Combinatorial approach to the interpolation method and scaling limits in sparse random graphs

Annals of Probability

Volumn 41, Issue 6, 2013, Pages 4080-4115

  Bayati, Mohsen ^a   Gamarnik, David ^b   Tetali, Prasad ^c  

^a STANFORD UNIVERSITY   (United States)

^b MASSACHUSETTS INSTITUTE OF TECHNOLOGY   (United States)

^c Georgia Institute of Technology   (United States)

Author keywords

Constraint satisfaction problems; Partition function; Random graphs

Indexed keywords

EID: 84888409119     PISSN: 00911798     EISSN: None     Source Type: Journal    
DOI: 10.1214/12-AOP816     Document Type: Article

References (27)

1. ALDOUS, D. Open problems. Preprint. Available at: http://www.stat.berkeley.edu/̃Aldous/Research/OP/sparse_graph.html.
2. ALDOUS, D. Some open problems. Preprint. Available at: http://stat-www.berkeley.edu/users/aldous/Research/problems.ps.
3. ALDOUS, D. and STEELE, J. M. (2004). The objective method: Probabilistic combinatorial optimization and local weak convergence. In Probability on Discrete Structures (H. Kesten, ed.) 1-72. Springer, Berlin.
4. ALON, N. and SPENCER, J. H. (1992). The Probabilistic Method. Wiley, New York. MR1140703
5. BOLLOBÁS, B. (1980). A probabilistic proof of an asymptotic formula for the number of labelled regular graphs. European J. Combin. 1 311-316. MR0595929
6. BOLLOBÁS, B. (2001). Random Graphs. Cambridge Univ. Press, Cambridge.
7. BOLLOBÁS, B. and RIORDAN, O. (2006). Percolation. Cambridge Univ. Press, New York. MR2283880
8. BOLLOBÁS, B. and RIORDAN, O. (2011). Sparse graphs: Metrics and random models. Random Structures Algorithms 39 1-38. MR2839983
9. COJA-OGHLAN, A. (2012). Personal communication.
10. COPPERSMITH, D., GAMARNIK, D., HAJIAGHAYI, M. T. and SORKIN, G. B. (2004). Random MAX SAT, random MAX CUT, and their phase transitions. Random Structures Algorithms 24 502-545. MR2060633
11. DE BRUIJN, N. G. and ERDÖS, P. (1951). Some linear and some quadratic recursion formulas. I. Indag. Math. (N.S.) 13 374-382. MR0047161
12. DE BRUIJN, N. G. and ERDÖS, P. (1952). Some linear and some quadratic recursion formulas. II. Indag. Math. (N.S.) 14 152-163. MR0047162
13. DEMBO, A. andMONTANARI, A. (2010). Ising models on locally tree-like graphs. Ann. Appl. Probab. 20 565-592.
14. DEMBO, A. and MONTANARI, A. (2010). Gibbs measures and phase transitions on sparse random graphs. Braz. J. Probab. Stat. 24 137-211. MR2643563
15. DEMBO, A., MONTANARI, A. and SUN, N. (2011). Factor models on locally tree-like graphs. Available at arXiv:1110.4821.
16. FRANZ, S. and LEONE, M. (2003). Replica bounds for optimization problems and diluted spin systems. J. Stat. Phys. 111 535-564. MR1972121
17. FRANZ, S. and LEONE, M. (2003). Replica bounds for optimization problems and diluted spin systems. J. Phys. A Math. Gen. 36 10967-10985.
18. FRANZ, S. andMONTANARI, A. (2009). Personal communication.
19. FRIEDGUT, E. (1999). Sharp thresholds of graph properties, and the k-sat problem. J. Amer. Math. Soc. 12 1017-1054. With an appendix by Jean Bourgain. MR1678031
20. GALLAGER, R. G. (1963). Low-Density Parity-Check Codes. MIT Press, Cambridge, MA.
21. GAMARNIK, D. (2004). Linear phase transition in random linear constraint satisfaction problems. Probab. Theory Related Fields 129 410-440. MR2128240
22. GAMARNIK, D., NOWICKI, T. and SWIRSZCZ, G. (2006). Maximum weight independent sets and matchings in sparse random graphs. Exact results using the local weak convergence method. Random Structures Algorithms 28 76-106. MR2187483
23. GUERRA, F. and TONINELLI, F. L. (2002). The thermodynamic limit in mean field spin glass models. Comm. Math. Phys. 230 71-79. MR1930572
24. JANSON, S., LUCZAK, T. and RUCINSKI, A. (2000). Random Graphs. Wiley, New York. MR1782847
25. JANSON, S. and THOMASON, A. (2008). Dismantling sparse random graphs. Combin. Probab. Comput. 17 259-264. MR2396351
26. MONTANARI, A. (2005). Tight bounds for LDPC and LDGM codes under MAP decoding. IEEE Trans. Inform. Theory 51 3221-3246. MR2239148
27. PANCHENKO, D. and TALAGRAND, M. (2004). Bounds for diluted mean-fields spin glass models. Probab. Theory Related Fields 130 319-336. MR2095932