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What does a random independent set in a graph look like? In what physicists call the "hard-core lattice gas model" independent sets sometimes exhibit phase transitions, but embarrassingly little is known even when the graph is an ordinary d-dimensional grid. With Jim Propp (MIT) and Jeffrey Steif (Chalmers), working in August in the Sierras, we were able at least to pin down the behavior of random independent sets in a k-ary tree.
The rest is up to you.