scholarly journals The growth constants of lattice trees and lattice animals in high dimensions

2011 ◽  
Vol 16 (0) ◽  
pp. 129-136 ◽  
Author(s):  
Yuri Mejia Miranda ◽  
Gordon Slade
Keyword(s):  
Lattice Animals ◽  
High Dimensions ◽  
Lattice Trees ◽  
Growth Constants

scholarly journals Expansion in High Dimension for the Growth Constants of Lattice Trees and Lattice Animals

2013 ◽  
Vol 22 (4) ◽  
pp. 527-565 ◽  
Author(s):  
YURI MEJÍA MIRANDA ◽  
GORDON SLADE
Keyword(s):  
Error Estimates ◽  
High Dimension ◽  
Integer Lattice ◽  
Nearest Neighbour ◽  
Lattice Animals ◽  
Lace Expansion ◽  
Lattice Trees ◽  
Growth Constants ◽  
The One ◽  
Rigorous Error

We compute the first three terms of the 1/d expansions for the growth constants and one-point functions of nearest-neighbour lattice trees and lattice (bond) animals on the integer lattice $\mathbb{Z}^d$, with rigorous error estimates. The proof uses the lace expansion, together with a new expansion for the one-point functions based on inclusion–exclusion.

On the upper critical dimension of lattice trees and lattice animals

1990 ◽  
Vol 59 (5-6) ◽  
pp. 1469-1510 ◽  
Author(s):  
Takashi Hara ◽  
Gordon Slade
Keyword(s):  
Critical Dimension ◽  
Lattice Animals ◽  
Upper Critical Dimension ◽  
Lattice Trees

Some properties of the numbers of lattice animals and lattice trees

1990 ◽  
Vol 5 (3) ◽  
pp. 307-308 ◽  
Author(s):  
Christine E. Soteros ◽  
Stuart G. Whittington
Keyword(s):  
Lattice Animals ◽  
Lattice Trees

scholarly journals Collapsing lattice animals and lattice trees in two dimensions

2005 ◽  
Vol 2005 (06) ◽  
pp. P06003 ◽  
Author(s):  
Hsiao-Ping Hsu ◽  
Peter Grassberger
Keyword(s):  
Two Dimensions ◽  
Lattice Animals ◽  
Lattice Trees

scholarly journals Counting lattice animals in high dimensions

2011 ◽  
Vol 2011 (09) ◽  
pp. P09026 ◽  
Author(s):  
Sebastian Luther ◽  
Stephan Mertens
Keyword(s):  
Lattice Animals ◽  
High Dimensions

scholarly journals Correction to: NoBLE for Lattice Trees and Lattice Animals

2022 ◽  
Vol 186 (2) ◽  
Author(s):  
Robert Fitzner ◽  
Remco van der Hofstad
Keyword(s):  
Lattice Animals ◽  
Lattice Trees

scholarly journals NoBLE for Lattice Trees and Lattice Animals

2021 ◽  
Vol 185 (2) ◽  
Author(s):  
Robert Fitzner ◽  
Remco van der Hofstad
Keyword(s):  
Random Walk ◽  
Nearest Neighbor ◽  
Mean Field ◽  
Critical Dimension ◽  
Simple Random Walk ◽  
Computer Assisted ◽  
High Dimensions ◽  
Lace Expansion ◽  
Upper Critical Dimension ◽  
Lattice Trees

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

Lattice trees and super-Brownian motion

1997 ◽  
Vol 40 (1) ◽  
pp. 19-38 ◽  
Author(s):  
Eric Derbez ◽  
Gordon Slade
Keyword(s):  
Brownian Motion ◽  
Mean Field Theory ◽  
Scaling Limit ◽  
Mean Field ◽  
High Dimensional ◽  
High Dimensions ◽  
Lace Expansion ◽  
Brownian Excursion ◽  
Lattice Trees ◽  
Super Brownian Motion

AbstractThis article discusses our recent proof that above eight dimensions the scaling limit of sufficiently spread-out lattice trees is the variant of super-Brownian motion calledintegrated super-Brownian excursion(ISE), as conjectured by Aldous. The same is true for nearest-neighbour lattice trees in sufficiently high dimensions. The proof, whose details will appear elsewhere, uses the lace expansion. Here, a related but simpler analysis is applied to show that the scaling limit of a mean-field theory is ISE, in all dimensions. A connection is drawn between ISE and certain generating functions and critical exponents, which may be useful for the study of high-dimensional percolation models at the critical point.

The Scaling Limit of Lattice Trees in High Dimensions

1998 ◽  
Vol 193 (1) ◽  
pp. 69-104 ◽  
Author(s):  
Eric Derbez ◽  
Gordon Slade
Keyword(s):  
Scaling Limit ◽  
High Dimensions ◽  
Lattice Trees
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