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The critical points of lattice trees and lattice animals in high dimensions Mejía Miranda, Yuri

Abstract

We study lattice trees and lattice animals in high dimensions. Lattice trees and animals are interesting combinatorial objects used to model branched polymers in polymer science. They are also of interest in combinatorics and in the study of critical phenomena in statistical physics. [Abstract portion beginning here modified and differs from the print copy]. Our study takes place in the nearest-neighbor and spread-out models on the d-dimensional integer lattice. On either graph, a lattice animal is a finite connected subgraph, and a lattice tree is an animal without cycles.   Let t_n and a_n be the number of lattice trees and animals with n bonds that contain the origin, respectively. Standard subadditivity arguments provide the existence of the  growth constants τ and α, which are the limit, as the dimension d goes to infinity,  of the n-th root of t_n and a_n, respectively.] We are interested in the critical points of these models, which are the reciprocals of the corresponding growth constants. We rigorously calculate the first three terms of a 1/d-expansion for the critical points of nearest-neighbor lattice trees and animals. The proof follows a recursive argument similar to the one used by Hara and Slade (1995), van der Hofstad and Slade (2006), to obtain analogous results for the critical points of self-avoiding walks and percolation.   To provide the leading terms in the expansions, we use a mean-field model, related to the Galton-Watson branching process with critical Poisson offspring distribution, and results obtained with the lace expansion. The leading terms are also calculated in the spread-out model. Then we develop expansions for the nearest-neighbor generating functions and, together with the lace expansion, obtain the first and second correction terms. Our result gives a rigorous proof for previous work on the subject [11],[21, 36]. Given the algorithmic nature of the proof, it can be extended, with sufficient labor, to compute higher degree terms. It may provide the starting point for proving the existence of an asymptotic expansion with rational coefficients, for the critical point of nearest-neighbor lattice trees.

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