In this paper we report simulation studies of equilibrium features, namely circular islands on model surfaces, using Monte-Carlo methods. In particular, we are interested in studying the relationship between the density of vapour around a curved island and its curvature. The "classical" form of this relationship is the Gibbs-Thomson formula, which assumes the vapour surrounding the island to be an ideal gas. Numerical simulations of a lattice gas model, performed for various sizes of islands, don't fit very well to the Gibbs-Thomson formula. We show how corrections to this form arise at high vapour densities, wherein a knowledge of the exact equation of state (as opposed to the ideal gas approximation) is necessary to predict this relationship.By exploiting a mapping of the lattice gas to the Ising model one can compute the corrections to the Gibbs-Thomson formula using high field series expansions. The corrected Gibbs-Thomson formula matches very well with the Monte Carlo data. We also investigate finite size effects on the stability of the islands both theoretically and through simulations. Finally the simulations are used to study the microscopic origins of the Gibbs-Thomson formula. It is found that smaller islands have a greater adatom detachment rate per unit length of island perimeter. This is principally due to a lower coordination of edge atoms and a greater availability of detachment moves relative to edge 1 moves. A heuristic argument is suggested in which these effects are partially attributed to geometric constraints on the island edge. 05.50.+q, 64.60.Qb, 82.60.Nh, 82.65.Dp

Abstract-There are many contexts in distributed wireless networks where there is a critical threshold, corresponding to a minimum amount of the communication effort or power expenditure by individual nodes, above which a desirable global property exists with high probability. When this individual node effort is below the threshold the desired global property exists with a low probability. This "phase transition" is typically seen to become sharper as the number of nodes in the network increases. We discuss in this paper some examples of properties that exhibit such critical behavior: node reachability with probabilistic flooding, ad-hoc network connectivity, and sensor network coordination. We discuss the connections between these phenomena and the phase transitions that have been shown to arise in random graphs. We argue that a good understanding of these phase transition phenomena can provide useful design principles for engineering distributed wireless networks.

A theory based on the thermodynamic Gibbs-Thomson relation is presented which provides the framework for understanding the time evolution of isolated nanoscale features (i.e., islands and pits) on surfaces. Two limiting cases are predicted, in which either diffusion or interface transfer is the limiting process. These cases correspond to similar regimes considered in previous works addressing the Ostwald ripening of ensembles of features. A third possible limiting case is noted for the special geometry of "stacked" islands. In these limiting cases, isolated features are predicted to decay in size with a power law scaling in time: A ∝ (t 0 − t) n , where A is the area of the feature, t 0 is the time at which the feature disappears, and n = 2/3 or 1. The constant of proportionality is related to parameters describing both the kinetic and equilibrium properties of the surface. A continuous time Monte Carlo simulation is used to test the application of this theory to generic surfaces with atomic scale features. A new method is described to obtain macroscopic kinetic parameters describing interfaces in such simulations. Simulation and analytic theory are compared directly, using measurements of the simulation to determine the constants of the analytic theory. Agreement between the * Communicating author. Present Address:

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