School of Mathematics and Statistics - Theses
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ItemStatistical mechanics of twist-storing polymers( 2015)When double stranded DNA is turned it undergoes a conformational transition which is believed to be similar to the transition that occurs when a rubber coated cable is twisted. In this thesis we use lattice polymers to model such experiments. We consider a lattice ribbon model that was introduced in the 1990s and conclude that the critical structure of these experiments can be obtained by weighting the writhe of self-avoiding walks. We study the latter via exactly solvable toy models and extensively via simulations using the flatPERM and Wang-Landau algorithm. For the physically more relevant situation, one needs to restrict the knot type of the self-avoiding walk. In particular, we will consider unknotted self-avoiding walks. The restriction of the knot type poses several problems. First, the critical structure cannot be obtained from the renormalization group. Second, it becomes significantly more challenging to obtain good data from simulations.
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ItemOne-dimensional stochastic models with open boundaries: integrability, applications, and q-deformed Knizhnik–Zamolodchikov equations( 2014)This thesis contains work on three separate topics, but with common themes running throughout. These themes are drawn together in the asymmetric exclusion process (ASEP) – a stochastic process describing particles hopping on a one-dimensional lattice. The open boundary ASEP is set on a finite length lattice, with particles entering and exiting at both boundaries. The transition matrix of the open boundary ASEP provides a representation of the two boundary Temperley–Lieb algebra, and the integrability of the system allows the diagonalisation of the transition matrix through the Bethe ansatz method. We study the ASEP in the reverse bias regime, where the boundary injection and extraction rates oppose the preferred direction of flow in the bulk. We find the exact asymptotic relaxation rate along the coexistence line by analysing solutions of the Bethe equations. The Bethe equations are first solved numerically, then the form of the resulting root distribution is used as the basis for an asymptotic analysis. The reverse bias induces the appearance of isolated roots, which introduces a modified length scale in the system. We describe the careful treatment of the isolated roots that is required in both the numerical procedure, and in the asymptotic analysis. The second topic of this thesis is the study of a priority queueing system, modelled as an exclusion process with moving boundaries. We call this model the prioritising exclusion process (PEP). In the PEP, the hopping of particles corresponds to high priority customers overtaking low priority customers in order to gain service sooner. Although the PEP is not integrable, techniques from the ASEP allow calculation of exact density profiles in certain phases, and the calculation of approximate average waiting times when the expected queue length is finite. The final topic of this thesis is a study of polynomial solutions of a q-deformed Knizhnik–Zamolodchikov (qKZ) equation with mixed boundaries. The qKZ equation studied here is given in terms of the one boundary Temperley–Lieb algebra, and its solutions have a factorised form in terms of Baxterized elements of the type B Hecke algebra. We find an integral form for certain components of the qKZ solution, along with a factorised expression for a generalised sum rule. The representation of the Temperley– Lieb algebra that we study is related to the O(n) Temperley–Lieb loop model, and a specialization of the sum rule gives the normalisation of the ground state vector for the O(n = 1) model.
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ItemDiscretely holomorphic observables in statistical mechanics( 2014)In this thesis we investigate applications of discretely holomorphic observables in two-dimensional statistical mechanical lattice models. Discretely holomorphic observables or parafermionic ob- servables are functions defined on a graph embedding which depend on a real-valued parameter known as the parafermionic spin. Such observables were introduced in the context of lattice models in order to rigorously prove that the scaling limit of a given model is conformally invariant. This approach has been successfully carried out for the dimer model, percolation and the Ising model. However, these observables also have important applications at the lattice level. For example, Cardy and Ikhlef showed that they are naturally related to the Yang-Baxter integrability of the model, providing a straightforward method for obtaining the integrable weights. We further this connection by defining discretely holomorphic observables for loop models in domains with a boundary, showing that for a simple set of boundary conditions, the integrable boundary weights are obtained. This is true for models with diagonal as well as off-diagonal boundary weights, in which case the observables of Cardy and Ikhlef must be further generalised. Duminil-Copin and Smirnov made use of the discrete holomorphicity property in order to prove Nienhuis’ conjectured value for the connective constant of self-avoiding walks on the honeycomb lattice. We show that by relaxing the discrete holomorphicity condition we obtain an off-critical condition which allows us to relate certain critical exponents of different classes of self-avoiding walks in the dilute O(n) model. We are also able to derive an exponent equality and several exponent inequalities related to the winding angle distribution of the O(n) model, whose full distribution was first predicted using Conformal Field Theory techniques by Duplantier and Saleur. Finally we define parafermionic observables for the Andrews-Baxter-Forrester heights mod- els at the critical point. These models remain integrable away from the critical point. We therefore expect that the off-critical integrable weights should arise from an off-critical discrete holomorphicity condition, similar to that which we defined in the context of the dilute O(n) model.
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ItemCritical dense polymers on the strip and cylinder( 2013)This thesis is concerned with the study of solvable critical dense polymers using Yang-Baxter techniques in two distinct topologies, namely the strip and the cylinder. Critical dense polymers is the first member ${\cal LM}(1,2)$ of the Yang-Baxter integrable series of logarithmic minimal models ${\cal LM}(p,p')$. The associated logarithmic conformal field theory admits an infinite family of Kac representations labelled by the Kac labels $r,s=1,2,\ldots$. We begin by explicitly constructing the conjugate boundary conditions on the strip. The boundary operators are labelled by the Kac fusion \mbox{labels} $(r,s)=(r,1)\otimes (1,s)$ and involve a boundary field $\xi$. Tuning the field $\xi$ appropriately, we solve exactly for the transfer matrix eigenvalues on arbitrary finite-width strips and obtain the conformal spectra using the Euler-Maclaurin formula. The key to the solution is an inversion identity satisfied by the commuting double-row transfer matrices. The transfer matrix eigenvalues are classified by the physical combinatorics of the patterns of zeros in the complex spectral-parameter plane. This yields selection rules for the physically relevant solutions to the inversion identity which takes the form of a decomposition into irreducible blocks corresponding combinatorially to finitized characters given by generalized $q$-Catalan polynomials. This decomposition is in accord with the decomposition of the Kac characters into irreducible characters. In the scaling limit, we confirm the central charge $c=-2$ and the Kac formula for the conformal weights in the infinitely extended Kac table $\Delta_{r,s}=\frac{(2r-s)^2-1}{8}$ for $r,s=1,2,3,\ldots$. The cylinder topology allows for non-contractible loops with fugacity $\alpha$ that wind around the cylinder or for an arbitrary number $\ell$ of defects that propagate along the full length of the cylinder. Using an enlarged periodic Temperley-Lieb algebra, we set up commuting transfer matrices acting on states whose links are considered distinct with respect to connectivity around the front or back of the cylinder. These transfer matrices satisfy a functional equation in the form of an inversion identity. For even $N$, this involves a non-diagonalizable braid operator $\vec J$. The number of defects $\ell$ thus separates the theory into Ramond ($\ell/2$ even), Neveu-Schwarz ($\ell/2$ odd) and $\mathbb{Z}_4$ ($\ell$ odd) sectors. For the case of loop fugacity $\alpha=2$, the inversion identity is solved exactly sector by sector for the eigenvalues in finite geometry. The eigenvalues are classified by the physical combinatorics of the patterns of zeros in the complex spectral-parameter plane. This yields selection rules for the physically relevant solutions to the inversion identity. The finite-size corrections are obtained from Euler-Maclaurin formulas. In the scaling limit, we obtain the conformal partition functions as sesquilinear forms and confirm the central charge $c=-2$ and conformal weights $\Delta,\bar\Delta=\Delta_t=(t^2-1)/8$. Here $t=\ell/2$ and $t=2r-s\in\mathbb{N}$ in the $\ell$ even sectors with Kac labels $r=1,2,3,\ldots; s=1,2$ while $t\in\mathbb{Z}-\half$ in the $\ell$ odd sectors. Strikingly, the $\ell/2$ odd sectors exhibit a ${\cal W}$-extended symmetry but the $\ell/2$ even sectors do not. Moreover, the naive trace summing over all $\ell$ even sectors does not yield a modular invariant.
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ItemCombinatorics of lattice paths and polygons( 2012)We consider the enumeration of self-avoiding walks and polygons on regular lattices. Such objects are connected with many other problems in combinatorics, as well as in fields as diverse as physics and chemistry. We examine the general models of walks and polygons and methods we can use to study them; subclasses whose properties enable a rather deeper analysis; and extensions of these models which allow us to model physical phenomena like polymer collapse and adsorption. While the general models of self-avoiding walks and polygons are certainly not considered to be ‘solved’, recently a great deal of progress has been made in developing new methods for studying these objects and proving rigorous results about their enumerative properties, particularly on the honeycomb lattice. We consider these recent results and show that in some cases they can be extended or generalised so as to enable further proofs, conjectures and estimates. In particular, we find that properties shared by all two-dimensional lattices allow us to develop new methods for estimating the growth constants and certain amplitudes for the square and triangular lattices. The subclasses of self-avoiding walks and polygons that we consider are typically defined by imposing restrictions on the way in which a walk or polygon can be constructed. Ideally the restrictions should be as weak as possible, so as to result in a model that closely resembles the unrestricted case, while still enabling some manner of simple recursive construction. These recursions can sometimes lead to solutions for generating functions or other quantities of interest. Some of the models we consider display quite unusual asymptotic properties despite the relatively simple restrictions which lead to their construction. We approach the modelling of polymer adsorption in several ways. Firstly, we adapt some of the new methods for studying self-avoiding walks on the honeycomb lattice to account for interactions with an impenetrable surface. In this way we are able to prove the exact value of the critical surface fugacity for adsorbing walks, confirming an existing conjecture. Then, we show that some key identities for the honeycomb lattice model lead to a new method for estimating the critical surface fugacities for adsorption models on the square and triangular lattices. Many of the estimates we obtain in this way are new; for the cases where previous estimates did already exist, our results are several orders of magnitude more precise. Finally, we define some new solvable models of polymer adsorption which generalise existing models, and find that some of these models exhibit interesting and unexpected critical behaviour.