Judy-anne H. Osborn

Forcing Adsorption of a Tethered Polymer by Pulling

We present an analysis of a partially directed walk model of a polymer which at one end is tethered to a sticky surface and at the other end is subjected to a pulling force at fixed angle away from the point of tethering. Using the kernel method, we derive the full generating function for this model in two and three dimensions and obtain the respective phase diagrams. We observe adsorbed and desorbed phases with a thermodynamic phase transition in between. In the absence of a pulling force this model has a second-order thermal desorption transition which merely gets shifted by the presence of a lateral pulling force. On the other hand, if the pulling force contains a non-zero vertical component this transition becomes first order. Strikingly, we find that, if the angle between the pulling force and the surface is below a critical value, a sufficiently strong force will induce polymer adsorption, no matter how large the temperature of the system. Our findings are similar in two and three dimensions, an additional feature in three dimensions being the occurrence of a re-entrance transition at constant pulling force for low temperature, which has been observed previously for this model in the presence of pure vertical pulling. Interestingly, the re-entrance phenomenon vanishes under certain pulling angles, with details depending on how the three-dimensional polymer is modeled.

Lattice Paths and the Constant Term

We firstly review the constant term method (CTM), illustrating its combinatorial connections and show how it can be used to solve a certain class of lattice path problems. We show the connection between the CTM, the transfer matrix method (eigenvectors and eigenvalues), partial difference equations, the Bethe Ansatz and orthogonal polynomials. Secondly, we solve a lattice path problem first posed in 1971. The model stated in 1971 was only solved for a special case - we solve the full model. This paper is in two parts. The first part is mostly review of the Constant Term Method (CTM) and the second part of the paper is a new application of the method where is used to solve a model which has remained unsolved since 1971. The Constant Term Method has been used to provide elegant solutions to a number of lattice path problems, which have arisen variously in the study of several polymer models (4, 8, 9, 6, 11) and in the calculation of the stationary state of the asymmetric exclusion model (ASEP) (7, 5). Most problems may be solved by alternative means, but the form of the solution obtained by the CTM is of interest in that it is suggestive of a purely combinatorial proof. The CTM is a good example of a bridge between Statistical Mechanics and Pure Combinatorics. Often, in Statistical Mechanics, the desire to understand some physical system will motivate a mathematical model, the solution of which requires new combinatorial methods. The techniques thereby developed open up new possibilities in Combinatorics, as well as often suggesting new, tractable, physical models. The history of the CTM illustrates this interplay, having been originally developed to solve certain polymer models, but having lead to new combinatorics with wider application. We conclude by using the CTM to solve a previously open problem. The full details of the CTM and its generalisation may be found in Brak and Osborn (10).

Bounds on minors of binary matrices

We prove an upper bound on sums of squares of minors of f+1; 1g-matrices. The bound is sharp for Hadamard matrices, a result due to de Launey and Levin [‘(1; 1)-matrices with near-extremal properties’, SIAM J. Discrete Math. 23 (2009), 1422‐1440], but our proof is simpler. We give several corollaries relevant to minors of Hadamard matrices.