NoBLE for Lattice Trees and Lattice Animals

Abstract We study lattice trees (LTs) and animals (LAs) on the nearest-neighbor $${\mathbb {Z}}^d$$ Z d in high dimensions. prove that LTs LAs display mean-field behavior above dimension $$16$$ 16 $$17$$ 17 , respectively. Such results have previously been obtained by Hara Slade sufficiently The which their apply was not yet specified. rely non-backtracking lace expansion (NoBLE) method we recently developed. NoBLE makes use of an alternative for perturbs around random walk rather than simple walk, leading to smaller corrections. then provides a careful computational analysis improves result applies. Universality arguments predict upper critical dimension, our apply, is equal $$d_c=8$$ c = 8 both models, as known spread-out models mentioned earlier. main ingredients this paper are (a) derivation LT LA two-point functions; (b) bounds lace-expansion coefficients, thus showing general methodology can be applied; (c) sharp numerical coefficients. Our proof complemented computer-assisted verifies necessary used satisfied.