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Thomas Prellberg

Publications and source records attributed to Thomas Prellberg.

At least 37 records · Page 2Linked to original sources
arXiv2017

Area-width scaling in generalised Motzkin paths

We consider a generalised version of Motzkin paths, where horizontal steps have length $\ell$, with $\ell$ being a fixed positive integer. We first give the general functional equation for the area-length generating function of this model. Using a heuristic ansatz, we derive the area-length scaling behaviour in terms of a scaling function in one variable for the special cases of Dyck, (standard) Motzkin and Schröder paths, before generalising our approach to arbitrary $\ell$. We then derive an expression for the generating function of Schröder paths and analyse the scaling behaviour of this function rigorously in the vicinity of the tri-critical point of the model by applying the method of steepest descents for the case of two coalescing saddle points. Our results show that for Dyck and Schröder paths, the heuristic scaling ansatz reproduces the rigorous results.

cond-mat.stat-mech↗
arXiv2016

Higher-order Airy scaling in deformed Dyck paths

We introduce a deformed version of Dyck paths (DDP), where additional to the steps allowed for Dyck paths, 'jumps' orthogonal to the preferred direction of the path are permitted. We consider the generating function of DDP, weighted with respect to their half-length, area and number of jumps. This represents the first example of an exactly solvable lattice path model showing a higher-order multicritical point. Applying the generalized method of steepest descents, we see that the associated two-variable scaling function is given by the logarithmic derivative of a generalized (higher-order) Airy integral.

math-ph↗
arXiv2016

Writhe induced phase transition in unknotted self-avoiding polygons

Recently it has been argued that weighting the writhe of unknotted self-avoiding polygons can be related to possible experiments that turn double stranded DNA. We first solve exactly a directed model and demonstrate that in such a subset of polygons the problem of weighting their writhe is associated with a phase transition. We then analyse simulations using the Wang-Landau algorithm to observe scaling in the fluctuations of the writhe that is compatible with a second-order phase transition in a undirected self-avoiding polygon model. Crucially, we conclude that the transition becomes apparent when the polygon is stretched sufficiently with a pulling force.

cond-mat.stat-mech↗
arXiv2016

Phase Diagram of Twist Storing Lattice Polymers in Variable Solvent Quality

When double stranded DNA is turned in experiments it undergoes a transition. We use an interacting self-avoiding walk on a three-dimensional fcc lattice weighted by writhe to relate to these experiments and treat this problem via simulations. We provide evidence for the existence of a thermodynamic phase transition induced by writhe and examine related phase diagrams taking solvent quality and stretching into account.

cond-mat.stat-mech↗
arXiv2015

Winding angle distributions for two-dimensional collapsing polymers

We provide numerical support for a long-standing prediction of universal scaling of winding angle distributions. Simulations of interacting self-avoiding walks show that the winding angle distribution for $N$-step walks is compatible with the theoretical prediction of a Gaussian with a variance growing asymptotically as $C\log N$, with $C=2$ in the swollen phase (previously verified), and $C=24/7$ at the $θ$-point. At low temperatures weaker evidence demonstrates compatibility with the same scaling and a value of $C=4$ in the collapsed phase, also as theoretically predicted.

cond-mat.stat-mech↗
arXiv2014

Uniform asymptotics of area-weighted Dyck paths

Using the generalized method of steepest descents for the case of two coalescing saddle points, we derive an asymptotic expression for the bivariate generating function of Dyck paths, weighted according to their length and their area in the limit of the area generating variable tending towards 1. The result is valid uniformly for a range of the length generating variable, including the tricritical point of the model.

math-ph↗
arXiv2014

On the Number of Walks in a Triangular Domain

We consider walks on a triangular domain that is a subset of the triangular lattice. We then specialise this by dividing the lattice into two directed sublattices with different weights. Our central result is an explicit formula for the generating function of walks starting at a fixed point in this domain and ending anywhere within the domain. Intriguingly, the specialisation of this formula to walks starting in a fixed corner of the triangle shows that these are equinumerous to two-coloured Motzkin paths, and two-coloured three-candidate Ballot paths, in a strip of finite height.

math.CO↗
arXiv2014

A Bijection on Bilateral Dyck Paths

It is known that both the number of Dyck paths with $2n$ steps and $k$ peaks, and the number of Dyck paths with $2n$ steps and $k$ steps at odd height follow the Narayana distribution. In this paper we present a bijection which explicitly illustrates this equinumeracy. Moreover, we extend this bijection to bilateral Dyck paths. The restriction to Dyck paths preserves the number of contacts.

math.CO↗
arXiv2013

Pressure exerted by a vesicle on a surface

Several recent works have considered the pressure exerted on a wall by a model polymer. We extend this consideration to vesicles attached to a wall, and hence include osmotic pressure. We do this by considering a two-dimensional directed model, namely that of area-weighted Dyck paths. Not surprisingly, the pressure exerted by the vesicle on the wall depends on the osmotic pressure inside, especially its sign. Here, we discuss the scaling of this pressure in the different regimes, paying particular attention to the crossover between positive and negative osmotic pressure. In our directed model, there exists an underlying Airy function scaling form, from which we extract the dependence of the bulk pressure on small osmotic pressures.

cond-mat.stat-mech↗
arXiv2013

Lattice polymers with two competing collapse interactions

There have been separate studies of the polymer collapse transition, where the collapse was induced by two different types of attraction. In each case, the configurations of the polymer were given by the same subset of random walks being self-avoiding trails on the square lattice. Numerical evidence shows that when interacting via nearest-neighbour contacts, this transition is different from the collapse transition in square-lattice trails interacting via multiply visited sites. While both transitions are second-order, when interacting via nearest-neighbour contacts, the transition is relatively weak with a convergent specific heat, while when interacting via multiply visited sites, the specific heat diverges strongly. Moreover, an estimation of the crossover exponent for the nearest-neighbour contact interaction provides a value close to that of the canonical polymer collapse model of interacting self-avoiding walks, which also interact via nearest-neighbour contacts. From computer simulations using the flatPERM algorithm, we extend these studies by considering a model of self-avoiding trails on the square lattice containing both types of interaction, and which therefore contains all three of the models discussed above as special cases. We find that the strong multiply-visited site collapse is a singular point in the phase diagram and corresponds to a higher order multi-critical point separating a line of weak second-order transitions from a line of first-order transitions.

cond-mat.stat-mech↗
arXiv2012

Exact power spectra of Brownian motion with solid friction

We study a Langevin equation describing the Brownian motion of an object subjected to a viscous drag, an external constant force, and a solid friction force of the Coulomb type. In a previous work [H. Touchette, E. Van der Straeten, W. Just, J. Phys. A: Math. Theor. 43, 445002, 2010], we have presented the exact solution of the velocity propagator of this equation based on a spectral decomposition of the corresponding Fokker-Planck equation. Here, we present an alternative, exact solution based on the Laplace transform of this equation, which has the advantage of being expressed in closed form. From this solution, we also obtain closed-form expressions for the Laplace transform of the velocity autocorrelation function and for the power spectrum, i.e., the Fourier transform of the autocorrelation function. The behavior of the power spectrum as a function of the dry friction force and external forcing shows a clear crossover between stick and slip regimes known to occur in the presence of solid friction.

cond-mat.stat-mech↗
arXiv2012

Exact solution of a model of a vesicle attached to a wall subject to mechanical deformation

Area-weighted Dyck-paths are a two-dimensional model for vesicles attached to a wall. We model the mechanical response of a vesicle to a pulling force by extending this model. We obtain an exact solution using two different approaches, leading to a q-deformation of an algebraic functional equation, and a q-deformation of a linear functional equation with a catalytic variable, respectively. While the non-deformed linear functional equation is solved by substitution of special values of the catalytic variable (the so-called "kernel method"), the q-deformed case is solved by iterative substitution of the catalytic variable. Our model shows a non-trivial phase transition when a pulling force is applied. As soon as the area is weighted with non-unity weight, this transition vanishes.

cond-mat.stat-mech↗
arXiv2012

From Rosenbluth Sampling to PERM - rare event sampling with stochastic growth algorithms

We discuss uniform sampling algorithms that are based on stochastic growth methods, using sampling of extreme configurations of polymers in simple lattice models as a motivation. We shall show how a series of clever enhancements to a fifty-odd year old algorithm, the Rosenbluth method, led to a cutting-edge algorithm capable of uniform sampling of equilibrium statistical mechanical systems of polymers in situations where competing algorithms failed to perform well. Examples range from collapsed homo-polymers near sticky surfaces to models of protein folding.

cond-mat.stat-mech↗
arXiv2011

Anomalous critical behaviour in the polymer collapse transition of three-dimensional lattice trails

Trails (bond-avoiding walks) provide an alternative lattice model of polymers to self-avoiding walks, and adding self-interaction at multiply visited sites gives a model of polymer collapse. Recently, a two-dimensional model (triangular lattice) where doubly and triply visited sites are given different weights was shown to display a rich phase diagram with first and second order collapse separated by a multi-critical point. A kinetic growth process of trails (KGT) was conjectured to map precisely to this multi-critical point. Two types of low temperature phases, globule and crystal-like, were encountered. Here, we investigate the collapse properties of a similar extended model of interacting lattice trails on the simple cubic lattice with separate weights for doubly and triply visited sites. Again we find first and second order collapse transitions dependent on the relative sizes of the doubly and triply visited energies. However we find no evidence of a low temperature crystal-like phase with only the globular phase in existence. Intriguingly, when the ratio of the energies is precisely that which separates the first order from the second-order regions anomalous finite-sized scaling appears. At the finite size location of the rounded transition clear evidence exists for a first order transition that persists in the thermodynamic limit. This location moves as the length increases, with its limit apparently at the point that maps to a KGT. However, if one fixes the temperature to sit at exactly this KGT point then only a critical point can be deduced from the data. The resolution of this apparent contradiction lies in the breaking of crossover scaling and the difference in the shift and transition width (crossover) exponents.

cond-mat.stat-mech↗
arXiv2011

Decoding generalised hyperoctahedral groups and asymptotic analysis of correctible error patterns

We demonstrate a majority-logic decoding algorithm for decoding the generalised hyperoctahedral group $C_m \wr S_n$ when thought of as an error-correcting code. We also find the complexity of this decoding algorithm and compare it with that of another, more general, algorithm. Finally, we enumerate the number of error patterns exceeding the correction capability that can be successfully decoded by this algorithm, and analyse this asymptotically.

math.CO↗
arXiv2011

An infinite family of adsorption models and restricted Lukasiewicz paths

We define $(k,\ell)$-restricted Lukasiewicz paths, $k\le\ell\in\mathbb{N}_0$, and use these paths as models of polymer adsorption. We write down a polynomial expression satisfied by the generating function for arbitrary values of $(k,\ell)$. The resulting polynomial is of degree $\ell+1$ and hence cannot be solved explicitly for sufficiently large $\ell$. We provide two different approaches to obtain the phase diagram. In addition to a more conventional analysis, we also develop a new mathematical characterization of the phase diagram in terms of the discriminant of the polynomial and a zero of its highest degree coefficient. We then give a bijection between $(k,\ell)$-restricted Lukasiewicz paths and "rise"-restricted Dyck paths, identifying another family of path models which share the same critical behaviour. For $(k,\ell)=(1,\infty)$ we provide a new bijection to Motzkin paths. We also consider the area-weighted generating function and show that it is a q-deformed algebraic function. We determine the generating function explicitly in particular cases of $(k,\ell)$-restricted Lukasiewicz paths, and for $(k,\ell)=(0,\infty)$ we provide a bijection to Dyck paths.

math.CO↗
arXiv2011

Counting the number of elements in the mutation classes of \tilde{A}_n-quivers

In this article we prove explicit formulae for the number of non-isomorphic cluster-tilted algebras of type \tilde{A}_n in the derived equivalence classes. In particular, we obtain the number of elements in the mutation classes of quivers of type \tilde{A}_n. As a by-product, this provides an alternative proof for the number of quivers of Dynkin type D_n which was first determined by Buan and Torkildsen.

math.CO↗