No abstract

Reviewed by David Aldous Unlike most books reviewed in the Intelligencer this is definitely a textbook. It assumes knowledge one might acquire in the first two years of an undergraduate mathematics program-basic mathematical probability, plus linear algebra, a little graph theory and the infamous concept of "mathematical maturity". It has the theorem-proof style of pure mathematics, but with friendly explanations of intuition and motivation.

A. We study the Glauber dynamics for the Ising model on the complete graph, also known as the Curie-Weiss Model. For β < 1, we prove that the dynamics exhibits a cut-off: the distance to stationarity drops from near 1 to near 0 in a window of order n centered at [2(1 − β)] −1 n log n. For β = 1, we prove that the mixing time is of order n 3/2 . For β > 1, we study metastability. In particular, we show that the Glauber dynamics restricted to states of non-negative magnetization has mixing time O(n log n).1. I 1.1. Ising model and Glauber dynamics. Let G = (V, E) be a finite graph. Elements of the state space Ω := {−1, 1} V will be called configurations, and for σ ∈ Ω, the value σ(v) will be called the spin at v. The nearest-neighbor energy H(σ) of a configuration σ ∈ {−1, 1} V is defined bywhere w ∼ v means that {w, v} ∈ E. The parameters J(v, w) measure the interaction strength between vertices; we will always take J(v, w) ≡ J, where J is a positive constant. For β ≥ 0, the Ising model on the graph G with parameter β is the probability measure µ on Ω given by µ(σ) = e −βH(σ) Z(β) ,where Z(β) = σ∈Ω e −βH(σ) is a normalizing constant.

Bacterial metabolisms are responsible for critical chemical transformations in nearly all environments, including oceans, freshwater, and soil. Despite the ubiquity of bacteria in the atmosphere, little is known about the metabolic functioning of atmospheric bacterial communities. To gain a better understanding of the metabolism of bacterial communities in the atmosphere, we used a combined empirical and model-based approach to investigate the structure and composition of potentially active bacterial communities in air sampled at a high elevation research station. We found that the composition of the putatively active bacterial community (assayed via rRNA) differed significantly from the total bacterial community (assayed via rDNA). Rare taxa in the total (rDNA) community were disproportionately active relative to abundant taxa, and members of the order Rhodospirillales had the highest potential for activity. We developed theory to explore the effects of random sampling from the rRNA and rDNA communities on observed differences between the communities. We found that random sampling, particularly in cases where active taxa are rare in the rDNA community, will give rise to observed differences in community composition including the occurrence of “phantom taxa”, taxa which are detected in the rRNA community but not the rDNA community. We show that the use of comparative rRNA/rDNA techniques can reveal the structure and composition of the metabolically active portion of bacterial communities. Our observations suggest that metabolically active bacteria exist in the atmosphere and that these communities may be involved in the cycling of organic compounds in the atmosphere.

Abstract. We consider symmetric Markov chains on the integer lattice in d dimensions, where α ∈ (0, 2) and the conductance between x and y is comparable to |x − y| −(d+α) . We establish upper and lower bounds for the transition probabilities that are sharp up to constants.