Dying percolation conjecture

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The dying percolation conjecture (also known as θ ( p c ) = 0 {\displaystyle \theta (p_{c})=0} conjecture) is a prominent conjecture in percolation theory. It states that critical Bernoulli bond percolation on Z d {\displaystyle \mathbb {Z} ^{d}} for d ≥ 2 {\displaystyle d\geq 2} almost surely has no infinite clusters. The conjecture is proved for the cases d = 2 {\displaystyle d=2} by Kesten, for d ≥ 19 {\displaystyle d\geq 19} by Hara and Slade using the technique called "lace expansion". The later result was then improved to d...

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