Many phenomena in physics, chemistry, and biology can be modelled by spatial random processes. One such process is continuum percolation, which is used when the phenomenon being modelled is made up of individual events that overlap, for example, the way individual raindrops eventually make the ground evenly wet. This is a systematic rigorous account of continuum percolation. Two models, the Boolean model and the random connection model, are treated in detail, and related continuum models are discussed. All important techniques and methods are explained and applied to obtain results on the existence of phase transitions, equality and continuity of critical densities, compressions, rarefaction, and other aspects of continuum models. This self-contained treatment, assuming only familiarity with measure theory and basic probability theory, will appeal to students and researchers in probability and stochastic geometry.

Double-gate FinFET devices with asymmetric and symmetric poly-silicon gates have been fabricated. Symmetric gate devices show drain currents competitive with fully optimized bulk silicon technologies. Asymmetric-gate devices show IV,I-O.lV, with off-currents less than 100nA/um at Vgs= 0.

We study the asymptotic properties of a minimal spanning tree formed by n points uniformly distributed in the unit square, where the minimality is amongst all rooted spanning trees with a direction of growth. We show that the number of branches from the root of this tree, the total length of these branches, and the length of the longest branch each converges weakly. This model is related to the study of record values in the theory of extreme-value statistics and this relation is used to obtain our results. The results also hold when the tree is formed from a Poisson point process of intensity n in the unit square.

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Consider the d-dimensional lattice Z d where each vertex is "open" or "closed" with probability p or 1 − p, respectively. An open vertex v is connected by an edge to the closest open vertex w such that the dth co-ordinates of v and w satisfy w(d) = v(d) − 1. In case of nonuniqueness of such a vertex w, we choose any one of the closest vertices with equal probability and independently of the other random mechanisms. It is shown that this random graph is a tree almost surely for d = 2 and 3 and it is an infinite collection of distinct trees for d ≥ 4. In addition, for any dimension, we show that there is no bi-infinite path in the tree and we also obtain central limit theorems of (a) the number of vertices of a fixed degree ν and (b) the number of edges of a fixed length l.

It is shown that this random graph is a tree almost surely for d = 2 and 3 and it is an infinite collection of disjoint trees for d ≥ 4. In addition, for d = 2, we show that when properly scaled, the family of its paths converge in distribution to the Brownian web.

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