The Low-Temperature Expansion of the Wulff Crystal in the 3D Ising Model

We compute the expansion of the surface tension of the 3D random cluster model for q≥ 1 in the limit where p goes to 1. We also compute the asymptotic shape of a plane partition of n as n goes to ∞. This same shape determines the Wulff crystal to order o(ɛ) in the 3D Ising model (and more generally in the 3D random cluster model for q≥ 1) at temperature ɛ. Received: 15 February 2001/ Accepted: 11 May 2001     

On the Transition from Tin-Rich to Antimony-Rich European White Soda-Glass Trade Beads for the Senecas of Northeastern North America

It has been shown that several modifications occurred, over the span of the 17th to 19th centuries, in the agents used to opacify European-made white soda-glass beads that were transmitted as trade goods to northeastern North America. Tin was used at the beginning of the 17th century, followed by Sb later in the century, and then by As during the 18th and 19th centuries. In an attempt to define more closely the transition from Sn-rich to Sb-rich white beads, we analyzed 198 white glass beads from a number of archaeological sites in western New York State. Chemical analysis shows that the arrival of Sb-white soda-glass trade beads began in this region during the period from approximately A.D. 1625–1640, and that they had completely replaced Sn-white beads by A.D. 1675. Specific bead chemistries link a number of the archaeological sites.     

Operator stable laws: Series representations and domains of normal attraction

LetX,X ₁,X ₂,... be i.i.d. random vectors in ^d. The limit laws that can arise by suitable affine normalizations of the partial sums,S _n=X ₁+...+X _n, are calledoperator-stable laws. These laws are a natural extension to ^d of the stable laws on. Thegeneralized domain of attraction of [GDOA()] is comprised of all random vectorsX whose partial sums can be affinely normalized to converge to . If the linear part of the affine transformation is restricted to take the formn ^–B for some exponent operatorB naturally associated to thenX is in thegeneralized domain of normal attraction of [GDONA()]. This paper extends the theory of operator-stable laws and their domains of attraction and normal attraction.     

The Laplacian and Dirac operators on critical planar graphs

On a periodic planar graph whose edge weights satisfy a certain simple geometric condition, the discrete Laplacian and Dirac operators have the property that their determinants and inverses only depend on the local geometry of the graph. We obtain explicit expressions for the logarithms of the (normalized) determinants, as well as the inverses of these operators. We relate the logarithm of the determinants to the volume plus mean curvature of an associated hyperbolic ideal polyhedron. In the associated dimer and spanning tree models, for which the determinants of the Dirac operator and the Laplacian respectively play the role of the partition function, this allows us to compute the entropy and correlations in terms of the local geometry. In addition, we define a continuous family of special discrete holomorphic functions which, via convolutions, gives a general process for constructing discrete holomorphic functions and discrete harmonic functions on critical planar graphs. Oblatum 6-III-2002 & 12-VI-2002?Published online: 6 August 2002     

Optimal Upper and Lower Bounds for the Upper Tails of Compound Poisson Processes

A compound Poisson process is of the form where Z, Z ₁, Z ₂, are arbitrary i.i.d. random variables and N is an independent Poisson random variable with parameter . This paper identifies the degree of precision that can be achieved when using exponential bounds together with a single truncation to approximate . The truncation level introduced depends only on and Z and not on the overall exceedance level a.