September  2008, 7(5): 1145-1178. doi: 10.3934/cpaa.2008.7.1145

Convergent expansions for random cluster model with $q>0$ on infinite graphs


Department of Mathematics, Universidade Federal de Minas Gerais, 30161-970 Belo Horizonte


Dipartimento di Matematica Universitá “Tor Vergata” di Roma, V.le della ricerca scientifica, 00100 Roma, Italy

Received  August 2007 Revised  April 2008 Published  June 2008

In this paper we extend our previous results on the connectivity functions and pressure of the Random Cluster Model in the highly subcritical phase and in the highly supercritical phase, originally proved only on the cubic lattice $\mathbb Z^d$, to a much wider class of infinite graphs. In particular, concerning the subcritical regime, we show that the connectivity functions are analytic and decay exponentially in any bounded degree graph. In the supercritical phase, we are able to prove the analyticity of finite connectivity functions in a smaller class of graphs, namely, bounded degree graphs with the so called minimal cut-set property and satisfying a (very mild) isoperimetric inequality. On the other hand we show that the large distances decay of finite connectivity in the supercritical regime can be polynomially slow depending on the topological structure of the graph. Analogous analyticity results are obtained for the pressure of the Random Cluster Model on an infinite graph, but with the further assumptions of amenability and quasi-transitivity of the graph.
Citation: A. Procacci, Benedetto Scoppola. Convergent expansions for random cluster model with $q>0$ on infinite graphs. Communications on Pure and Applied Analysis, 2008, 7 (5) : 1145-1178. doi: 10.3934/cpaa.2008.7.1145

Gerhard Keller, Carlangelo Liverani. Coupled map lattices without cluster expansion. Discrete and Continuous Dynamical Systems, 2004, 11 (2&3) : 325-335. doi: 10.3934/dcds.2004.11.325


Takashi Hara and Gordon Slade. The incipient infinite cluster in high-dimensional percolation. Electronic Research Announcements, 1998, 4: 48-55.


Peigen Cao, Fang Li, Siyang Liu, Jie Pan. A conjecture on cluster automorphisms of cluster algebras. Electronic Research Archive, 2019, 27: 1-6. doi: 10.3934/era.2019006


Inês Cruz, M. Esmeralda Sousa-Dias. Reduction of cluster iteration maps. Journal of Geometric Mechanics, 2014, 6 (3) : 297-318. doi: 10.3934/jgm.2014.6.297


Valentin Ovsienko, MichaeL Shapiro. Cluster algebras with Grassmann variables. Electronic Research Announcements, 2019, 26: 1-15. doi: 10.3934/era.2019.26.001


Octav Cornea and Francois Lalonde. Cluster homology: An overview of the construction and results. Electronic Research Announcements, 2006, 12: 1-12.


Shuping Li, Zhen Jin. Impacts of cluster on network topology structure and epidemic spreading. Discrete and Continuous Dynamical Systems - B, 2017, 22 (10) : 3749-3770. doi: 10.3934/dcdsb.2017187


Xiwei Liu, Tianping Chen, Wenlian Lu. Cluster synchronization for linearly coupled complex networks. Journal of Industrial and Management Optimization, 2011, 7 (1) : 87-101. doi: 10.3934/jimo.2011.7.87


David E. Bernholdt, Mark R. Cianciosa, Clement Etienam, David L. Green, Kody J. H. Law, Jin M. Park. Corrigendum to "Cluster, classify, regress: A general method for learning discontinuous functions [1]". Foundations of Data Science, 2020, 2 (1) : 81-81. doi: 10.3934/fods.2020005


Michael Gekhtman, Michael Shapiro, Serge Tabachnikov, Alek Vainshtein. Higher pentagram maps, weighted directed networks, and cluster dynamics. Electronic Research Announcements, 2012, 19: 1-17. doi: 10.3934/era.2012.19.1


David E. Bernholdt, Mark R. Cianciosa, David L. Green, Jin M. Park, Kody J. H. Law, Clement Etienam. Cluster, classify, regress: A general method for learning discontinuous functions. Foundations of Data Science, 2019, 1 (4) : 491-506. doi: 10.3934/fods.2019020


Tianhu Yu, Jinde Cao, Chuangxia Huang. Finite-time cluster synchronization of coupled dynamical systems with impulsive effects. Discrete and Continuous Dynamical Systems - B, 2021, 26 (7) : 3595-3620. doi: 10.3934/dcdsb.2020248


Fang Li, Jie Pan. On inner Poisson structures of a quantum cluster algebra without coefficients. Electronic Research Archive, 2021, 29 (5) : 2959-2972. doi: 10.3934/era.2021021


Thomas Zaslavsky. Quasigroup associativity and biased expansion graphs. Electronic Research Announcements, 2006, 12: 13-18.


Gillala Rekha, V Krishna Reddy, Amit Kumar Tyagi. A novel approach for solving skewed classification problem using cluster based ensemble method. Mathematical Foundations of Computing, 2020, 3 (1) : 1-9. doi: 10.3934/mfc.2020001


Feyza Gürbüz, Panos M. Pardalos. A decision making process application for the slurry production in ceramics via fuzzy cluster and data mining. Journal of Industrial and Management Optimization, 2012, 8 (2) : 285-297. doi: 10.3934/jimo.2012.8.285


Juan Cao, Fengli Ren, Dacheng Zhou. Asymptotic and finite-time cluster synchronization of neural networks via two different controllers. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022005


Li-Xin Zhang. A note on the cluster set of the law of the iterated logarithm under sub-linear expectations. Probability, Uncertainty and Quantitative Risk, , () : -. doi: 10.3934/puqr.2022006


Joachim von Below, José A. Lubary. Isospectral infinite graphs and networks and infinite eigenvalue multiplicities. Networks and Heterogeneous Media, 2009, 4 (3) : 453-468. doi: 10.3934/nhm.2009.4.453


Robert Carlson. Dirichlet to Neumann maps for infinite quantum graphs. Networks and Heterogeneous Media, 2012, 7 (3) : 483-501. doi: 10.3934/nhm.2012.7.483

2021 Impact Factor: 1.273


  • PDF downloads (97)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]