Upper bound on the decay of correlations in the plane rotator model with long-range random interaction

1984 ◽  
Vol 36 (3-4) ◽  
pp. 489-516 ◽  
Author(s):  
P. Picco
Keyword(s):  
Long Range ◽  
Upper Bound ◽  
Decay Of Correlations ◽  
Rotator Model ◽  
Random Interaction ◽  
Plane Rotator

scholarly journals Power-law bounds for critical long-range percolation below the upper-critical dimension

2021 ◽  
Author(s):  
Tom Hutchcroft
Keyword(s):  
Long Range ◽  
Power Law ◽  
Upper Bound ◽  
Mean Field ◽  
Maximum Size ◽  
Finite Graph ◽  
Critical Dimension ◽  
Point Function ◽  
Upper Critical Dimension ◽  
Infinite Cluster

AbstractWe study long-range Bernoulli percolation on $${\mathbb {Z}}^d$$ Z d in which each two vertices x and y are connected by an edge with probability $$1-\exp (-\beta \Vert x-y\Vert ^{-d-\alpha })$$ 1 - exp ( - β ‖ x - y ‖ - d - α ) . It is a theorem of Noam Berger (Commun. Math. Phys., 2002) that if $$0<\alpha <d$$ 0 < α < d then there is no infinite cluster at the critical parameter $$\beta _c$$ β c . We give a new, quantitative proof of this theorem establishing the power-law upper bound $$\begin{aligned} {\mathbf {P}}_{\beta _c}\bigl (|K|\ge n\bigr ) \le C n^{-(d-\alpha )/(2d+\alpha )} \end{aligned}$$ P β c ( | K | ≥ n ) ≤ C n - ( d - α ) / ( 2 d + α ) for every $$n\ge 1$$ n ≥ 1 , where K is the cluster of the origin. We believe that this is the first rigorous power-law upper bound for a Bernoulli percolation model that is neither planar nor expected to exhibit mean-field critical behaviour. As part of the proof, we establish a universal inequality implying that the maximum size of a cluster in percolation on any finite graph is of the same order as its mean with high probability. We apply this inequality to derive a new rigorous hyperscaling inequality $$(2-\eta )(\delta +1)\le d(\delta -1)$$ ( 2 - η ) ( δ + 1 ) ≤ d ( δ - 1 ) relating the cluster-volume exponent $$\delta $$ δ and two-point function exponent $$\eta $$ η .

Dynamical Plane Rotator Model. III

1982 ◽  
Vol 51 (2) ◽  
pp. 390-396 ◽  
Author(s):  
Minoru Takahashi
Keyword(s):  
Rotator Model ◽  
Dynamical Plane ◽  
Plane Rotator

scholarly journals Monte Carlo Simulation and Static and Dynamic Critical Behavior of the Plane Rotator Model

1978 ◽  
Vol 60 (6) ◽  
pp. 1669-1685 ◽  
Author(s):  
S. Miyashita ◽  
H. Nishimori ◽  
A. Kuroda ◽  
M. Suzuki
Keyword(s):  
Monte Carlo Simulation ◽  
Monte Carlo ◽  
Critical Behavior ◽  
Rotator Model ◽  
Plane Rotator
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