Left and right convergence of graphs with bounded degree

Random Structures and Algorithms

Volumn 42, Issue 1, 2013, Pages 1-28

  Borgs, Christian ^a   Chayes, Jennifer ^a,b,c,d   Kahn, Jeff ^e   Lovász, László ^a,f  

^a MICROSOFT RESEARCH   (United States)

^b UNIVERSITY OF CAMBRIDGE   (United Kingdom)

^c UNIVERSITY OF MEMPHIS   (United States)

^d UNIVERSITY OF OXFORD   (United Kingdom)

^e RUTGERS UNIVERSITY   (United States)

^f EÖTVÖS LORÁND UNIVERSITY   (Hungary)

Author keywords

Cluster expansion; Dobrushin Uniqueness; Graph limits; Left convergence; Right convergence

Indexed keywords

EID: 84869877581     PISSN: 10429832     EISSN: 10982418     Source Type: Journal    
DOI: 10.1002/rsa.20414     Document Type: Article

References (24)

1. M. Abért, T. Hubai, Benjamini-Schramm convergence and the distribution of chromatic roots for sparse graphs. Available at:http://arxiv.org/abs/1201.3861.
2. A. Bandyopadhyay, D. Gamarnik, Counting without sampling. Asymptotics of the log-partition function for certain statistical physics models, Rand Struct Alg 33 ( 2008), 452-479.
3. I. Benjamini, O. Schramm, Recurrence of distributional limits of finite planar graphs, Electron J Probab 6 ( 2001), 1-13, paper no. 23.
4. I. Benjamini, O. Schramm, A. Shapira, Every minor-closed property of sparse graphs is testable, In 40th Ann. ACM Symp. Th. Comp., ACM, 2008, pp. 393-402.
5. A. Bertoni, P. Campadelli, R. Posenato, An upper bound for the maximum cut mean value, In Graph-theoretic concepts in computer science, Lect Notes Comput Sci, Vol. 1335, Springer, Berlin, 1997, pp. 78-84.
6. B. Bollobas, O. Riordan, Sparse graphs: Metrics and random models, Rand Struct Alg 39 ( 2011), 1-38.
7. C. Borgs, Absence of zeros for the chromatic polynomial on bounded degree graphs, Combin Probab Comput 15 ( 2006), 63-74.
8. C. Borgs, J. T. Chayes, L. Lovász, V. T. Sós, K. Vesztergombi, Convergent graph sequences I: Subgraph frequencies, metric properties, and testing, Adv Math 219 ( 2008), 1801-1851.
9. C. Borgs, J. T. Chayes, L. Lovász, V. T. Sós, K. Vesztergombi, Convergent graph sequences II: Multiway cuts and statistical physics, Annals of Math (to appear).
10. R. L. Dobrushin, Estimates of semi-invariants for the Ising model at low temperatures, In Topics in statistical and theoretical physics, Vol. 177 of American Mathematical Society translations, Series 2, American Mathematical Society, Providence, RI, 1996, pp. 59-81.
11. G. Elek, On limits of finite graphs, Combinatorica 27 ( 2007), 503-507.
12. P. Erds, L. Lovász, J. Spencer, Strong independence of graphcopy functions, In Graph theory and related topics, Academic Press, New York-London, 1979, pp. 165-172.
13. H.-O. Georgii, Gibbs measures and phase transitions, de Gruyter, Berlin, 1988.
14. J. Jonasson, Uniqueness of uniform random colorings of regular trees, Stat Probab Lett 57 ( 2002), 243-248.
15. D. Levin, Y. Peres, E. Wilmer, Markov chains and mixing times, AMS, Providence, 2008.
16. L. Lovász, Very large graphs, Current developments in mathematics 2008, D. Jerison, B. Mazur, T. Mrowka, W. Schmid, R. Stanley, S. T. Yau (Editors), International Press, Somerville, MA, 2009, pp. 67-128.
17. R. Lyons, Asymptotic enumeration of spanning trees, Combin Probab Comput 14 ( 2005), 491-522.
18. V. A. Malyshev, Uniform cluster estimates for lattice models, Commun Math Phys 64 ( 1979), 131-157.
19. M. Mezard, G. Parisi, The cavity method at zero temperature, J Stat Phys 111 ( 2003), 1-34.
20. G.-C. Rota, On the foundations of combinatorial theory, I. Theory of Möbius functions, Z Wahrsch Verw Gebiete 2 ( 1964), 340-368.
21. O. Schramm, Hyperfinite graph limits, Elect Res Announce Math Sci 15 ( 2008), 17-23.
22. E. Seiler, Gauge theories as a problem of constructive field theory and statistical mechanics, Lect Notes Phys, Springer, Berlin, Heidelberg, New York, 1982.
23. R. Stanley, Enumerative combinatorics, Vol. 2, Cambridge University Press, Cambridge, 1999.
24. D. Weitz, Counting independent sets up to the tree threshold, In Proc 38th Ann ACM Symp Theory Comput, ACM, New York, 2006, pp. 140-149.