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About this paper
Nikita Gladkov, Igor Pak, Aleksandr Zimin
Researchers from UCLA and MIT have disproved the 38-year-old Bunkbed Conjecture in percolation theory by constructing an explicit planar graph with 7222 vertices and 14442 edges, demonstrating a counterintuitive reversal in connection probabilities. The counterexample reveals that the conjectured inequality fails for Bernoulli bond percolation, with an infinitesimally small probability difference of approximately 10^-4331.
Key insights
- Hollom's hypergraph counterexample, originally for an alternative hypergraph percolation model, can be extended to a more general WZ hypergraph percolation model under a specific inequality condition on hyperedge connection probabilities.
- A carefully designed planar graph gadget can effectively simulate the complex probabilistic behavior of a 3-uniform hyperedge within standard Bernoulli bond percolation.
- The disproof highlights a fundamental distinction between Bernoulli bond percolation and other probabilistic models (like the Ising model analogue) where similar correlation inequalities hold.
Summary by alphaXiv.
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