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We consider random‐access networks where nodes represent servers with a queue and can be either active or inactive. A node deactivates at unit rate, while it activates at a rate that depends on its queue length, provided none of its neighbors is active. We consider arbitrary bipartite graphs in the limit as the initial queue lengths become large and identify the transition time between the two states where one half of the network is active and the other half is inactive. The transition path is decomposed into a succession of transitions on complete bipartite subgraphs. We formulate a randomized greedy algorithm that takes the graph as input and gives as output the set of transition paths the network is most likely to follow. Along each path we determine the mean transition time and its law on the scale of its mean. Depending on the activation rates, we identify three regimes of behavior. [ABSTRACT FROM AUTHOR]

Abstract: We obtain bounds for the expected loss of torsional rigidity of a cylinder C L of length L and planar cross‐section Ω due to a Brownian fracture that starts at a random point in C L and runs until the first time it exits C L. These bounds are expressed in terms of the geometry of the cross‐section Ω ⊂ R 2. It is shown that if Ω is a disc with radius R, then in the limit as L → ∞ the expected loss of torsional rigidity equals c R 5 for some c ∈ ( 0 , ∞ ). We derive bounds for c in terms of the expected Newtonian capacity of the trace of a Brownian path that starts at the centre of a ball in R 3 with radius 1, and runs until the first time it exits this ball. [ABSTRACT FROM AUTHOR]