Steven Finch

Universality of finite-size corrections to the number of critical percolation clusters

Monte-Carlo simulations on a variety of 2d percolating systems at criticality suggest that the excess number of clusters in finite systems over the bulk value of nc is a universal quantity, dependent upon the system shape but independent of the lattice and percolation type. Values of nc are found to high accuracy, and for bond percolation confirm the theoretical predictions of Temperley and Lieb, and Baxter, Temperley, and Ashley, which we have evaluated explicitly in terms of simple algebraic numbers. Predictions for the fluctuations are also verified for the first time.

Several constants arising in statistical mechanics

This is a brief survey of certain constants associated with random lattice models, including self-avoiding walks, polyominoes, the Lenz-Ising model, monomers and dimers, ice models, hard squares and hexagons, and percolation models.

ROOTS OF UNITY AND NULLITY MODULO n

For a fixed positive integer l, we consider the function of n that counts the number of elements of order l in ℤ * n . We show that the average growth rate of this function is C l (log n) d(l)―1 for an explicitly given constant C l , where d(l) is the number of divisors of l. From this we conclude that the average growth rate of the number of primitive Dirichlet characters modulo n of order l is (d(l) ― 1)C l (log n) d(l)―2 for l ≥ 2. We also consider the number of elements of ℤ n whose lth power equals 0, showing that its average growth rate is D l (log n) l―1 for another explicit constant D l . Two techniques for evaluating sums of multiplicative functions, the Wirsing—Odoni and Selberg— Delange methods, are illustrated by the proofs of these results.

Random convex hulls: a variance revisited

An exact expression is determined for the asymptotic constant c 2 in the limit theorem by P. Groeneboom (1988), which states that the number of vertices of the convex hull of a uniform sample of n random points from a circular disk satisfies a central limit theorem, as n → ∞, with asymptotic variance 2πc 2 n 1/3.