Mathieu, Pierre; Temmel, Christoph \(k\)-independent percolation on trees. (English) Zbl 1245.82026 Stochastic Processes Appl. 122, No. 3, 1129-1153 (2012). Summary: Consider the class of \(k\)-independent bond or site percolations with parameter \(p\) on a tree \(\mathbb T\). We derive tight bounds on \(p\) for both almost sure percolation and almost sure nonpercolation. The bounds are continuous functions of \(k\) and the branching number of \(\mathbb T\). This extends previous results by Lyons for the independent case \((k=0)\) and by Balister & Bollobás for 1-independent bond percolations. Central to our argumentation are moment method bounds à la Lyons supplemented by explicit percolation models à la Balister & Bollobás. An indispensable tool is the minimality and explicit construction of Shearer’s measure on the \(k\)-fuzz of \(\mathbb Z\). Cited in 3 Documents MSC: 82B43 Percolation 60K35 Interacting random processes; statistical mechanics type models; percolation theory 82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics 05C05 Trees Keywords:\(k\)-independent; \(k\)-dependent; tree percolation; critical value; percolation kernel; second moment method; Shearer’s measure × Cite Format Result Cite Review PDF Full Text: DOI arXiv References: [1] Aaronson, J.; Gilat, D.; Keane, M.; de Valk, V., An algebraic construction of a class of one-dependent processes, Ann. Probab., 17, 128-143 (1989) · Zbl 0681.60038 [2] Billingsley, P., Probability and measure, (Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics (1995), John Wiley & Sons Inc.: John Wiley & Sons Inc. New York) · Zbl 0172.21201 [3] N.P. Balister, B. Bollobás, Random geometric graphs and dependent percolation, 2006, Presentation notes taken by Mathieu, Pierre in Paris during the IHP trimester Phenomena in High Dimensions.; N.P. Balister, B. Bollobás, Random geometric graphs and dependent percolation, 2006, Presentation notes taken by Mathieu, Pierre in Paris during the IHP trimester Phenomena in High Dimensions. [4] Erdős, P.; Lovász, L., Problems and results on 3-chromatic hypergraphs and some related questions, (Infinite and Finite Sets (Colloq., Keszthely, 1973; dedicated to P. Erdős on his 60th birthday), Vol. II. Infinite and Finite Sets (Colloq., Keszthely, 1973; dedicated to P. Erdős on his 60th birthday), Vol. II, Colloquia Mathematica Societatis János Bolyai, vol. 10 (1975), North-Holland: North-Holland Amsterdam), 609-627 · Zbl 0315.05117 [5] Ford, L. R.; Fulkerson, D. R., Flows in Networks (1962), Princeton University Press: Princeton University Press Princeton, NJ · Zbl 0106.34802 [6] Liggett, T. M.; Schonmann, R. H.; Stacey, A. M., Domination by product measures, Ann. Probab., 25, 71-95 (1997) · Zbl 0882.60046 [7] Lyons, R., Random walks and percolation on trees, Ann. Probab., 18, 931-958 (1990) · Zbl 0714.60089 [8] Lyons, R., Random walks, capacity and percolation on trees, Ann. Probab., 20, 2043-2088 (1992) · Zbl 0766.60091 [9] R. Lyons, Y. Peres, Probability on trees and networks, Cambridge University Press, 2011, in preparation and available online http://mypage.iu.edu/ rdlyons/prbtree/prbtree.html; R. Lyons, Y. Peres, Probability on trees and networks, Cambridge University Press, 2011, in preparation and available online http://mypage.iu.edu/ rdlyons/prbtree/prbtree.html · Zbl 1376.05002 [10] Scott, A. D.; Sokal, A. D., The repulsive lattice gas, the independent-set polynomial, and the Lovász local lemma, J. Stat. Phys., 118, 1151-1261 (2005) · Zbl 1107.82013 [11] Shearer, J. B., On a problem of spencer, Combinatorica, 5, 241-245 (1985) · Zbl 0587.60012 [12] C. Temmel, \(K\); C. Temmel, \(K\) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.