## The Annals of Probability

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

#### Abstract

We establish the existence of free energy limits for several combinatorial models on Erdös–Rényi graph $\mathbb{G} (N,\lfloor cN\rfloor)$ and random $r$-regular graph $\mathbb{G} (N,r)$. For a variety of models, including independent sets, MAX-CUT, coloring and K-SAT, we prove that the free energy both at a positive and zero temperature, appropriately rescaled, converges to a limit as the size of the underlying graph diverges to infinity. In the zero temperature case, this is interpreted as the existence of the scaling limit for the corresponding combinatorial optimization problem. For example, as a special case we prove that the size of a largest independent set in these graphs, normalized by the number of nodes converges to a limit w.h.p. This resolves an open problem which was proposed by Aldous (Some open problems) as one of his six favorite open problems. It was also mentioned as an open problem in several other places: Conjecture 2.20 in Wormald [In Surveys in Combinatorics, 1999 (Canterbury) (1999) 239–298 Cambridge Univ. Press]; Bollobás and Riordan [Random Structures Algorithms 39 (2011) 1–38]; Janson and Thomason [Combin. Probab. Comput. 17 (2008) 259–264] and Aldous and Steele [In Probability on Discrete Structures (2004) 1–72 Springer].

Our approach is based on extending and simplifying the interpolation method of Guerra and Toninelli [Comm. Math. Phys. 230 (2002) 71–79] and Franz and Leone [J. Stat. Phys. 111 (2003) 535–564]. Among other applications, this method was used to prove the existence of free energy limits for Viana–Bray and K-SAT models on Erdös–Rényi graphs. The case of zero temperature was treated by taking limits of positive temperature models. We provide instead a simpler combinatorial approach and work with the zero temperature case (optimization) directly both in the case of Erdös–Rényi graph $\mathbb{G} (N,\lfloor cN\rfloor)$ and random regular graph $\mathbb{G} (N,r)$. In addition we establish the large deviations principle for the satisfiability property of the constraint satisfaction problems, coloring, K-SAT and NAE-K-SAT, for the $\mathbb{G} (N,\lfloor cN\rfloor)$ random graph model.

#### Article information

Source
Ann. Probab., Volume 41, Number 6 (2013), 4080-4115.

Dates
First available in Project Euclid: 20 November 2013

https://projecteuclid.org/euclid.aop/1384957783

Digital Object Identifier
doi:10.1214/12-AOP816

Mathematical Reviews number (MathSciNet)
MR3161470

Zentralblatt MATH identifier
1280.05115

Subjects
Secondary: 82-08: Computational methods

#### Citation

Bayati, Mohsen; Gamarnik, David; Tetali, Prasad. Combinatorial approach to the interpolation method and scaling limits in sparse random graphs. Ann. Probab. 41 (2013), no. 6, 4080--4115. doi:10.1214/12-AOP816. https://projecteuclid.org/euclid.aop/1384957783

#### References

• [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.
• [5] Bollobás, B. (1980). A probabilistic proof of an asymptotic formula for the number of labelled regular graphs. European J. Combin. 1 311–316.
• [6] Bollobás, B. (2001). Random Graphs. Cambridge Univ. Press, Cambridge.
• [7] Bollobás, B. and Riordan, O. (2006). Percolation. Cambridge Univ. Press, New York.
• [8] Bollobás, B. and Riordan, O. (2011). Sparse graphs: Metrics and random models. Random Structures Algorithms 39 1–38.
• [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.
• [11] de Bruijn, N. G. and Erdös, P. (1951). Some linear and some quadratic recursion formulas. I. Indag. Math. (N.S.) 13 374–382.
• [12] de Bruijn, N. G. and Erdös, P. (1952). Some linear and some quadratic recursion formulas. II. Indag. Math. (N.S.) 14 152–163.
• [13] Dembo, A. and Montanari, 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.
• [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.
• [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. and Montanari, 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.
• [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.
• [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.
• [23] Guerra, F. and Toninelli, F. L. (2002). The thermodynamic limit in mean field spin glass models. Comm. Math. Phys. 230 71–79.
• [24] Janson, S., Łuczak, T. and Rucinski, A. (2000). Random Graphs. Wiley, New York.
• [25] Janson, S. and Thomason, A. (2008). Dismantling sparse random graphs. Combin. Probab. Comput. 17 259–264.
• [26] Montanari, A. (2005). Tight bounds for LDPC and LDGM codes under MAP decoding. IEEE Trans. Inform. Theory 51 3221–3246.
• [27] Panchenko, D. and Talagrand, M. (2004). Bounds for diluted mean-fields spin glass models. Probab. Theory Related Fields 130 319–336.
• [28] Wormald, N. C. (1999). Models of random regular graphs. In Surveys in Combinatorics, 1999 (Canterbury). London Mathematical Society Lecture Note Series 267 239–298. Cambridge Univ. Press, Cambridge.