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 beneath 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 reentrance transition at constant pulling force for small temperature, which has been observed previously for this model in the presence of pure vertical pulling. Interestingly, the reentrance phenomenon vanishes under certain pulling angles, with details depending on how the three-dimensional polymer is modeled.

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.

Let D(n) be the maximal determinant for n × n {±1}-matrices, and R(n) = D(n)/n n/2 be the ratio of D(n) to the Hadamard upper bound. Using the probabilistic method, we prove new lower bounds on D(n) and R(n) in terms of d = n − h, where h is the order of a Hadamard matrix and h is maximal subject to h ≤ n. For example,By a recent result of Livinskyi, d 2 /h 1/2 → 0 as n → ∞, so the second bound is close to (πe/2) −d/2 for large n. Previous lower bounds tended to zero as n → ∞ with d fixed, except in the cases d ∈ {0, 1}. For d ≥ 2, our bounds are better for all sufficiently large n. If the Hadamard conjecture is true, then d ≤ 3, so the first bound above shows that R(n) is bounded below by a positive constant (πe/2) −3/2 > 0.1133.

We show that the maximal determinant D(n) for n × n {±1}-Here n n/2 is the Hadamard upper bound, and κ d depends only on d := n − h, where h is the maximal order of a Hadamard matrix with h ≤ n. Previous lower bounds on R(n) depend on both d and n. Our bounds are improvements, for all sufficiently large n, if d > 1.We give various lower bounds on R(n) that depend only on d. For example, R(n) ≥ 0.07 (0.352) d > 3 −(d+3) . For any fixed d ≥ 0 we have R(n) ≥ (2/(πe)) d/2 for all sufficiently large n (and conjecturally for all positive n). If the Hadamard conjecture is true, then d ≤ 3 and κ d ≥ (2/(πe)) d/2 > 1/9.

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