A. Second Moment Method: The second moment method is a simple probabilistic tool to establish existence of non-rare events in a complex setup.
1 Frieze, Alan, and Nicholas C. Wormald. “Random k-SAT: A Tight Threshold for Moderately Growing k.” Combinatorica 25, no. 3 (2005): 297-305. (PDF)
2 Achlioptas, Dimitris, and Assaf Naor. “The Two Possible Values of the Chromatic Number of a Random Graph.” Annals of Mathematics 162, no. 3 (2005): 1335-1351. (PDF)
B. Local Weak Convergence and Correlation Decay: The notion of local weak convergence allows one to study asymptotic spatial structural properties of probabilistic combinatorial optimization problems. The property of correlation decay facilitates one to establish these properties.
3 Kelly, F. P. “Stochastic Models of Computer Communication Systems (in A Symposium on Stochastic Networks).” Journal of the Royal Statistical Society. Series B (Methodological) 47, no. 3 (1985): 379-395.
4 Weitz, D. “Counting Independent Sets up to the Tree Threshold.” (PDF)
5 Aldous, D., and J. Steele. “The Objective Method: Probabilistic Combinatorial Optimization and Local Weak Convergence.” In Probability on Discrete Structures. Edited by Harry Kesten. New York, NY: Springer-Verlag, 2003. ISBN: 3540008454. (PS)
6 Gamarnik, David, Tomasz Nowicki, and Grzegorz Swirszcz. “Maximum Weight Independent Sets and Matchings in Sparse Random Graphs. Exact Results Using the Local Weak Convergence Method.” Random Structures and Algorithms 28, no. 1 (2006): 76-106.
C. Belief Propagation and LDPC Codes: Belief propagation is an iterative distributed heuristic algorithm for combinatorial optimization problem. It is expected to perform very well in the context of probabilistic setup.
7-8 Wainwright, M. J., and M. I. Jordan. “Graphical Models, Exponential Families and Variational Inference.” UC Berkeley, Dept. of Statistics, Technical Report 649 (2003). (PS - 1.7 MB)
9 Bayati, M., D. Shah, and M. Sharma. “Maximum Weight Matching via Max-product Belief Propagation.” International Symposium on Information Theory, Proceedings (2005): 1763-1767.
10 Richardson, T., and R. Urbanke. Modern Coding Theory. Cambridge, UK: Cambridge University Press, 2006, chapters 1 and 2. (Forthcoming)