Superbranching processes and projections of random Cantor sets

We study sequences (X ₀, X ₁, ...) of random variables, taking values in the positive integers, which grow faster than branching processes in the sense that , for m, n0, where the X _n (m, i) are distributed as X _n and have certain properties of independence. We prove that, under appropriate conditions, X _n ^1/n almost surely and in L ¹, where =sup E(X _n )^1/n . Our principal application of this result is to study the Lebesgue measure and (Hausdorff) dimension of certain projections of sets in a class of random Cantor sets, being those obtained by repeated random subdivisions of the M-adic subcubes of [0, 1] ^d . We establish a necessary and sufficient condition for the Lebesgue measure of a projection of such a random set to be non-zero, and determine the box dimension of this projection.Work done partly whilst visiting Cornell University with the aid of a Fulbright travel grant     

Random walk on the infinite cluster of the percolation model

Summary We consider random walk on the infinite cluster of bond percolation on ^d . We show that, in the supercritical regime whend3, this random walk is a.s. transient. This conclusion is achieved by considering the infinite percolation cluster as a random electrical network in which each open edge has unit resistance. It is proved that the effective resistance of this network between a nominated point and the points at infinity is almost surely finite.G.R.G. acknowledges support from Cornell University, and also partial support by the U.S. Army Research Office through the Mathematical Sciences Institute of Cornell UniversityH.K. was supported in part by the N.S.F. through a grant to Cornell University     

Percolation in half-spaces: equality of critical densities and continuity of the percolation probability

Renormalization arguments are developed and applied to independent nearest-neighbor percolation on various subsets of ^d ,d2, yielding:

Bond percolation on isoradial graphs: criticality and universality

In an investigation of percolation on isoradial graphs, we prove the criticality of canonical bond percolation on isoradial embeddings of planar graphs, thus extending celebrated earlier results for homogeneous and inhomogeneous square, triangular, and other lattices. This is achieved via the star–triangle transformation, by transporting the box-crossing property across the family of isoradial graphs. As a consequence, we obtain the universality of these models at the critical point, in the sense that the one-arm and $2j$ -alternating-arm critical exponents (and therefore also the connectivity and volume exponents) are constant across the family of such percolation processes. The isoradial graphs in question are those that satisfy certain weak conditions on their embedding and on their track system. This class of graphs includes, for example, isoradial embeddings of periodic graphs, and graphs derived from rhombic Penrose tilings.     

A linear cell-size dependent branching process

We consider a cell-size dependent branching process in which each cell grows at a linear rate and divides into a pair of daughter cells, preserving total size, at a rate proportional to its size. Such processes expand exponentially fast. If, on division, each possible combination of daughter sizes occurs with equal probability, then conventional analysis provides explicit values for the limiting distribution of the size of a typical cell, together with the distributions of its size just after its birth and just before its division.     

Percolation of arbitrary words in one dimension

We consider a type of long‐range percolation problem on the positive integers, motivated by earlier work of others on the appearance of (in)finite words within a site percolation model. The main issue is whether a given infinite binary word appears within an iid Bernoulli sequence at locations that satisfy certain constraints. We settle the issue in some cases, and we provide partial results in others. © 2010 Wiley Periodicals, Inc. Random Struct. Alg., 2010     

A method for determination of selenium in organic tissues using microwave digestion and liquid chromatography

An existing laboratory procedure for selenium analysis using open-vessel wet digestion and liquid chromatographic fluorescence determination was modified for use with microwave digestion. The proposed microwave digestion method eliminated the hazards associated with the use of HClO4 while maintaining excellent recoveries of selenium. A 2-step HNO3/H2O2 digestion procedure was developed. Digested samples were derivatized with 2,3-diaminonaphthalene, and the resultant piazselenol complex was measured fluorometrically using a liquid chromatograph. Measured values were in agreement with 9 different certified reference materials. The detection limit for this method was 0.54 ng Se/g tissue (3 sigma), and the calibration curve remained linear (r2 = 0.9968) up to 2 microg Se/g.     

Weak convergence using higher-order cumulants

Denote byc _j (F) thejth cumulant (or semi-invariant) of the distribution functionF. We say thatF is specified by its higher-order cumulants if it is the unique distribution functionG having the following property: there exists a positive integerJ such thatc _j (G)=c _j (F) forj=1,2 andjJ. Let (F _n n1) be a sequence of distribution functions, and suppose that there existsJ such thatc _j (F _n )c _j (F) asn, forj=1,2 andjJ. It is proved thatF _n F so long asF is specified by its higher-order cumulants. It is an open problem to characterize the family of distributions which are specified by their higher-order cumulants.