# Spatial mixing and the connective constant: optimal bounds

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## Abstract

We study the problem of deterministic approximate counting of matchings and independent sets in graphs of bounded *connective constant*. More generally, we consider the problem of evaluating the partition functions of the *monomer-dimer model* (which is defined as a weighted sum over all matchings where each matching is given a weight \(\gamma ^{|V| - 2 |M|}\) in terms of a fixed parameter \(\gamma \) called the *monomer activity*) and the *hard core model* (which is defined as a weighted sum over all independent sets where an independent set *I* is given a weight \(\lambda ^{|I|}\) in terms of a fixed parameter \(\lambda \) called the *vertex activity*). The *connective constant* is a natural measure of the average degree of a graph which has been studied extensively in combinatorics and mathematical physics, and can be bounded by a constant even for certain unbounded degree graphs such as those sampled from the sparse Erdős–Rényi model \(\mathcal {G}(n, d/n)\). Our main technical contribution is to prove the best possible rates of decay of correlations in the natural probability distributions induced by both the hard core model and the monomer-dimer model in graphs with a given bound on the connective constant. These results on decay of correlations are obtained using a new framework based on the so-called *message* approach that has been extensively used recently to prove such results for bounded degree graphs. We then use these optimal decay of correlations results to obtain fully polynomial time approximation schemes (FPTASs) for the two problems on graphs of bounded connective constant. In particular, for the monomer-dimer model, we give a deterministic FPTAS for the partition function on all graphs of bounded connective constant for any given value of the monomer activity. The best previously known deterministic algorithm was due to Bayati et al. (Proc. 39th ACM Symp. Theory Comput., pp. 122–127, 2007), and gave the same runtime guarantees as our results but only for the case of bounded degree graphs. For the hard core model, we give an FPTAS for graphs of connective constant \(\varDelta \) whenever the vertex activity \(\lambda < \lambda _c(\varDelta )\), where \(\lambda _c(\varDelta ) :=\frac{\varDelta ^\varDelta }{(\varDelta - 1)^{\varDelta + 1}}\); this result is optimal in the sense that an FPTAS for any \(\lambda > \lambda _c(\varDelta )\) would imply that NP=RP (Sly and Sun, Ann. Probab. 42(6):2383–2416, 2014). The previous best known result in this direction was in a recent manuscript by a subset of the current authors (Proc. 54th IEEE Symp. Found. Comput. Sci., pp 300–309, 2013), where the result was established under the sub-optimal condition \(\lambda < \lambda _c(\varDelta + 1)\). Our techniques also allow us to improve upon known bounds for decay of correlations for the hard core model on various regular lattices, including those obtained by Restrepo et al. (Probab Theory Relat Fields 156(1–2):75–99, 2013) for the special case of \(\mathbb {Z}^2\) using sophisticated numerically intensive methods tailored to that special case.

## Mathematics Subject Classification

82B20 60J10 68W25 68W40## Notes

### Acknowledgments

We thank Elchanan Mossel, Allan Sly, Eric Vigoda and Dror Weitz for helpful discussions. We also thank the anonymous referees for detailed helpful comments.

## References

- 1.Alm, S.E.: Upper bounds for the connective constant of self-avoiding walks. Comb. Probab. Comput.
**2**(02), 115–136 (1993). doi: 10.1017/S0963548300000547 MathSciNetCrossRefMATHGoogle Scholar - 2.Alm, S.E.: Upper and lower bounds for the connective constants of self-avoiding walks on the Archimedean and Laves lattices. J. Phys. A
**38**(10), 2055–2080 (2005). doi: 10.1088/0305-4470/38/10/001 MathSciNetCrossRefMATHGoogle Scholar - 3.Andrews, G.E.: The hard-hexagon model and Rogers–Ramanujan type identities. Proc. Nat. Acad. Sci.
**78**(9), 5290–5292 (1981). http://www.pnas.org/content/78/9/5290. PMID: 16593082 - 4.Bandyopadhyay, A., Gamarnik, D.: Counting without sampling: asymptotics of the log-partition function for certain statistical physics models random structures and algorithms. Random Struct. Algorithms
**33**(4), 452–479 (2008)CrossRefMATHGoogle Scholar - 5.Baxter, R.J.: Hard hexagons: exact solution. J. Phys. A: Math. Gen.
**13**(3), L61–L70 (1980). doi: 10.1088/0305-4470/13/3/007. http://iopscience.iop.org/0305-4470/13/3/007 - 6.Baxter, R.J., Enting, I.G., Tsang, S.K.: Hard-square lattice gas. J. Stat. Phys.
**22**(4), 465–489 (1980). doi: 10.1007/BF01012867. http://link.springer.com/article/10.1007/BF01012867 - 7.Bayati, M., Gamarnik, D., Katz, D., Nair, C., Tetali, P.: Simple deterministic approximation algorithms for counting matchings. In: Proc. 39th ACM Symp. Theory Comput., pp. 122–127. ACM (2007). doi: 10.1145/1250790.1250809
- 8.Broadbent, S.R., Hammersley, J.M.: Percolation processes I. Crystals and mazes. Math. Proc. Camb. Philos. Soc.
**53**(03), 629–641 (1957). doi: 10.1017/S0305004100032680 MathSciNetCrossRefMATHGoogle Scholar - 9.Duminil-Copin, H., Smirnov, S.: The connective constant of the honeycomb lattice equals \(\sqrt{2+\sqrt{2}}\). Ann. Math.
**175**(3), 1653–1665 (2012). doi: 10.4007/annals.2012.175.3.14 MathSciNetCrossRefMATHGoogle Scholar - 10.Dyer, M., Greenhill, C.: On Markov chains for independent sets. J. Algorithms
**35**(1), 17–49 (2000)MathSciNetCrossRefMATHGoogle Scholar - 11.Efthymiou, C.: MCMC sampling colourings and independent sets of \(\cal G(n,d/n)\) near uniqueness threshold. In: Proc. 25th ACM-SIAM Symp. Discret. Algorithms, pp. 305–316. SIAM (2014). Full version available at arXiv:1304.6666
- 12.Galanis, A., Ge, Q., Štefankovič, D., Vigoda, E., Yang, L.: Improved inapproximability results for counting independent sets in the hard-core model. Random Struct. Algorithms
**45**(1), 78–110 (2014). doi: 10.1002/rsa.20479 MathSciNetCrossRefMATHGoogle Scholar - 13.Gamarnik, D., Katz, D.: Correlation decay and deterministic FPTAS for counting list-colorings of a graph. In: Proc. 18th ACM-SIAM Symp. Discret. Algorithms, pp. 1245–1254. SIAM (2007). http://dl.acm.org/citation.cfm?id=1283383.1283517
- 14.Gaunt, D.S., Fisher, M.E.: Hard-sphere lattice gases. I. Plane-square lattice. J. Chem. Phys.
**43**(8), 2840–2863 (1965). doi: 10.1063/1.1697217. http://scitation.aip.org/content/aip/journal/jcp/43/8/10.1063/1.1697217 - 15.Georgii, H.O.: Gibbs Measures and Phase Transitions. De Gruyter Studies in Mathematics, Walter de Gruyter Inc, (1988)Google Scholar
- 16.Godsil, C.D.: Matchings and walks in graphs. J. Graph Th.
**5**(3), 285–297 (1981). http://onlinelibrary.wiley.com/doi/10.1002/jgt.3190050310/abstract - 17.Goldberg, L.A., Jerrum, M., Paterson, M.: The computational complexity of two-state spin systems. Random Struct. Algorithms
**23**, 133–154 (2003)MathSciNetCrossRefMATHGoogle Scholar - 18.Goldberg, L.A., Martin, R., Paterson, M.: Strong spatial mixing with fewer colors for lattice graphs. SIAM J. Comput.
**35**(2), 486–517 (2005). doi: 10.1137/S0097539704445470. http://link.aip.org/link/?SMJ/35/486/1 - 19.Hammersley, J.M.: Percolation processes II. The connective constant. Math. Proc. Camb. Philos. Soc.
**53**(03), 642–645 (1957). doi: 10.1017/S0305004100032692 MathSciNetCrossRefMATHGoogle Scholar - 20.Hammersley, J.M., Morton, K.W.: Poor man’s Monte Carlo. J. Royal Stat. Soc. B
**16**(1), 23–38 (1954). doi: 10.2307/2984008 MathSciNetMATHGoogle Scholar - 21.Hayes, T.P., Vigoda, E.: Coupling with the stationary distribution and improved sampling for colorings and independent sets. Ann. Appl. Probab.
**16**(3), 1297–1318 (2006)MathSciNetCrossRefMATHGoogle Scholar - 22.Jensen, I.: Enumeration of self-avoiding walks on the square lattice. J. Phys. A
**37**(21), 5503 (2004). doi: 10.1088/0305-4470/37/21/002 MathSciNetCrossRefMATHGoogle Scholar - 23.Jerrum, M., Sinclair, A.: Approximating the permanent. SIAM J. Comput.
**18**(6), 1149–1178 (1989). doi: 10.1137/0218077. http://epubs.siam.org/doi/abs/10.1137/0218077 - 24.Jerrum, M., Valiant, L.G., Vazirani, V.V.: Random generation of combinatorial structures from a uniform distribution. Theor. Comput. Sci.
**43**, 169–188 (1986)MathSciNetCrossRefMATHGoogle Scholar - 25.Kahn, J., Kim, J.H.: Random matchings in regular graphs. Combinatorica
**18**(2), 201–226 (1998). doi: 10.1007/PL00009817. http://link.springer.com/article/10.1007/PL00009817 - 26.Kesten, H.: On the number of self-avoiding walks. II. J. Math. Phys.
**5**(8), 1128–1137 (1964). doi: 10.1063/1.1704216 MathSciNetCrossRefMATHGoogle Scholar - 27.Li, L., Lu, P., Yin, Y.: Approximate counting via correlation decay in spin systems. In: Proc. 23rd ACM-SIAM Symp. Discret. Algorithms, pp. 922–940. SIAM (2012)Google Scholar
- 28.Li, L., Lu, P., Yin, Y.: Correlation decay up to uniqueness in spin systems. In: Proc. 24th ACM-SIAM Symp. Discret. Algorithms, pp. 67–84. SIAM (2013)Google Scholar
- 29.Luby, M., Vigoda, E.: Approximately counting up to four. In: Proc. 29th ACM Symp. Theory. Comput., pp. 682–687. ACM (1997). doi: 10.1145/258533.258663
- 30.Lyons, R.: The Ising model and percolation on trees and tree-like graphs. Commun. Math. Phys.
**125**(2), 337–353 (1989)MathSciNetCrossRefMATHGoogle Scholar - 31.Lyons, R.: Random walks and percolation on trees. Ann. Probab.
**18**(3), 931–958 (1990). doi: 10.1214/aop/1176990730 MathSciNetCrossRefMATHGoogle Scholar - 32.Madras, N., Slade, G.: The Self-Avoiding Walk. Birkhäuser (1996)Google Scholar
- 33.Martinelli, F., Olivieri, E.: Approach to equilibrium of Glauber dynamics in the one phase region I. The attractive case. Comm. Math. Phys.
**161**(3), 447–486 (1994). doi: 10.1007/BF02101929. http://link.springer.com/article/10.1007/BF02101929 - 34.Martinelli, F., Olivieri, E.: Approach to equilibrium of Glauber dynamics in the one phase region II. The general case. Comm. Math. Phys.
**161**(3), 487–514 (1994). doi: 10.1007/BF02101930. http://link.springer.com/article/10.1007/BF02101930 - 35.Mossel, E.: Survey: Information flow on trees. In: Graphs, Morphisms and Statistical Physics, DIMACS Series in Discrete Mathematics and Theoretical Computer Science, vol. 63, pp. 155–170. American Mathematical Society (2004)Google Scholar
- 36.Mossel, E., Sly, A.: Rapid mixing of Gibbs sampling on graphs that are sparse on average. Random Struct. Algorithms
**35**(2), 250–270 (2009). doi: 10.1002/rsa.20276 MathSciNetCrossRefMATHGoogle Scholar - 37.Mossel, E., Sly, A.: Gibbs rapidly samples colorings of \(\cal G(n, d/n)\). Probab. Theory Relat. Fields
**148**(1–2), 37–69 (2010). doi: 10.1007/s00440-009-0222-x MathSciNetCrossRefMATHGoogle Scholar - 38.Mossel, E., Sly, A.: Exact thresholds for Ising-Gibbs samplers on general graphs. Ann. Probab.
**41**(1), 294–328 (2013). doi: 10.1214/11-AOP737 MathSciNetCrossRefMATHGoogle Scholar - 39.Nienhuis, B.: Exact critical point and critical exponents of \(O(n)\) models in two dimensions. Phys. Rev. Let.
**49**(15), 1062–1065 (1982). doi: 10.1103/PhysRevLett.49.1062 MathSciNetCrossRefGoogle Scholar - 40.Pemantle, R., Steif, J.E.: Robust phase transitions for Heisenberg and other models on general trees. Ann. Probab.
**27**(2), 876–912 (1999)MathSciNetCrossRefMATHGoogle Scholar - 41.Pönitz, A., Tittmann, P.: Improved upper bounds for self-avoiding walks in \(\mathbb{Z}^{d}\). Electron. J. Comb.,
**7**, Research Paper 21 (2000)Google Scholar - 42.Restrepo, R., Shin, J., Tetali, P., Vigoda, E., Yang, L.: Improved mixing condition on the grid for counting and sampling independent sets. Probab. Theory Relat. Fields
**156**(1–2), 75–99 (2013). Extended abstract in Proc. IEEE Symp. Found. Comput. Sci., 2011MathSciNetCrossRefMATHGoogle Scholar - 43.Sinclair, A., Srivastava, P., Thurley, M.: Approximation algorithms for two-state anti-ferromagnetic spin systems on bounded degree graphs. J. Stat. Phys.
**155**(4), 666–686 (2014)MathSciNetCrossRefMATHGoogle Scholar - 44.Sinclair, A., Srivastava, P., Yin, Y.: Spatial mixing and approximation algorithms for graphs with bounded connective constant. In: Proc. 54th IEEE Symp. Found. Comput. Sci., pp. 300–309. IEEE Computer Society (2013). Full version available at arXiv:1308.1762v1
- 45.Sly, A.: Computational transition at the uniqueness threshold. In: Proc. 51st IEEE Symp. Found. Comput. Sci., pp. 287–296. IEEE Computer Society (2010)Google Scholar
- 46.Sly, A., Sun, N.: Counting in two-spin models on \(d\)-regular graphs. Ann. Probab.
**42**(6), 2383–2416 (2014). doi: 10.1214/13-AOP888. http://projecteuclid.org/euclid.aop/1412083628 - 47.Vera, J.C., Vigoda, E., Yang, L.: Improved bounds on the phase transition for the hard-core model in 2-dimensions. In: Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, Lecture Notes in Computer Science, vol 8096, pp. 699–713. Springer Berlin Heidelberg (2013). doi: 10.1007/978-3-642-40328-6_48
- 48.Vigoda, E.: A note on the Glauber dynamics for sampling independent sets. Electron. J. Combin.
**8**(1), R8 (2001). http://www.combinatorics.org/ojs/index.php/eljc/article/view/v8i1r8 - 49.Weisstein, E.W.: Self-avoiding walk connective constant. From MathWorld–a Wolfram Web Resource. http://mathworld.wolfram.com/Self-AvoidingWalkConnectiveConstant.html
- 50.Weitz, D.: Counting independent sets up to the tree threshold. In: Proc. 38th ACM Symp. Theory Comput., pp. 140–149. ACM (2006). doi: 10.1145/1132516.1132538