Mean-Field Behaviour and the Lace Expansion

1994 ◽  
pp. 87-122 ◽  
Author(s):  
Takashi Hara ◽  
Gordon Slade
Keyword(s):  
Mean Field ◽  
Lace Expansion ◽  
Field Behaviour

Structural studies of nematic and smectic phases

1983 ◽  
Vol 309 (1507) ◽  
pp. 145-153 ◽  
Keyword(s):  
Liquid Crystal ◽  
Mean Field ◽  
Structural Studies ◽  
X Ray ◽  
Smectic A ◽  
Smectic C ◽  
X Ray Scattering ◽  
Smectic Phases ◽  
Crystal Phases ◽  
Field Behaviour

The information about liquid crystal phases that can be obtained by light scattering and by high-resolution X-ray scattering is reviewed. Results for the nematic-smectic A transition suggest the de Gennes-McMillan model is correct, but adequate theoretical solutions to the model remain elusive. Recent results on the smectic A to smectic C transition are presented that show unambiguously that it exhibits classic mean-field behaviour and this is explained by a Ginzburg criterion argument. Some preliminary results of a study of a nematic-smectic A transition in a lyotropic material are given and indicate similarity to thermotropic materials.

Phase transitions in perovskites near the tricritical point: an experimental study of KMn1–xCaxF3 and SrTiO3

2000 ◽  
Vol 64 (6) ◽  
pp. 971-982 ◽  
Author(s):  
M. C. Gallardo ◽  
F. J. Romero ◽  
S. A. Hayward ◽  
E. K. H. Salje ◽  
J. del Cerro
Keyword(s):  
Experimental Data ◽  
Experimental Study ◽  
Phase Transitions ◽  
Order Parameter ◽  
Tricritical Point ◽  
Mean Field ◽  
Calorimetric Data ◽  
X Ray ◽  
First Order ◽  
Field Behaviour

AbstractWe present experimental data for the Pm3m-I4/mcm phase transitions in the perovskite crystals KMn1-xCaxF3 and SrTiO3. Comparison of calorimetric data (latent heat and specific heat) with order parameter data (measured with X-ray rocking methods) indicates that these transitions follow mean-field behaviour, and may be described using Landau potentials where the free energy expansion includes terms up to Q6. This potential is characteristic of transitions close to the tricritical point. Comparison of the behaviour of SrTiO3 and KMnF3 indicates that KMnF3 is closer to the tricritical point; a small amount of substitution of Ca for Mn causes the transition to cross the tricritical point from first order to second order behaviour.

scholarly journals Mean-Field Behaviour in a Local Low-Dimensional Model

1995 ◽  
Vol 30 (6) ◽  
pp. 319-324 ◽  
Author(s):  
H.-M Bröker ◽  
P Grassberger
Keyword(s):  
Mean Field ◽  
Dimensional Model ◽  
Low Dimensional ◽  
Field Behaviour

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 Individual-based modelling and control of bovine brucellosis

2018 ◽  
Vol 5 (5) ◽  
pp. 180200 ◽  
Author(s):  
Erivelton G. Nepomuceno ◽  
Alípio M. Barbosa ◽  
Marcos X. Silva ◽  
Matjaž Perc
Keyword(s):  
Compartmental Model ◽  
Mean Field ◽  
Bovine Brucellosis ◽  
Paulo State ◽  
Control Actions ◽  
The Mean ◽  
And Control ◽  
Population Effects ◽  
Field Behaviour ◽  
Good Agreement

We present a theoretical approach to control bovine brucellosis. We have used individual-based modelling, which is a network-type alternative to compartmental models. Our model thus considers heterogeneous populations, and spatial aspects such as migration among herds and control actions described as pulse interventions are also easily implemented. We show that individual-based modelling reproduces the mean field behaviour of an equivalent compartmental model. Details of this process, as well as flowcharts, are provided to facilitate the reproduction of the presented results. We further investigate three numerical examples using real parameters of herds in the São Paulo state of Brazil, in scenarios which explore eradication, continuous and pulsed vaccination and meta-population effects. The obtained results are in good agreement with the expected behaviour of this disease, which ultimately showcases the effectiveness of our theory.

On the Crossover from Ising to Mean-Field Behaviour in Compatible Binary-Polymer Blends

1993 ◽  
Vol 22 (8) ◽  
pp. 577-583 ◽  
Author(s):  
G Meier ◽  
D Schwahn ◽  
K Mortensen ◽  
S Janssen
Keyword(s):  
Polymer Blends ◽  
Mean Field ◽  
Binary Polymer ◽  
Field Behaviour
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