scholarly journals Percolation results for the continuum random cluster model

2018 ◽  
Vol 50 (01) ◽  
pp. 231-244 ◽  
Author(s):  
Pierre Houdebert
Keyword(s):  
Phase Transition ◽  
Cluster Model ◽  
Boolean Model ◽  
Connected Components ◽  
Random Structure ◽  
Stochastic Domination ◽  
Random Cluster Model ◽  
Random Cluster ◽  
Fixed Parameter ◽  
The Continuum

Abstract The continuum random cluster model is a Gibbs modification of the standard Boolean model with intensity z > 0 and law of radii Q. The formal unnormalised density is given by q N cc , where q is a fixed parameter and N cc is the number of connected components in the random structure. We prove for a large class of parameters that percolation occurs for large enough z and does not occur for small enough z. We provide an application to the phase transition of the Widom–Rowlinson model with random radii. Our main tools are stochastic domination properties, a detailed study of the interaction of the model, and a Fortuin–Kasteleyn representation.

scholarly journals Random-Cluster Dynamics on Random Regular Graphs in Tree Uniqueness

2021 ◽  
Author(s):  
Antonio Blanca ◽  
Reza Gheissari
Keyword(s):  
Phase Transition ◽  
Potts Model ◽  
Cluster Model ◽  
Iterative Scheme ◽  
Mixing Time ◽  
Glauber Dynamics ◽  
Regular Graphs ◽  
Random Cluster Model ◽  
Regular Tree ◽  
Random Cluster

AbstractWe establish rapid mixing of the random-cluster Glauber dynamics on random $$\varDelta $$ Δ -regular graphs for all $$q\ge 1$$ q ≥ 1 and $$p<p_u(q,\varDelta )$$ p < p u ( q , Δ ) , where the threshold $$p_u(q,\varDelta )$$ p u ( q , Δ ) corresponds to a uniqueness/non-uniqueness phase transition for the random-cluster model on the (infinite) $$\varDelta $$ Δ -regular tree. It is expected that this threshold is sharp, and for $$q>2$$ q > 2 the Glauber dynamics on random $$\varDelta $$ Δ -regular graphs undergoes an exponential slowdown at $$p_u(q,\varDelta )$$ p u ( q , Δ ) . More precisely, we show that for every $$q\ge 1$$ q ≥ 1 , $$\varDelta \ge 3$$ Δ ≥ 3 , and $$p<p_u(q,\varDelta )$$ p < p u ( q , Δ ) , with probability $$1-o(1)$$ 1 - o ( 1 ) over the choice of a random $$\varDelta $$ Δ -regular graph on n vertices, the Glauber dynamics for the random-cluster model has $$\varTheta (n \log n)$$ Θ ( n log n ) mixing time. As a corollary, we deduce fast mixing of the Swendsen–Wang dynamics for the Potts model on random $$\varDelta $$ Δ -regular graphs for every $$q\ge 2$$ q ≥ 2 , in the tree uniqueness region. Our proof relies on a sharp bound on the “shattering time”, i.e., the number of steps required to break up any configuration into $$O(\log n)$$ O ( log n ) sized clusters. This is established by analyzing a delicate and novel iterative scheme to simultaneously reveal the underlying random graph with clusters of the Glauber dynamics configuration on it, at a given time.

Numerical analysis of Pt utilization in PEMFC catalyst layer using random cluster model

2006 ◽  
Vol 51 (15) ◽  
pp. 3091-3096 ◽  
Author(s):  
Z.D. Wei ◽  
H.B. Ran ◽  
X.A. Liu ◽  
Y. Liu ◽  
C.X. Sun ◽  
...  
Keyword(s):  
Numerical Analysis ◽  
Cluster Model ◽  
Catalyst Layer ◽  
Random Cluster Model ◽  
Random Cluster ◽  
Pt Utilization

scholarly journals Connectivity properties of the random-cluster model

2016 ◽  
Vol 681 ◽  
pp. 012014
Author(s):  
Martin Weigel ◽  
Eren Metin Elci ◽  
Nikolaos G. Fytas
Keyword(s):  
Cluster Model ◽  
Random Cluster Model ◽  
Random Cluster

Percolation of a random network by statistical physics method

2019 ◽  
Vol 30 (02n03) ◽  
pp. 1950009
Author(s):  
Hai Lin ◽  
Jingcheng Wang
Keyword(s):  
Ising Model ◽  
Cluster Model ◽  
Statistical Physics ◽  
Random Network ◽  
Critical Phenomenon ◽  
Analytical Framework ◽  
Critical Value ◽  
Giant Component ◽  
Random Cluster Model ◽  
Random Cluster

In this paper, we develop an analytical framework and analyze the percolation properties of a random network by introducing statistical physics method. To adequately apply the statistical physics method on the research of a random network, we establish an exact mapping relation between a random network and Ising model. Based on the mapping relation and random cluster model (RCM), we obtain the partition function of the random network and use it to compute the size of the giant component and the critical value of the present probability. We extend this approach to investigate the size of remaining giant component and the critical phenomenon in the random network which is under a certain random attack. Numerical simulations show that our approach is accurate and effective.

scholarly journals Potts q-color field theory and scaling random cluster model

2011 ◽  
Vol 852 (1) ◽  
pp. 149-173 ◽  
Author(s):  
Gesualdo Delfino ◽  
Jacopo Viti
Keyword(s):  
Field Theory ◽  
Cluster Model ◽  
Random Cluster Model ◽  
Random Cluster ◽  
Color Field

scholarly journals Single-cluster dynamics for the random-cluster model

2009 ◽  
Vol 80 (3) ◽  
Author(s):  
Youjin Deng ◽  
Xiaofeng Qian ◽  
Henk W. J. Blöte
Keyword(s):  
Cluster Model ◽  
Random Cluster Model ◽  
Random Cluster ◽  
Single Cluster
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