## The Annals of Probability

### Approximating the hard square entropy constant with probabilistic methods

Ronnie Pavlov

#### Abstract

For any two-dimensional nearest neighbor shift of finite type $X$ and any integer $n\geq1$, one can define the horizontal strip shift $H_{n}(X)$ to be the set of configurations on $\mathbb{Z}\times\{1,\ldots,n\}$ which do not contain any forbidden pairs of adjacent letters for $X$. It is always the case that the sequence $h^{\mathrm{top} }(H_{n}(X))/n$ of normalized topological entropies of the strip shifts converges to $h^{\mathrm{top} }(X)$, the topological entropy of $X$. In this paper, we combine ergodic theoretic techniques with methods from percolation theory and interacting particle systems to show that for the two-dimensional hard square shift $\mathcal{H}$, the sequence $h^{\mathrm{top} }(H_{n+1}(\mathcal{H}))-h^{\mathrm{top} }(H_{n}(\mathcal{H}))$ also converges to $h^{\mathrm{top} }(\mathcal{H})$, and that the rate of convergence is at least exponential. As a corollary, we show that $h^{\mathrm{top} }(\mathcal{H})$ is computable to any tolerance $\varepsilon$ in time polynomial in $1/\varepsilon$. We also show that this phenomenon is not true in general by defining a block gluing two-dimensional nearest neighbor shift of finite type $Y$ for which $h^{\mathrm{top} }(H_{n+1}(Y))-h^{\mathrm{top} }(H_{n}(Y))$ does not even approach a limit.

#### Article information

Source
Ann. Probab., Volume 40, Number 6 (2012), 2362-2399.

Dates
First available in Project Euclid: 26 October 2012

https://projecteuclid.org/euclid.aop/1351258729

Digital Object Identifier
doi:10.1214/11-AOP681

Mathematical Reviews number (MathSciNet)
MR3050506

Zentralblatt MATH identifier
06114702

#### Citation

Pavlov, Ronnie. Approximating the hard square entropy constant with probabilistic methods. Ann. Probab. 40 (2012), no. 6, 2362--2399. doi:10.1214/11-AOP681. https://projecteuclid.org/euclid.aop/1351258729

#### References

• [1] Baxter, R. J. (1980). Hard hexagons: Exact solution. J. Phys. A 13 L61–L70.
• [2] Baxter, R. J. (1999). Planar lattice gases with nearest-neighbor exclusion. Ann. Comb. 3 191–203.
• [3] Berger, R. (1966). The undecidability of the domino problem. Mem. Amer. Math. Soc. 66 1–72.
• [4] Boyle, M., Pavlov, R. and Schraudner, M. (2010). Multidimensional sofic shifts without separation and their factors. Trans. Amer. Math. Soc. 362 4617–4653.
• [5] Burton, R. and Steif, J. E. (1994). Non-uniqueness of measures of maximal entropy for subshifts of finite type. Ergodic Theory Dynam. Systems 14 213–235.
• [6] Burton, R. and Steif, J. E. (1995). New results on measures of maximal entropy. Israel J. Math. 89 275–300.
• [7] Calkin, N. J. and Wilf, H. S. (1998). The number of independent sets in a grid graph. SIAM J. Discrete Math. 11 54–60 (electronic).
• [8] Campanino, M. and Russo, L. (1985). An upper bound on the critical percolation probability for the three-dimensional cubic lattice. Ann. Probab. 13 478–491.
• [9] Capobianco, S. (2008). Multidimensional cellular automata and generalization of Fekete’s lemma. Discrete Math. Theor. Comput. Sci. 10 95–104.
• [10] Desai, A. (2006). Subsystem entropy for $\mathbb{Z}^{d}$ sofic shifts. Indag. Math. (N.S.) 17 353–359.
• [11] Desai, A. (2009). A class of $\mathbb{Z}^{d}$ shifts of finite type which factors onto lower entropy full shifts. Proc. Amer. Math. Soc. 137 2613–2621.
• [12] Engel, K. (1990). On the Fibonacci number of an $m\times n$ lattice. Fibonacci Quart. 28 72–78.
• [13] Forchhammer, S. and Justesen, J. (1999). Entropy bounds for constrained two-dimensional random fields. IEEE Trans. Inform. Theory 45 118–127.
• [14] Friedland, S. (1997). On the entropy of $\mathbf{Z}^{d}$ subshifts of finite type. Linear Algebra Appl. 252 199–220.
• [15] Gamarnik, D. and Katz, D. (2009). Sequential cavity method for computing free energy and surface pressure. J. Stat. Phys. 137 205–232.
• [16] Grimmett, G. (1999). Percolation, 2nd ed. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 321. Springer, Berlin.
• [17] Häggström, O. (1995). A subshift of finite type that is equivalent to the Ising model. Ergodic Theory Dynam. Systems 15 543–556.
• [18] Häggström, O. (1996). On phase transitions for subshifts of finite type. Israel J. Math. 94 319–352.
• [19] Higuchi, Y. (1982). Coexistence of the infinite $(^{\ast})$ clusters: A remark on the square lattice site percolation. Z. Wahrsch. Verw. Gebiete 61 75–81.
• [20] Hochman, M. and Meyerovitch, T. (2010). A characterization of the entropies of multidimensional shifts of finite type. Ann. of Math. (2) 171 2011–2038.
• [21] Holley, R. (1974). Remarks on the $\textrm{FKG}$ inequalities. Comm. Math. Phys. 36 227–231.
• [22] Johnson, A. and Madden, K. (2005). Factoring higher-dimensional shifts of finite type onto the full shift. Ergodic Theory Dynam. Systems 25 811–822.
• [23] Kamae, T., Krengel, U. and O’Brien, G. L. (1977). Stochastic inequalities on partially ordered spaces. Ann. Probab. 5 899–912.
• [24] Kasteleyn, P. W. (1961). The statistics of dimers on a lattice I: The number of dimer arrangements on a quadratic lattice. Physica 27 1209–1225.
• [25] Ko, K.-I. (1991). Complexity Theory of Real Functions. Birkhäuser, Boston, MA.
• [26] Lieb, E. H. (1967). Exact solution of the problem of the entropy of two-dimensional ice. Phys. Rev. Lett. 18 692–694.
• [27] Liggett, T. M. (1985). Interacting Particle Systems. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 276. Springer, New York.
• [28] Lightwood, S. J. (2003). Morphisms from non-periodic $\mathbb{Z}^{2}$-subshifts. I. Constructing embeddings from homomorphisms. Ergodic Theory Dynam. Systems 23 587–609.
• [29] Lightwood, S. J. (2004). Morphisms from non-periodic $\mathbb{Z}^{2}$ subshifts. II. Constructing homomorphisms to square-filling mixing shifts of finite type. Ergodic Theory Dynam. Systems 24 1227–1260.
• [30] Lind, D. and Marcus, B. (1995). An Introduction to Symbolic Dynamics and Coding. Cambridge Univ. Press, Cambridge.
• [31] Men’shikov, M. V. (1986). Coincidence of critical points in percolation problems. Dokl. Akad. Nauk SSSR 288 1308–1311.
• [32] Men’shikov, M. V. and Pelikh, K. D. (1989). Percolation with several types of defects. A bound for the critical probability of a square lattice. Mat. Zametki 46 38–47.
• [33] Misiurewicz, M. (1975). A short proof of the variational principle for a $\mathbb{Z}_{+}^{n}$ action on a compact space. Asterisque 40 147–157.
• [34] Parry, W. (1964). Intrinsic Markov chains. Trans. Amer. Math. Soc. 112 55–66.
• [35] Pierce, L. (2008). Computing entropy for $\mathbb{Z}^{2}$-actions. Ph.D. thesis, Oregon State Univ.
• [36] Rudolph, D. J. (1990). Fundamentals of Measurable Dynamics: Ergodic Theory on Lebesgue Spaces. Oxford Univ. Press, New York.
• [37] Strassen, V. (1965). The existence of probability measures with given marginals. Ann. Math. Statist. 36 423–439.
• [38] Temperley, H. N. V. and Fisher, M. E. (1961). Dimer problem in statistical mechanics—an exact result. Philos. Mag. (8) 6 1061–1063.
• [39] Tóth, B. (1985). A lower bound for the critical probability of the square lattice site percolation. Z. Wahrsch. Verw. Gebiete 69 19–22.
• [40] van den Berg, J. and Ermakov, A. (1996). A new lower bound for the critical probability of site percolation on the square lattice. Random Structures Algorithms 8 199–212.
• [41] van den Berg, J. and Steif, J. E. (1994). Percolation and the hard-core lattice gas model. Stochastic Process. Appl. 49 179–197.
• [42] Walters, P. (1982). An Introduction to Ergodic Theory. Graduate Texts in Mathematics 79. Springer, New York.
• [43] Wang, H. (1961). Proving theorems by pattern recognition II. AT&T Bell Labs. Tech. J. 40 1–41.
• [44] Zuev, S. A. (1988). A lower bound for a percolation threshold for a square lattice. Vestnik Moskov. Univ. Ser. I Mat. Mekh. 5 59–61.