A J Guttmann

Vicious walkers, friendly walkers and Young tableaux: II. With a wall

We derive new results for the number of star and watermelon configurations of vicious walkers in the presence of an impenetrable wall by showing that these follow from standard results in the theory of Young tableaux and combinatorial descriptions of symmetric functions. For the problem of n friendly walkers, we derive exact asymptotics for the number of stars and watermelons, both in the absence of a wall and in the presence of a wall.

Vicious walkers and Young tableaux I: without walls

We rederive previously known results for the number of star and watermelon configurations by showing that these follow immediately from standard results in the theory of Young tableaux and integer partitions. In this way we provide a proof of a result, previously only conjectured, for the total number of stars.

Self-avoiding polygons on the square lattice

We have developed an improved algorithm that allows us to enumerate the number of self-avoiding polygons on the square lattice to perimeter length 90. Analysis of the resulting series yields very accurate estimates of the connective constant =2.638 158 529 27(1) (biased) and the critical exponent = 0.500 0005(10) (unbiased). The critical point is indistinguishable from a root of the polynomial 581x4 + 7x2 - 13 = 0. An asymptotic expansion for the coefficients is given for all n. There is strong evidence for the absence of any non-analytic correction-to-scaling exponent.

The Susceptibility of the Square Lattice Ising Model: New Developments

We have made substantial advances in elucidating the properties of the susceptibility of the square lattice Ising model. We discuss its analyticity properties, certain closed form expressions for subsets of the coefficients, and give an algorithm of complexity O(N6) to determine its first N coefficients. As a result, we have generated and analyzed series with more than 300 terms in both the high- and low-temperature regime. We quantify the effect of irrelevant variables to the scaling-amplitude functions. In particular, we find and quantify the breakdown of simple scaling, in the absence of irrelevant scaling fields, arising first at order |T−Tc|9/4, though high-low temperature symmetry is still preserved. At terms of order |T−Tc|17/4 and beyond, this symmetry is no longer present. The short-distance terms are shown to have the form (T−Tc)p (log |T−Tc|)q with p≥q2. Conjectured exact expressions for some correlation functions and series coefficients in terms of elliptic theta functions also foreshadow future developments.

Algebraic techniques for enumerating self-avoiding walks on the square lattice

The authors describe a new algebraic technique for enumerating self-avoiding walks on the rectangular lattice. The computational complexity of enumerating walks of N steps is of order 3N/4 times a polynomial in N, and so the approach is greatly superior to direct counting techniques. They have enumerated walks of up to 39 steps. As a consequence, they are able to accurately estimate the critical point, critical exponent, and critical amplitude.

Lattice Green's functions in all dimensions

We give a systematic treatment of lattice Green’s functions (LGF) on the d-dimensional diamond, simple cubic, body-centred cubic and face-centred cubic lattices for arbitrary dimensionality d 2 for the first three lattices, and for 2 d 5 for the hyper-fcc lattice. We show that there is a close connection between the LGF of the d-dimensional hyper-cubic lattice and that of the (d − 1)-dimensional diamond lattice. We give constant-term formulations of LGFs for each of these lattices in all dimensions. Through a still under-developed connection with Mahler measures, we point out an unexpected connection between the coefficients of the sc, bcc and diamond LGFs and some Ramanujan-type formulae for 1/π.

Self-avoiding walks on the simple cubic lattice

We have substantially extended the series for the number of self-avoiding walks and the mean-square end-to-end distance on the simple cubic lattice. Our analysis of the series gives refined estimates for the critical point and critical exponents. Our estimates of the exponents γ and ν are in good agreement with recent high-precision Monte Carlo estimates, and also with recent renormalization group estimates. Critical amplitude estimates are also given. A new, improved rigorous upper bound for the connective constant µ<4.7114 is obtained.

Two-dimensional lattice vesicles and polygons

We consider a lattice model of two-dimensional vesicles, in which the boundary of the vesicle is the perimeter of a self-avoiding polygon embeddable in the square lattice. With fixed boundary length m we incorporate an osmotic pressure difference by associating a fugacity with the area enclosed by the polygon. We derive rigorous results concerning the behaviour of the associated free energy and the form of the phase diagram. By deriving exact values ofthe numbers of polygons with m edges which enclose area n, and analysing the resulting series, we obtain the free energy, the phase boundary and various scaling exponents and amplitudes numerically.

A collapse transition in a directed walk model

The authors consider a directed walk model of linear polymers in dilute solution, with an energy associated with the number of near-neighbour contacts in the walk. For this model they can derive an exact expression for the generating function in two variables conjugate to the number of steps and the number of contacts. They discuss the analytic structure of this generating function and identify the transition corresponding to collapse.

Series expansion of the percolation probability for the directed square lattice

By extrapolation from finite lattices, the authors extend the known series for the percolation probability on the directed square lattice from eight terms to 41. Analysing the series, they obtain the estimates qc=0.355 299+or-0.000 001, beta =0.2764+or-0.0001 for the critical probability and the critical exponent. From this, together with scaling relations and previous results on the moments of the pair-connectedness function, the authors conjecture that beta may be exactly 199/720.