Finite size scaling for the core of large random hypergraphs

Annals of Applied Probability

Volumn 18, Issue 5, 2008, Pages 1993-2040

  Dembo, Amir ^a   Montanari, Andrea ^a  

^a STANFORD UNIVERSITY   (United States)

Author keywords

Core; Finite size scaling; Low density parity check codes; Random graph; Random hypergraph; XOR SAT

Indexed keywords

EID: 61349102591     PISSN: 10505164     EISSN: 10505164     Source Type: Journal    
DOI: 10.1214/07-AAP514     Document Type: Article

References (35)

1. ALDOUS, D. (1997). Brownian excursions, critical random graphs and the multiplicative coalescent. Ann. Probab. 25 812-854. MR1434128
2. ARONSON, J., FRIEZE, A. and PITTEL, B. G. (1998). Maximum matchings in sparse random graphs: Karp-Sipser revisited. Random Structures Algorithms 12 111-177. MR1637403
3. AMRAOUI, A., MONTANARI, A., RICHARDSON, T. and URBANKE, R. (2005). Finite-length scaling for iteratively decoded LDPC ensembles. IEEE Trans. Inform. Theory. Available at http://arxiv.org/ abs/cs.IT/0406050.
4. AMRAOUI, A., MONTANARI, A. and URBANKE, R. (2007). How to find good finite-length codes: From art towards science. European Transactions on Telecommunications 18 491-508.
5. ACHLIOPTAS, D. and RICCI-TERSENGHI, F. (2006). On the solution-space geometry of random constraint satisfaction problems. In STOC'06: Proceedings of the 38th Annual ACM Symposium on Theory of Computing 130-139. ACM, New York. MR2277138
6. ABRAMOWITZ, M. and STEGUN, I. A. (1964). Handbook of Mathematical Functions. National Bureau of Standards, Washington, DC.
7. BHATTACHARYA, R. N. and RANGA RAO, R. (1976). Normal Approximation and Asymptotic Expansions. Wiley, New York. MR0436272
8. BOLLOBÁS, B. (1980). A probabilistic proof of an asymptotic formula for the number of labelled regular graphs. European J. Combin. 1 311-316. MR595929
9. BOLLOBÁS, B., BORGS, C., CHAYES, J. T., KIM, J. H. and WILSON, D. B. (2001). The scaling window of the 2-SAT transition. Random Structures Algorithms 18 201-256. MR1824274
10. CHAYES, J. T., CHAYES, L., FISHER, D. S. and SPENCER, T. (1986). Finite-size scaling and correlation lengths for disordered systems. Phys. Rev. Lett. 57 2999-3002. MR925751
11. CREIGNOU, N. and DAUDE, H. (1999). Satisfiability threshold for random XOR-CNF formulas. Discrete Appl. Math. 96/97 41-53. MR1724715
12. COCEO, S., DUBOIS, O., MANDLER, J. and MONASSON, R. (2003). Rigorous decimation-based construction of ground pure states for spin glass models on random lattices. Phys. Rev. Lett. 90 47205.
13. DARLING, R. W. R., LEVIN, D. A. and NORRIS, J. R. (2004). Continuous and discontinuous phase transitions in hypergraph processes. Random Structures Algorithms 24 397-419. MR2060628
14. DARLING, R. W. R. and NORRIS, J. R. (2005). Structure of large random hypergraphs. Ann. Appl. Probab. 15 125-152. MR2115039
15. DAUDÉ, H., MÉZARD, M., MORA, T. and ZECCHINA, R. (2008). Pairs of SAT assignments in random Boolean formulae. Theoret. Comput. Sci. 393 266-279.
16. DYER, M., FRIEZE, A. and PITTEL, B. (1993). The average performance of the greedy matching algorithm. Ann. Appl. Probab. 3 526-552. MR1221164
17. FRIEZE, A. and SUEN, S. (1996). Analysis of two simple heuristics on a random instance of Jt-SAT. J. Algorithms 20 312-355. MR1379227
18. GOLDSCHMIDT, C. and NORRIS, J. (2004). Essential edges in Poisson random hypergraphs. Random Structures Algorithms 24 381-396. MR2060627
19. GOLDSCHMIDT, C. (2005). Critical random hypergraphs: The emergence of a giant set of identifiable vertices. Ann. Probab. 33 1573-1600. MR2150199
20. GROENEBOOM, P. (1989). Brownian motion with a parabolic drift and Airy functions. Probab. Theory Related Fields 81 79-109. MR981568
21. HALL, P. (1982). Rates of Convergence in the Central Limit Theorem. Research Notes in Mathematics 62. Pitman, Boston. MR668197
22. JANSON, S., KNUTH, D. E., LUCZAK, T. and PITTEL, B. (1993). The birth of the giant component. Random Structures Algorithms 4 231-358. MR1220220
23. KOMLÓS, J., MAJOR, P. and TUSNÁ DY, G. (1975). An approximation of partial sums of independent RVs and the sample DF. I. Z. Wahrsch. Verw. Gebiete 32 111-131. MR0375412
24. KARP, R. M. and SIPSER, M. (1981). Maximum matching in sparse random graphs. In Proc. 22nd Annual IEEE Symposium on Foundations of Computing 364-375. IEEE, New York.
25. KURTZ, T. G. (1970). Solutions of ordinary differential equations as limits of pure jump Markov processes. J. Appl. Probab. 149-58. MR0254917
26. LUBY, M. G., MITZENMACHER, M., SHOKROLLAHI, M. A. and SPIELMAN, D. A. (2001). Efficient erasure correcting codes. IEEE Trans. Inform. Theory 47 569-584. MR1820477
27. 1/2). Combinatorica 11 369-382. MR1137769
28. MOLLOY, M. (2005). Cores in random hypergraphs and Boolean formulas. Random Structures Algorithms 27 124-135. MR2150018
29. MEZARD, M., RICCI-TERSENGHI, F. and ZECCHINA, R. (2003). Two solutions to diluted p-spin models and XORSAT problems. J. Statist. Phys. 111 505-533. MR1972120
30. ORLITSKY, A., VISWANATHAN, K. and ZHANG, J. (2005). Stopping set distribution of LDPC code ensembles. IEEE Trans. Inform. Theory 51 929-953. MR2237961
31. PITTEL, B., SPENCER, J. and WORMALD, N. (1996). Sudden emergence of a giant k-core in a random graph. J. Combin. Theory Ser. B 67 111-151. MR1385386
32. SAKHANENKO, A. I. (1985). Rate of convergence in the invariance principle for variables with exponential moments that are not identically distributed. In Advances in Probability: Limit Theorems for Sums of Random Variables (A. A. Borovkov and A. Balakrishnan, eds.) 2-73. Optimization Software.
33. SELLITTO, M., BIROLI, G. and TONINELLI, C. (2005). Facilitated spin models on Bethe lattice: Bootstrap percolation, mode-coupling transition and glassy dynamics. Europhys. Lett. 69 496-501.
34. SHAO, Q. M. (1995). Strong approximation theorems for independent random variables and their applications. J. Multivariate Anal. 52 107-130. MR1325373
35. WILSON, D. B. (2002). On the critical exponents of random k-SAT. Random Structures Algorithms 21182-195. MR1917357