L. S. Schulman

Infinite Clusters in Percolation Models

The qualitative nature of infinite clusters in percolation models is investigated. The results, which apply to both independent and correlated percolation in any dimension, concern the number and density of infinite clusters, the size of their external surface, the value of their (total) surface-to-volume ratio, and the fluctuations in their density. In particular it is shown thatN0, the number of distinct infinite clusters, is either 0, 1, or ∞ and the caseN0=∞ (which might occur in sufficiently high dimension) is analyzed.

Path Integrals in Curved Spaces

In this paper we present a simplification of the path integral solution of the Schrodinger equation in terms of coordinates which need not be Cartesian. After presenting the existing formula, we discuss the relationship between the distance and time differentials. Making this relationship precise through the technique of stationary phase, we are able to simplify the path integral. The resulting expression can be used to obtain a Hamiltonian path integral. Finally, we comment on a similar phenomenon involving differentials in the Ito integral.

One dimensional 1/|j − i| S percolation models: The existence of a transition forS≦2

AbstractConsider a one-dimensional independent bond percolation model withpj denoting the probability of an occupied bond between integer sitesi andi±j,j≧1. Ifpj is fixed forj≧2 and

Statistical mechanics of a dynamical system based on Conway's game of Life

\mathop {\lim }\limits_{j \to \infty }

Explicit time-dependent Schrodinger propagators

j2pj>1, then (unoriented) percolation occurs forp1 sufficiently close to 1. This result, analogous to the existence of spontaneous magnetization in long range one-dimensional Ising models, is proved by an inductive series of bounds based on a renormalization group approach using blocks of variable size. Oriented percolation is shown to occur forp1 close to 1 if

Zeno dynamics yields ordinary constraints

\mathop {\lim }\limits_{j \to \infty }