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We investigate unimodular random networks. Our motivations include their characterization via reversibility of an associated random walk and their similarities to unimodular quasi-transitive graphs. We extend various theorems concerning random walks, percolation, spanning forests, and amenability from the known context of unimodular quasi-transitive graphs to the more general context of unimodular random networks. We give properties of a trace associated to unimodular random networks with applications to stochastic comparison of continuous-time random walk. Contents2000 Mathematics Subject Classification. Primary 60C05. Secondary 60K99, 05C80.

JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org. This content downloaded from 130.237.165.40 on Thu, internet: http://www.stat.berkeley.edu/users/aldous Consider N particles, which merge into clusters according to the following rule: a cluster of size x and a cluster of size y merge at (stochastic) rate K(x, y)/N, were K is a specified rate kernel. This Marcus-Lushnikov model of stochastic coalescence and the underlying deterministic approximation given by the Smoluchowski coagulation equations have an extensive scientific literature. Some mathematical literature (Kingman's coalescent in population genetics; component sizes in random graphs) implicitly studies the special cases K(x, y) = 1 and K(x, y) = xy. We attempt a wide-ranging survey. General kernels are only now starting to be studied rigorously; so many interesting open problems appear.We included the conventional attributions of four of these formulae, which have been rediscovered many times. The remaining two continuous solutions arise by rescaling time in the corresponding discrete solutions and taking limits as t -o00 or t -? 1. These continuous solutions are more implicit than explicit in the SM literature. The continuous x + y solution has ml(t)= 1 but infinite cluster density mo(t). The continuous xy solution has both mo(t) and ml(t) infinite and is therefore often called "unphysical"; see Section 4.4 for its interpretation.There has also been considerable attention paid to the general bilinear kernel K(x, y) = A + B(x + y) + Cxy, for which some more complicated explicit solutions are available (Trubnikov 1971; Spouge 1983a,b; van Dongen and Ernst 1984).

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