentrations, but in the case of nontrivial knot types 31 and 41 it is negative for low concentrations. At high concentrations the osmoti...

We investigate the phase diagram of a self-avoiding walk model of a 3-star polymer in two dimensions, adsorbing at a surface and being desorbed by the action of a force. We show rigorously that there are four phases: a free phase, a ballistic phase, an adsorbed phase and a mixed phase where part of the 3-star is adsorbed and part is ballistic. We u...

We consider self-avoiding walks terminally attached to a surface at which they can adsorb. A force is applied, normal to the surface, to desorb the walk and we investigate how the behaviour depends on the vertex of the walk at which the force is applied. We use rigorous arguments to map out some features of the phase diagram, including bounds on th...

The osmotic pressure of a knotted ring polymer in a confining cavity is modelled by a lattice polygon confined in a cube in ${\mathbb Z}^3$. These polygons can be knotted and are called lattice knots. In this paper the GAS algorithm [17] is used to estimate the free energy of lattice knots of knot types the unknot, the trefoil knot, and the figure...

We investigate the phase diagram of a self-avoiding walk model of a 3-star polymer in two dimensions, adsorbing at a surface and being desorbed by the action of a force. We show rigorously that there are four phases, a free phase, a ballistic phase, an adsorbed phase and a mixed phase where part of the 3-star is adsorbed and part is ballistic. We u...

A numerical simulation shows that the osmotic pressure of compressed lattice knots is a function of knot type, and so of entanglements. The osmotic pressure for the unknot goes through a negative minimum at low concentrations, but in the case of non-trivial knot types $3_1$ and $4_1$ it is negative for low concentrations. At high concentrations the...

We analyze the phase diagrams of self-avoiding walk models of uniform branched polymers adsorbed at a surface and subject to an externally applied vertical pulling force which, at critical values, desorbs the polymer. In particular, models of adsorbed branched polymers with homeomorphism types, stars, tadpoles, dumbbells and combs are examined. The...

Flory-Huggins theory (Flory 1942 J. Chem. Phys. 10 51-61; Huggins 1942 J. Am. Chem. Soc. 64 2716-8) is a mean field theory for modelling the free energy of dense polymer solutions and polymer melts. In this paper we use Flory-Huggins theory as a model of a dense two-dimensional self-avoiding walk compressed in a square in the square lattice. The th...

We analyze the phase diagrams of self-avoiding walk models of uniform branched polymers adsorbed at a surface and subject to an externally applied vertical pulling force which, at critical values, desorbs the polymer. In particular, models of adsorbed branched polymers with homeomorphism types stars, tadpoles, dumbbells and combs are examined. Thes...

Flory-Huggins theory is a mean field theory for modelling the free energy of dense polymer solutions and polymer melts. In this paper we use Flory-Huggins theory as a model of a dense two dimensional self-avoiding walk confined to a square in the square lattice. The theory describes the free energy of the walk well, and we estimate the Flory intera...

We consider a cubic lattice self-avoiding walk model of 3-star polymers adsorbed at a surface and then desorbed by pulling with an externally applied force. We determine the free energy of the model, and show that the phase diagram includes 4 phases, namely a ballistic phase, an adsorbed phase and a mixed phase, in addition to the free phase where...

The zeros of the size-$n$ partition functions for a statistical mechanical model can be used to help understand the critical behaviour of the model as $n\to\infty$. Here we use weighted Dyck paths as a simple model of two-dimensional polymer adsorption, and study the behaviour of the partition function zeros, particularly in the thermodynamic limit...

We consider self-avoiding lattice polygons, in the square and cubic lattices, as a model of a ring polymer adsorbed at a surface and either being desorbed by the action of a force, or pushed towards the surface. We show that, when there is no interaction with the surface, then the response of the polygon to the applied force is identical (in the th...

The sampling of scale-free networks in Molecular Biology is usually achieved by growing networks from a seed using recursive algorithms with elementary moves which include the addition and deletion of nodes and bonds. These algorithms include the Barabási-Albert algorithm. Later algorithms, such as the Duplication-Divergence algorithm, the Solé alg...

We consider a self-avoiding walk on the $d$-dimensional hypercubic lattice, terminally attached to an impenetrable hyperplane at which it can adsorb. When a force is applied the walk can be pulled off the surface and we consider the situation where the force is applied at the middle vertex of the walk. We show that the temperature dependence of the...

The Lee-Yang theory of adsorbing self-avoiding walks is presented. It is shown that Lee-Yang zeros of the generating function of this model asymptotically accumulate uniformly on a circle in the complex plane, and that Fisher zeros of the partition function distribute in the complex plane such that a positive fraction are located in annular regions...

Subadditivity is an important tool for proving the existence of thermodynamic limits in the statistical mechanics of models of lattice clusters (such as the self-avoiding walk and percolation clusters). The partition functions of these models may satisfy a variety of super- or submultiplicative inequalities, and these may be reduced to a subadditiv...

We consider self-avoiding walks terminally attached to an impenetrable surface at which they can adsorb. We call the vertices farthest away from this plane the top vertices and we consider applying a force at the plane containing the top vertices. This force can be directed away from the adsorbing surface or towards it. In both cases we prove that...

We prove some theorems about self-avoiding walks attached to an impenetrable surface (i.e. positive walks) and subject to a force. Specifically we show the force dependence of the free energy is identical when the force is applied at the last vertex or at the top (confining) plane. We discuss the relevance of this result to numerical results and to...

A polymer in a confined space loses entropy and exerts an entropic force on the walls of the confining space. In this paper a partially directed walk model of the entropic forces in a confined polymer is introduced and analysed. The walk is a model of a 2-dimensional adsorbing polymer placed between confining plates (which are lines in the lattice)...

Linear polymers adsorbing on a wall can be modelled by half-space self-avoiding walks adsorbing on a line in the square lattice, or on a surface in the cubic lattice. In this paper a numerical approach based on the GAS algorithm is used to approximately enumerate states in the partition function of this model. The data are used to approximate the f...

This book is an account of the theory and mathematical approaches in polymer entropy, with particular emphasis on mathematical approaches to directed and undirected lattice models. Results in the scaling and critical behaviour of models of directed and undirected models of self-avoiding walks, paths, polygons, animals and networks are presented. Th...

This book is an account of the theory and mathematical approaches in polymer entropy, with particular emphasis on mathematical approaches to directed and undirected lattice models. Results in the scaling and critical behaviour of models of directed and undirected models of self-avoiding walks, paths, polygons, animals and networks are presented. Th...

This book is an account of the theory and mathematical approaches in polymer entropy, with particular emphasis on mathematical approaches to directed and undirected lattice models. Results in the scaling and critical behaviour of models of directed and undirected models of self-avoiding walks, paths, polygons, animals and networks are presented. Th...

This book is an account of the theory and mathematical approaches in polymer entropy, with particular emphasis on mathematical approaches to directed and undirected lattice models. Results in the scaling and critical behaviour of models of directed and undirected models of self-avoiding walks, paths, polygons, animals and networks are presented. Th...

This book is an account of the theory and mathematical approaches in polymer entropy, with particular emphasis on mathematical approaches to directed and undirected lattice models. Results in the scaling and critical behaviour of models of directed and undirected models of self-avoiding walks, paths, polygons, animals and networks are presented. Th...

This book is an account of the theory and mathematical approaches in polymer entropy, with particular emphasis on mathematical approaches to directed and undirected lattice models. Results in the scaling and critical behaviour of models of directed and undirected models of self-avoiding walks, paths, polygons, animals and networks are presented. Th...

This book is an account of the theory and mathematical approaches in polymer entropy, with particular emphasis on mathematical approaches to directed and undirected lattice models. Results in the scaling and critical behaviour of models of directed and undirected models of self-avoiding walks, paths, polygons, animals and networks are presented. Th...

This book is an account of the theory and mathematical approaches in polymer entropy, with particular emphasis on mathematical approaches to directed and undirected lattice models. Results in the scaling and critical behaviour of models of directed and undirected models of self-avoiding walks, paths, polygons, animals and networks are presented. Th...

This book is an account of the theory and mathematical approaches in polymer entropy, with particular emphasis on mathematical approaches to directed and undirected lattice models. Results in the scaling and critical behaviour of models of directed and undirected models of self-avoiding walks, paths, polygons, animals and networks are presented. Th...

This book is an account of the theory and mathematical approaches in polymer entropy, with particular emphasis on mathematical approaches to directed and undirected lattice models. Results in the scaling and critical behaviour of models of directed and undirected models of self-avoiding walks, paths, polygons, animals and networks are presented. Th...

This book is an account of the theory and mathematical approaches in polymer entropy, with particular emphasis on mathematical approaches to directed and undirected lattice models. Results in the scaling and critical behaviour of models of directed and undirected models of self-avoiding walks, paths, polygons, animals and networks are presented. Th...

This book is an account of the theory and mathematical approaches in polymer entropy, with particular emphasis on mathematical approaches to directed and undirected lattice models. Results in the scaling and critical behaviour of models of directed and undirected models of self-avoiding walks, paths, polygons, animals and networks are presented. Th...

Let ${\mathbb{L}}$ be the $d$-dimensional hypercubic lattice and let ${\mathbb{L}}_0$ be an $s$-dimensional sublattice, with $2 \leq s < d$. We consider a model of inhomogeneous bond percolation on ${\mathbb{L}}$ at densities $p$ and $\sigma$, in which edges in ${\mathbb{L}}\setminus {\mathbb{L}}_0$ are open with probability $p$, and edges in ${\ma...

The entropic pressure in the vicinity of a cubic lattice knot is examined as a model of the entropic pressure near a knotted ring polymer in a good solvent. A model for the scaling of the pressure is developed and this is tested numerically by sampling lattice knots using a Monte Carlo algorithm. Good agreement is found between scaling predictions...

We compute the cogrowth series for Baumslag-Solitar groups $\mathrm{BS}(N,N) = < a,b | a^N b = b a^N > $, which we show to be D-finite. It follows that their cogrowth rates are algebraic numbers.

We describe a novel algorithm for random sampling of freely reduced words equal to the identity in a finitely presented group. The algorithm is based on Metropolis Monte Carlo sampling. The algorithm samples from a stretched Boltzmann distribution \begin{align*}\pi(w) &= (|w|+1)^{\alpha} \beta^{|w|} \cdot Z^{-1} \end{align*} where $|w|$ is the leng...

The entropic pressure in the vicinity of a two dimensional square lattice polygon is examined as a model of the entropic pressure near a planar ring polymer. The scaling of the pressure as a function of distance from the polygon and length of the polygon is determined and tested numerically.

We consider a self-avoiding walk model of polymer adsorption where the adsorbed polymer can be desorbed by the application of a force. In this paper the force is applied normal to the surface at the last vertex of the walk. We prove that the appropriate limiting free energy exists where there is an applied force and a surface potential term, and pr...

We propose a numerical method for studying the cogrowth of finitely presented groups. To validate our numerical results we compare them against the corresponding data from groups whose cogrowth series are known exactly. Further, we add to the set of such groups by finding the cogrowth series for Baumslag-Solitar groups $\mathrm{BS}(N,N) = < a,b | a...

A directed path in the vicinity of a hard wall exerts pressure on the wall because of loss of entropy. The pressure at a particular point may be estimated by estimating the loss of entropy if the point is excluded from the path. In this paper we determine asymptotic expressions for the pressure on the X-axis in models of adsorbing directed paths in...

In this paper the number and lengths of minimal length lattice knots confined to slabs of width $L$, is determined. Our data on minimal length verify the results by Sharein et.al. (2011) for the similar problem, expect in a single case, where an improvement is found. From our data we construct two models of grafted knotted ring polymers squeezed be...

The (isothermic) compressibility of lattice knots can be examined as a model of the effects of topology and geometry on the compressibility of ring polymers. In this paper, the compressibility of minimal length lattice knots in the simple cubic, face centered cubic and body centered cubic lattices are determined. Our results show that the compressi...

In this paper we examine the phases of a directed path model of a copolymer attached to a surface under the influence of a pulling force. The simplest model of an adsorbing directed polymer, attached at the one end to a surface and pulled from the surface by the other end, is reviewed—its phase diagram includes free, adsorbed and ballistic phases....

In this paper we examine directed path models of polymers adsorbing at a linear boundary while subject to long ranged, short ranged, and periodic attractions. The generating functions of these models are infinite continued fractions, which may be analyzed to examine the free energies and bound the critical curves in the models. In each model the pa...

Self-avoiding polygons in the cubic lattice are models of ring polymers in dilute solution. The conformational entropy of a ring polymer is a dominant factor in its physical and chemical properties, and this is modeled by the large number of conformations of lattice polygons. Cubic lattice polygons are embeddings of the circle in three space and ma...

In this paper the generating function, free energy and phase diagram of a class of models of directed paths, related to the models of paths adsorbing in the boundary Y = −NX/(N + 2) (for ) and pulled by a vertical force in its endpoint, are determined. These models are related to the special classes of models of random walks in the quarter plane, a...

In this paper we examine numerically the properties of minimal length knotted lattice polygons in the simple cubic, face-centered cubic, and body-centered cubic lattices by sieving minimal length polygons from a data stream of a Monte Carlo algorithm, implemented as described in Arago de Carvalho and Caracciolo (1983 Phys. Rev. B 27 1635), Arago de...

Let pn denote the number of self-avoiding polygons of length n on a regular three-dimensional lattice, and let pn(K) be the number which have knot type K. The probability that a random polygon of length n has knot type K is pn(K)/pn and is known to decay exponentially with length (Sumners and Whittington 1988 J. Phys. A: Math. Gen. 21 1689–94, Pipp...

In this paper the elementary moves of the BFACF-algorithm for lattice polygons are generalised to elementary moves of BFACF-style algorithms for lattice polygons in the body-centred (BCC) and face-centred (FCC) cubic lattices. We prove that the ergodicity classes of these new elementary moves coincide with the knot types of unrooted polygons in the...

The adsorption of a Motzkin path with lifted endpoints onto an adsorbing line is examined as a model of polymer adsorption. The partition function of this model is determined exactly, and the free energy is derived by examining the partition function. The location of the critical point in the model is given exactly in one case, and as the solution...

An adsorbing directed path in a half-space is a model of a dilute polymer adsorbing onto a hard wall. The simplest adsorbing directed path is the adsorbing Dyck path, and it is a two-dimensional path in the positive half-space in the square lattice. This model is known to have a continuous phase transition where the path adsorbs onto the X axis (wh...

In this paper the models of pulled Dyck paths in Janse van Rensburg (2010 J. Phys. A: Math. Theor. 43 215001) are generalized to pulled Motzkin path models. The generating functions of pulled Motzkin paths are determined in terms of series over trinomial coefficients and the elastic response of a Motzkin path pulled at its endpoint (see Orlandini a...

The generating functions of models of directed walks pulled by an external force f are determined using the kernel method. These paths are models of linear polymers subject to an external force. The generating function is related to the generating function of ballot paths, and has an unexpected and non-physical singularity for the model of Dyck pat...

Algorithms for the approximate enumeration of lattice self-avoiding walks are reviewed. Innovations in the approximate counting of such walks started with the invention of PERM (the pruned enhanced Rosenbluth method) in 1997. The recent generalization of the underlying Rosenbluth method (RM) to the GARM (generalized atmospheric RM), and to the GAS...

The coil-globule collapse of dilute linear polymers in a poor solvent is generally thought to be a second order phase transition through θ-polymers at the critical point [10]. A common model for the collapse transition of polymers is a lattice self-avoiding walk with a nearest neighbour attraction [10, 38]. In this paper we consider an alternative...

In this paper, we introduce a new Monte Carlo method for sampling lattice self-avoiding walks. The method, which we call 'GAS' (generalized atmospheric sampling), samples walks along weighted sequences by implementing elementary moves generated by the positive, negative and neutral atmospheric statistics of the walks. A realized sequence is weighte...

The numerical simulation of self-avoiding walks remains a significant component in the study of random objects in lattices. In this review, I give a comprehensive overview of the current state of Monte Carlo simulations of models of self-avoiding walks. The self-avoiding walk model is revisited, and the motivations for Monte Carlo simulations of th...

We use rigorous arguments and Monte Carlo simulations to study the thermodynamics and the topological properties of self-avoiding walks on the cubic lattice subjected to an external force f. The walks are anchored at one or both endpoints to an impenetrable plane at Z = 0 and the force is applied in the Z-direction. If a force is applied to the fre...

It is frequently claimed that Monte Carlo simulation is a method of last resort. This may be true in the most general sense, but it remains surprising that a fairly simple statistical technique could, when applied appropriately, produce high quality data by sampling states randomly in a given model. Models of lattice polygons, which are related to...

Lattice knot statistics, or the study of knotted polygons in the cubic lattice, gained momentum in 1988 when the Frisch-Wasserman-Delbruck conjecture was proven by Sumners and Whittington (J Phys A Math Gen 21:L857–861, 1988), and independently in 1989 by Pippenger (Disc Appl Math 25:273–278, 1989). In this paper, aspects of lattice knot statistics...

A polymer in a layered environment is modeled as a directed path in a layered square lattice composed of alternating A-layers of width wa and B-layers of width wb. In this paper we consider general cases of this model, where edges in the path interact with the layers, and vertices in the path interact with interfaces between adjacent layers. The ph...

We show that the classical Rosenbluth method for sampling self-avoiding walks (Hammersley and Morton 1954 J. R. Stat. Soc. B 16 23, Rosenbluth and Rosenbluth 1955 J. Chem. Phys. 23 356) can be extended to a general algorithm for sampling many families of objects, including self-avoiding polygons. The implementation relies on an elementary move whic...

We show that the classical Rosenbluth method for sampling self-avoiding walks can be extended to a general algorithm for sampling many families of objects, including self-avoiding polygons. The implementation relies on an elementary move which is a generalisation of kinetic growth; rather than only appending edges to the endpoint, edges may be inse...

A polymer in a confined geometry may be modeled by a self-avoiding walk or a self-avoiding polygon confined between two parallel walls. In two dimensions, this model involves self-avoiding walks or self-avoiding polygons in the square lattice between two parallel confining lines. Interactions of the polymer with the confining walls are introduced b...

In this paper we define two statistics a+(ω) and a−(ω), the positive and negative atmospheres of a lattice polygon ω of fixed length n. These statistics have the property that a+(ω)/a−(ω) = pn+2/pn, where pn is the number of polygons of length n, counted modulo translations. We use the pivot algorithm to sample polygons and to compute the correspon...

We use Monte Carlo methods to study the knot probability of lattice polygons on the cubic lattice in the presence of an external force f. The force is coupled to the span of the polygons along a lattice direction, say the z-direction. If the force is negative polygons are squeezed (the compressive regime), while positive forces tend to stretch the...

The knotting in a lattice polygon model of ring polymers is examined when a stretching force is applied to the polygon. By examining the incidence of cut-planes in the polygon, we prove a pattern theorem in the stretching regime for large applied forces. This theorem can be used to examine the incidence of entanglements such as knotting and writhin...

Directed paths have been used extensively in the scientific literature as a model of a linear polymer. Such paths models in particular the conformational entropy of a linear polymer and the effects it has on the free energy. These directed models are simplified versions of the self-avoiding walk, but they do nevertheless give insight into the phase...

A model of knotted polymers in a confined space is studied by considering lattice polygons of fixed knot type in a slab geometry. If pn(K, w) is the number of lattice polygons of knot type K in a slab of width w, then the generating function of this lattice model is given by , where t is a generating variable conjugate with the length of the polygo...

The enumeration of lattice paths in wedges poses unique mathematical challenges. These models are not translationally invariant, and the absence of this symmetry complicates both the derivation of a functional recurrence for the generating function, and its solution. In this paper we consider a model of partially directed walks from the origin in t...

A polymer in the confined spaces between colloid particles loses entropy and exerts a repulsive entropic force on the confining particles. This situation can be modelled by a self-avoiding walk confined in a slab between two parallel planes in the lattice. In this paper, we prove the existence of a limiting free energy for the general case that the...

The enumeration of lattice paths in wedges poses unique mathematical challenges. These models are not translationally invariant, and the absence of this symmetry complicates both the derivation of a functional equation for the generating function, and solving for it. In this paper we consider a model of partially directed walks from the origin in t...

We examine partitions and their natural three-dimensional generalizations, plane partitions, as models of vesicles undergoing an inflation–deflation transition. The phase diagrams of these models include a critical point corresponding to an inflation–deflation transition, and exhibits multicritical scaling in the vicinity of a multicritical point l...

A directed path from the origin in the square lattice, and confined to a wedge, exerts a net entropic force on the wedge. If the wedge is formed by the Y -axis and the line Y = rX, then the moment of the force on the line Y = rX about the origin is given by where α is the vertex angle of the wedge formed by the lines X = 0 and Y = rX in the square...