Generalized approach to the non-backtracking lace expansion

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The lace expansion is a powerful perturbative technique to analyze the critical behavior of random spatial processes such as the self-avoiding walk, percolation and lattice trees and animals. The non-backtracking lace expansion (NoBLE) is a modification that allows us to improve its applicability in the nearest-neighbor setting on the Z d-lattice for percolation, lattice trees and lattice animals. The NoBLE gives rise to a recursive formula that we study in this paper at a general level. We state assumptions that guarantee that the solution of this recursive formula satisfies the infrared bound. In two related papers, we show that these conditions are satisfied for percolation in d≥ 11 , for lattice trees in d≥ 16 and for lattice animals in d≥ 18.

Original languageEnglish
Pages (from-to)1041–1119
Number of pages79
JournalProbability Theory and Related Fields
Issue number3-4
Publication statusPublished - 1 Dec 2017


  • Infrared bound
  • Lace expansion
  • Lattice animals
  • Lattice trees
  • Nearest-neighbor models
  • Percolation


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