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Convergence of lattice trees to super-brownian motion above the critical dimension Holmes, Mark


A lattice tree is a finite connected set of lattice bonds containing no cycles. Lattice trees are interesting combinatorial objects and an important model for branched polymers in polymer chemistry and physics. In addition they provide an interesting example of critical phenomena in statistical physics with similar properties to models such as self-avoiding walks and percolation. We use the lace expansion to prove asymptotic formulae for the Fourier transform of the r-point functions (quantities which count critically weighted trees containing r fixed points) for a spread-out model of lattice" trees in ℤ[sup d] for d > 8. Our results therefore provide additional evidence in support of the critical dimension d[sub c] = 8. The spread out model allows bonds between vertices x,y ∈ ℤ[sup d] with; x-y; ∞ ≤ L, providing a small parameter L [sup –d/2] needed for convergence of the lace expansion. We extend the inductive approach (to the lace expansion on an interval) of van der Hofstad and Slade [19] to prove convergence of the Fourier transform of the 2-point function (r = 2). We then proceed by induction on r, equipped with the lace expansion on a tree [21]. The asymptotic formulae for the r-point functions imply convergence of certain expectations of the spread out lattice trees model formulated as a measure valued process, to those of the canonical measure of super-Brownian motion. Appealing to the hypothesis of universality, we expect that the results also hold for the nearest neighbour model. Our results together with the convergence of the survival probability would imply convergence of the finite-dimensional distributions of our process to those of the canonical measure of super-Brownian motion. Convergence of the survival probability remains an open problem.

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