scholarly journals Critical Site Percolation in High Dimension

2020 ◽  
Vol 181 (3) ◽  
pp. 816-853
Author(s):  
Markus Heydenreich ◽  
Kilian Matzke
Keyword(s):  
High Dimension ◽  
Critical Exponents ◽  
Mean Field ◽  
High Dimensions ◽  
Lace Expansion ◽  
Site Percolation ◽  
Triangle Condition ◽  
Hypercubic Lattice ◽  
Infra Red ◽  
Critical Site

Abstract We use the lace expansion to prove an infra-red bound for site percolation on the hypercubic lattice in high dimension. This implies the triangle condition and allows us to derive several critical exponents that characterize mean-field behavior in high dimensions.

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.

scholarly journals Expansion for the critical point of site percolation: the first three terms

2021 ◽  
pp. 1-25
Author(s):  
Markus Heydenreich ◽  
Kilian Matzke
Keyword(s):  
Critical Point ◽  
Lace Expansion ◽  
Site Percolation ◽  
Hypercubic Lattice ◽  
Powers Of 2

Abstract We expand the critical point for site percolation on the d-dimensional hypercubic lattice in terms of inverse powers of 2d, and we obtain the first three terms rigorously. This is achieved using the lace expansion.

Nonclassical critical exponents out of mean-field results

1998 ◽  
Vol 260 (1-2) ◽  
pp. 99-105 ◽  
Author(s):  
Tânia Tomé ◽  
Mário J.de Oliveira
Keyword(s):  
Critical Exponents ◽  
Mean Field

scholarly journals Erratum: Critical exponents in mean-field classical spin systems [Phys. Rev. E 100 , 032131 (2019)]

2020 ◽  
Vol 102 (3) ◽  
Author(s):  
Yoshiyuki Y. Yamaguchi ◽  
Debraj Das ◽  
Shamik Gupta
Keyword(s):  
Critical Exponents ◽  
Mean Field ◽  
Spin Systems ◽  
Classical Spin

The Curie–Weiss–Néel model of a ferrimagnet

1981 ◽  
Vol 59 (7) ◽  
pp. 883-887 ◽  
Author(s):  
R. G. Bowers ◽  
S. L. Schofield
Keyword(s):  
Exact Solution ◽  
Critical Phenomena ◽  
Critical Exponents ◽  
Mean Field ◽  
The Other ◽  
New Model ◽  
Total Magnetization ◽  
Present Context ◽  
Elementary Methods ◽  
Parallel Fashion

An analogue, for ferrimagnetism, of the Curie–Weiss model ferromagnet is introduced. The resulting structure, the Curie–Weiss–Néel model, is based on a two sublattice description in which spins of one magnitude occupy one sublattice and spins of another magnitude occupy the other. Attention is concentrated on the case in which spins on the different sublattices tend to align in an anti-parallel fashion. Many properties of the new model are similar to those of the Curie–Weiss ferromagnet. Artificially long-ranged interactions connect spins on the different sublattices. The complete thermodynamics can be obtained exactly by relatively elementary methods. The exact solution of the model is essentially identical with the appropriate mean field results (of Néel). Attention is given to the Néel point and associated critical phenomena. Many standard critical exponents are calculated and, of course, classical exponent values result. Novel features of critical phenomena in ferrimagnets are considered. These are associated with the fact that, theoretically, the staggered magnetization and staggered fields are important while, experimentally, the total magnetization and uniform fields are usually employed. It is shown that, within the present context, corresponding staggered and uniform properties have identical exponent values.

scholarly journals Criticality in elastoplastic models of amorphous solids with stress-dependent yielding rates

Soft Matter ◽  
2019 ◽  
Vol 15 (44) ◽  
pp. 9041-9055 ◽  
Author(s):  
E. E. Ferrero ◽  
E. A. Jagla
Keyword(s):  
Critical Exponents ◽  
Mean Field ◽  
Amorphous Solids ◽  
Stress Dependent ◽  
Local Yielding

Elastoplastic models are analyzed at the yielding transition. Universality and critical exponents are discussed. The flowcurve exponent happens to be sensitive to the local yielding rule. An alternative mean-field description of yielding is explained.

Critical behavior of percolation and Markov fields on branching planes

1993 ◽  
Vol 30 (3) ◽  
pp. 538-547 ◽  
Author(s):  
C. Chris Wu
Keyword(s):  
Continuous Function ◽  
Critical Behavior ◽  
Mean Field ◽  
Percolation Model ◽  
Homogeneous Tree ◽  
Dimensional Lattice ◽  
One Dimensional ◽  
Percolation Probability ◽  
Triangle Condition ◽  
Markov Fields

For an independent percolation model on, whereis a homogeneous tree andis a one-dimensional lattice, it is shown, by verifying that the triangle condition is satisfied, that the percolation probabilityθ(p) is a continuous function ofpat the critical pointpc, and the critical exponents,γ,δ, and Δ exist and take their mean-field values. Some analogous results for Markov fields onare also obtained.

Sign in / Sign up

Export Citation Format

Share Document