On the upper critical dimension of lattice trees and lattice animals

Takashi Hara, Gordon Slade

Research output: Contribution to journalArticlepeer-review

53 Citations (Scopus)


We give a rigorous proof of mean-field critical behavior for the susceptibility (γ=1/2) and the correlation length (v=1/4) for models of lattice trees and lattice animals in two cases: (i) for the usual model with trees or animals constructed from nearest-neighbor bonds, in sufficiently high dimensions, and (ii) for a class of "spread-out" or long-range models in which trees and animals are constructed from bonds of various lengths, above eight dimensions. This provides further evidence that for these models the upper critical dimension is equal to eight. The proof involves obtaining an infrared bound and showing that a certain "square diagram" is finite at the critical point, and uses an expansion related to the lace expansion for the self-avoiding walk.

Original languageEnglish
Pages (from-to)1469-1510
Number of pages42
JournalJournal of Statistical Physics
Issue number5-6
Publication statusPublished - Jun 1990
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Mathematical Physics


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