Recently a uniqueness condition for Gibbs measures in terms of disagreement percolation (a type of dependent percolation involving two realizations) has been obtained. In general this condition is sufficient but not necessary for uniqueness. In the present paper we study the hard-core lattice gas model which we abbreviate as hard-core model. This model is not only relevant in Statistical Physics, but was recently rediscovered in Operations Research in the context of certain communication networks. First we show that the uniqueness result mentioned above implies that the critical activity for the hard-core model on a graph is at least PJ (1-PJ, where P< is the critical probability for site percolation on that graph. Then, for the hard-core model on bi-partite graphs. we study the probability that a given vertex ,. is occupied under the two extreme boundary conditions, and show that the difference can be written in terms of the probability of having a •path of disagreement' from 1• to the boundary. This is the key to a proof that, for this case, the uniqueness condition mentioned above is also necessary, i.e. roughly speaking, phase transition is equivalent with disagreement percolation in the product space. Finally, we discuss the hard-core model on "ll." with two different values of the activity, one for the even, and one for the odd vertices. It appears that the question whether this model has a unique Gibbs measure, can, in analogy with the standard ferromagnetic Ising model, be reduced to the question whether the third central moment of the surplus of odd occupied vertices for a certain class of finite boxes is negative.

Consider first-passage percolation on z_d. A classical result says, roughly speaking, that the shortest travel time from (0, 0, ... , 0) to (n, 0, ... , 0) is asymptotically equal to nµ., for some constant µ., which is called the time constant, and which depends on the distribution of the time coordinates. Except for very special cases, the value of µ. is not known. We show that certain changes of the time coordinate distribution lead to a decrease of µ.; usually µ. will strictly decrease. Two examples of our results are:(i) If F and G are distribution functions with F 5 G, F ;;S G, then, under mild conditions, the time constant for G is strictly smaller than that for F.(ii) For 0 < e 1 < e2 s a< b, the time constant for the uniform distribution on [a -e2, b + eil is strictly smaller than for the uniform distribution on [a, b].We assume throughout that all our distributions have finite first moments.

A probability measure μ on ℝn+ is defined to be strongly new better than used (SNBU) if for all increasing subsets . For n = 1 this is equivalent to being new better than used (NBU distributions play an important role in reliability theory). We derive an inequality concerning products of NBU probability measures, which has as a consequence that if μ1, μ2, ···, μn are NBU probability measures on ℝ+, then the product-measure μ = μ × μ2 × ··· × μn on ℝn+ is SNBU. A discrete analog (i.e., with N instead of ℝ+) also holds.Applications are given to reliability and percolation. The latter are based on a new inequality for Bernoulli sequences, going in the opposite direction to the FKG–Harris inequality. The main application (3.15) gives a lower bound for the tail of the cluster size distribution for bond-percolation at the critical probability. Further applications are simplified proofs of some known results in percolation. A more general inequality (which contains the above as well as the FKG-Harris inequality) is conjectured, and connections with an inequality of Hammersley [12] and others ([17], [19] and [7]) are indicated.

C e n t r u m v o o r W i s k u n d e e n I n f o r m a t i c a PNA Probability, Networks and Algorithms Probability, Networks and AlgorithmsABSTRACT Consider ordinary site percolation on an infinite graph in which the sites, independent of each other, are occupied with probability p and vacant with probability 1−p. Now suppose that, by some 'catastrophe', all sites which are in an infinite occupied cluster become vacant. Finally, each vacant site gets an extra enhancement to become occupied. More precisely, each site that was already vacant or that was made vacant by the catastrophe, becomes occupied with probability δ (independent of the other sites). When p is larger than but close to the critical value p c one might believe (for 'nice' graphs) that only a small δ is needed to have an infinite occupied cluster in the final configuration. This appears to be indeed the case for the binary tree. However, on the square lattice we strongly conjecture that this is not true. We discuss the background for these problems and also show that the conjecture, if true, has some remarkable consequences. Mathematics Subject Classification: 60K35

In this paper we present an integrated packet/flow level modelling approach for analysing flow throughputs and transfer times in 802.11 s. The packet level model captures the statistical characteristics of the transmission of individual packets at the layer, while the flow level model takes into account the system dynamics due to the initiation and completion of data flow transfers. The latter model is a processor sharing type of queueing model reflecting the 802.11 design principle of distributing the transmission capacity fairly among the active flows. The resulting integrated packet/flow level model is analytically tractable and yields a simple approximation for the throughput and flow transfer time. Extensive simulations show that the approximation is very accurate for a wide range of parameter settings. In addition, the simulation study confirms the attractive property following from our approximation that the expected flow transfer delay is insensitive to the flow size distribution (apart from its mean).