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The number of random 2-SAT solutions is asymptotically log-normal (2405.03302v2)
Published 6 May 2024 in cs.DM, math.CO, and math.PR
Abstract: We prove that throughout the satisfiable phase, the logarithm of the number of satisfying assignments of a random 2-SAT formula satisfies a central limit theorem. This implies that the log of the number of satisfying assignments exhibits fluctuations of order $\sqrt n$, with $n$ the number of variables. The formula for the variance can be evaluated effectively. By contrast, for numerous other random constraint satisfaction problems the typical fluctuations of the logarithm of the number of solutions are {\em bounded} throughout all or most of the satisfiable regime.
- E. Abbe, A. Montanari: On the concentration of the number of solutions of random satisfiability formulas. Random Structures and Algorithms 45 (2014) 362–382.
- D. Achlioptas, A. Coja-Oghlan: Algorithmic barriers from phase transitions. Proc. 49th FOCS (2008) 793–802.
- D. Achlioptas, C. Moore: Random k𝑘kitalic_k-SAT: two moments suffice to cross a sharp threshold. SIAM Journal on Computing 36 (2006) 740–762.
- D. Achlioptas, Y. Peres: The threshold for random k𝑘kitalic_k-SAT is 2kln2−O(k)superscript2𝑘2𝑂𝑘2^{k}\ln 2-O(k)2 start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT roman_ln 2 - italic_O ( italic_k ). Journal of the AMS 17 (2004) 947–973.
- D. Aldous, J. Steele: The objective method: probabilistic combinatorial optimization and local weak convergence. In: H. Kesten (ed.): Probability on Discrete Structures. Springer (2004).
- P. J. Bickel, P. A. Freedman. Some asymptotic theory for the bootstrap. Annals of Statistics 9 (1981) 1196–1217.
- S. Cao: Central limit theorems for combinatorial optimization problems on sparse Erdős-Rényi graphs. Annals of Applied Probability 31 (2021) 1687–1723.
- G. Bresler, B. Huang: The algorithmic phase transition of random k𝑘kitalic_k-SAT for low degree polynomials. Proc. 62nd FOCS (2021) 298–309.
- V. Chvátal, B. Reed: Mick gets some (the odds are on his side). Proc. 33th FOCS (1992) 620–627.
- A. Coja-Oghlan, K. Panagiotou: The asymptotic k𝑘kitalic_k-SAT threshold. Advances in Mathematics 288 (2016) 985–1068.
- O. Dubois, J. Mandler: The 3-XORSAT threshold. Proc. 43rd FOCS (2002) 769–778.
- G. Eagleson: Martingale convergence to mixtures of infinitely divisible laws. Annals of Probability 3 (1975) 557–562.
- C. Efthymiou: On sampling symmetric gibbs distributions on sparse random graphs and hypergraphs. Proc. 49th ICALP (2022) #57.
- E. Friedgut: Sharp thresholds of graph properties, and the k𝑘kitalic_k-SAT problem. Journal of the AMS 12 (1999) 1017–1054.
- A. Goerdt: A threshold for unsatisfiability. J. Comput. Syst. Sci. 53 (1996) 469–486
- P. Hall, C. Heyde: Martingale limit theory and its applications. Academic Press (1980).
- R. Monasson, R. Zecchina: The entropy of the k𝑘kitalic_k-satisfiability problem. Phys. Rev. Lett. 76 (1996) 3881.
- A. Montanari, D. Shah: Counting good truth assignments of random k𝑘kitalic_k-SAT formulae. Proc. 18th SODA (2007) 1255–1264.
- D. Panchenko: On the replica symmetric solution of the K𝐾Kitalic_K-sat model. Electron. J. Probab. 19 (2014) #67.
- D. Panchenko, M. Talagrand: Bounds for diluted mean-fields spin glass models. Probab. Theory Relat. Fields 130 (2004) 319–336.
- F. Rassmann: On the number of solutions in random graph k𝑘kitalic_k-colouring. Combinatorics, Probability and Computing 28 (2019) 130–158.
- R. Robinson, N. Wormald: Almost all regular graphs are Hamiltonian. Random Structures and Algorithms 5 (1994) 363–374.
- M. Talagrand: The high temperature case for the random K𝐾Kitalic_K-sat problem. Probab. Theory Related Fields 119 (2001) 187–212.
- L. Valiant: The complexity of enumeration and reliability problems. SIAM Journal on Computing 8 (1979) 410–421.
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