{"id":"openai-math/Critical-bond-and-site-percolation-on-the-cubic-lattice","pdf":"https://github.com/openai/math/blob/main/preprints/Critical-bond-and-site-percolation-on-the-cubic-lattice-September-24-2026/paper.pdf","url":"https://github.com/openai/math/tree/main/preprints/Critical-bond-and-site-percolation-on-the-cubic-lattice-September-24-2026","tags":["percolation","probability","lattice-models"],"score":0,"title":"Critical bond and site percolation on the cubic lattice","domain":"Probability and statistical mechanics","family":"213","origin":"ai-generated","source":"openai-math","authors":["OpenAI"],"reviews":0,"subject":"math.PR","summary":"We prove that nearest-neighbor Bernoulli bond and site percolation on ℤ 3 have no infinite cluster at their respective critical parameters. The proof combines a finite connection inequality for independent hyperedges with a finite-scale extension estimate and an adaptive exploration.","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":1,"published":"2026-09-24","reactions":{},"discussion":[{"url":"https://haggstrom.substack.com/p/what-i-write-about-the-day-after#:~:text=Do%20I%20explain%20the%20significance%20of%20the%20solutions%20to%20problems%20in%20percolation%20theory","date":"2026-10-07","kind":"web","site":"haggstrom.substack.com","text":"Do I explain the significance of the solutions to problems in percolation theory (such as this one , and this one , and this one , and this one , and this one ) that I used to put so much effort into solving (but of course failing) back in the 90s and early 00s? Do I attempt a bit of ego-boost by…","added":"2026-11-03","title":"What I write about the day after the most earth-shattering event in the history of mathematics","author":"Olle Häggström"}],"formalized":true,"family_title":"Critical percolation on every quasi-transitive graph"}