# American Institute of Mathematical Sciences

March  2013, 18(2): 575-600. doi: 10.3934/dcdsb.2013.18.575

## The monomer-dimer problem and moment Lyapunov exponents of homogeneous Gaussian random fields

 1 University of New South Wales, Canberra, ACT 2600, Australia

Received  October 2011 Published  November 2012

We consider an "elastic'' version of the statistical mechanical monomer-dimer problem on the $n$-dimensional integer lattice. Our setting includes the classical "rigid'' formulation as a special case and extends it by allowing each dimer to consist of particles at arbitrarily distant sites of the lattice, with the energy of interaction between the particles in a dimer depending on their relative position. We reduce the free energy of the elastic dimer-monomer (EDM) system per lattice site in the thermodynamic limit to the moment Lyapunov exponent (MLE) of a homogeneous Gaussian random field (GRF) whose mean value and covariance function are the Boltzmann factors associated with the monomer energy and dimer potential. In particular, the classical monomer-dimer problem becomes related to the MLE of a moving average GRF. We outline an approach to recursive computation of the partition function for "Manhattan'' EDM systems where the dimer potential is a weighted $l_1$-distance and the auxiliary GRF is a Markov random field of Pickard type which behaves in space like autoregressive processes do in time. For one-dimensional Manhattan EDM systems, we compute the MLE of the resulting Gaussian Markov chain as the largest eigenvalue of a compact transfer operator on a Hilbert space which is related to the annihilation and creation operators of the quantum harmonic oscillator and also recast it as the eigenvalue problem for a pantograph functional-differential equation.
Citation: Igor G. Vladimirov. The monomer-dimer problem and moment Lyapunov exponents of homogeneous Gaussian random fields. Discrete & Continuous Dynamical Systems - B, 2013, 18 (2) : 575-600. doi: 10.3934/dcdsb.2013.18.575
##### References:
 [1] V. V. Anh, N .N. Leonenko and N. R. Shieh, Multifractality of products of geometric Ornstein-Uhlenbeck type processes,, Adv. Appl. Prob., 40 (2008), 1129.  doi: 10.1239/aap/1231340167.  Google Scholar [2] R. J. Baxter, Corner transfer matrices,, Physica A, 106 (1981), 18.  doi: 10.1016/0378-4371(81)90203-X.  Google Scholar [3] R. J. Baxter, "Exactly Solved Models in Statistical Mechanics,", Academic Press, (1982).   Google Scholar [4] J. E. Besag, Spatial interaction and statistical analysis of lattice systems (with discussion),, J. Roy. Statist. Soc., 36 (1974), 192.   Google Scholar [5] L. Bogachev, G. Derfel, S. Molchanov and J. Ockendon, On bounded solutions of the balanced generalized pantograph equation,, in, 145 (2008), 29.   Google Scholar [6] F. Champagnat, J. Idier and Y. Goussard, Stationary Markov random fields on a finite rectangular lattice,, IEEE Trans. Inform. Theory, 44 (1998), 2901.  doi: 10.1109/18.737521.  Google Scholar [7] F. Champagnat and J. Idier, On the correlation structure of unilateral AR processes on the plane,, Adv. Appl. Prob., 32 (2000), 408.  doi: 10.1239/aap/1013540171.  Google Scholar [8] N. Cressie and J. L. Davidson, Image analysis with partially ordered Markov models,, Comput. Stat. Data Anal., 29 (1998), 1.   Google Scholar [9] T. M. Cover and J. A. Thomas, "Elements of Information Theory,", 2nd ed., (2006).   Google Scholar [10] G. A. Derfel, Probabilistic method for a class of functional-differential equations,, Ukr. Math. J., 41 (1989), 1137.   Google Scholar [11] V. Elser, Solution of the dimer problem on a hexagonal lattice with boundary,, J. Phys. A: Math. Gen., 17 (1984), 1509.   Google Scholar [12] M. E. Fisher, Statistical mechanics of dimers on a plane lattice,, Phys. Rev., 124 (1961), 1664.   Google Scholar [13] M. Fisher and H. Temperley, The dimer problem in statistical mechanics - an exact result,, Phil. Mag., 6 (1961), 1061.   Google Scholar [14] U. Frisch, "Turbulence: the Legacy of A. N. Kolmogorov,", Cambridge University Press, (1995).   Google Scholar [15] I. I. Gikhman and A. V. Skorokhod, "The Theory of Stochastic Processes,", Springer, (2004).   Google Scholar [16] J. K. Goutsias, Mutually compatible Gibbs random fields,, IEEE Trans. Inform. Theory, 35 (1989), 1233.   Google Scholar [17] R. M. Gray, "Entropy and Information Theory,", Springer-Verlag, (1990).   Google Scholar [18] R. Hayn and V. N. Plechko, Grassmann variable analysis for dimer problems in two dimensions,, J. Phys. A: Math. Gen., 27 (1994), 4753.   Google Scholar [19] R. A. Horn and C. R. Johnson, "Matrix Analysis,", Cambridge University Press, (2007).   Google Scholar [20] J. M. Hammersley and V. V. Menon, A lower bound for the monomer-dimer problem,, IMA J. Appl. Maths., 6 (1970), 341.   Google Scholar [21] K. Huang, "Statistical Mechanics,", 2nd ed., (1987).   Google Scholar [22] J. Idier and Y. Goussard, "Champs de Pickard tridimensionnels,", Tech. Rep., (1999).   Google Scholar [23] L. Isserlis, On a formula for the product-moment coefficient of any order of a normal frequency distribution in any number of variables,, Biometrika, 12 (1918), 134.   Google Scholar [24] S. Janson, "Gaussian Hilbert Spaces,'', Cambridge University Press, (1997).   Google Scholar [25] P. W. Kasteleyn, The statistics of dimers on a lattice I. The number of dimer arrangements on a quadratic lattice,, Physica, 27 (1961), 1209.  doi: 10.1016/0031-8914(61)90063-5.  Google Scholar [26] T. Kato and J. B. McLeod, The functional-differential equation $y'(x) = ay(\lambda x) + by(x)$,, Bull. Amer. Math. Soc., 77 (1971), 891.   Google Scholar [27] C. Kenyon, D. Randall and A. Sinclair, Approximating the number of monomer-dimer coverings of a lattice,, J. Stat. Phys., 83 (1996), 637.  doi: 10.1007/BF02183743.  Google Scholar [28] V. Kozyakin, N. Kuznetsov, A. Pokrovskii and I. Vladimirov, Some problems in analysis of discretizations of continuous dynamical systems,, Nonlin. Anal., 30 (1997), 767.   Google Scholar [29] E. H. Lieb, Solution of the dimer problem by the transfer matrix method,, J. Math. Phys., 8 (1967), 2339.   Google Scholar [30] M. Loebl, On the dimer problem and the Ising problem in finite 3-dimensional lattices,, Electr. J. Combinator., 9 (2002), 1.   Google Scholar [31] K. Mahler, On a special functional equation,, J. London Math. Soc., 15 (1940), 115.  doi: 10.1112/jlms/s1-15.2.115.  Google Scholar [32] P. Malliavin, "Integration and Probability,", Springer-Verlag, (1995).  doi: 10.1007/978-1-4612-4202-4.  Google Scholar [33] P. Malliavin, "Stochastic Analysis,", Springer, (1997).   Google Scholar [34] N. F. G. Martin and J. W. England, "Mathematical Theory of Entropy,", Addison-Wesley, (1981).   Google Scholar [35] P.-A. Meyer, "Quantum Probability for Probabilists,", 2nd ed., (1995).   Google Scholar [36] U. U. Müller, A. Schick and W. Wefelmeyer, Inference for alternating time series,, in:, (2007), 589.   Google Scholar [37] D. K. Pickard, A curious binary lattice process,, J. Appl. Prob., 14 (1977), 717.  doi: 10.2307/3213345.  Google Scholar [38] D. K. Pickard, Unilateral Markov fields,, Adv. Appl. Prob., 12 (1980), 655.  doi: 10.2307/1426425.  Google Scholar [39] A. V. Pokrovskii, A. J. Kent and J. G. McInerney, Mixed moments of random mappings and chaotic dynamical systems,, Proc. R. Soc. Lond. A, 456 (2000), 2465.  doi: 10.1098/rspa.2000.0621.  Google Scholar [40] H. Rue and L. Held, "Gaussian Markov Random Fields,", Chapman & Hall, (2006).   Google Scholar [41] D. Ruelle, "Thermodynamic Formalism,", 2nd ed., (2004).   Google Scholar [42] J. J. Sakurai, "Modern Quantum Mechanics,", Addison-Wesley, (1994).   Google Scholar [43] A. N. Shiryaev, "Probability,", 2nd ed., (1995).   Google Scholar [44] V. Spiridonov, Universal superpositions of coherent states and self-similar potentials,, Phys. Rev. A, 52 (1995), 1909.  doi: 10.1103/PhysRevA.52.1909.  Google Scholar [45] E. M. Tory and D. K. Pickard, Unilateral Gaussian fields,, Adv. Appl. Prob., 24 (1992), 95.  doi: 10.2307/1427731.  Google Scholar [46] I. Vladimirov, "Quantized Linear Systems on Integer Lattices: Frequency-Based Approach,", Center for Applied Dynamical Systems and Environmental Modeling, (1996), 96.   Google Scholar [47] I. Vladimirov, N. Kuznetsov and P. Diamond, Frequency measurability, algebras of quasiperiodic sets and spatial discretizations of smooth dynamical systems,, Math. Comp. Simul., 52 (2000), 251.  doi: 10.1016/S0378-4754(00)00154-3.  Google Scholar

show all references

##### References:
 [1] V. V. Anh, N .N. Leonenko and N. R. Shieh, Multifractality of products of geometric Ornstein-Uhlenbeck type processes,, Adv. Appl. Prob., 40 (2008), 1129.  doi: 10.1239/aap/1231340167.  Google Scholar [2] R. J. Baxter, Corner transfer matrices,, Physica A, 106 (1981), 18.  doi: 10.1016/0378-4371(81)90203-X.  Google Scholar [3] R. J. Baxter, "Exactly Solved Models in Statistical Mechanics,", Academic Press, (1982).   Google Scholar [4] J. E. Besag, Spatial interaction and statistical analysis of lattice systems (with discussion),, J. Roy. Statist. Soc., 36 (1974), 192.   Google Scholar [5] L. Bogachev, G. Derfel, S. Molchanov and J. Ockendon, On bounded solutions of the balanced generalized pantograph equation,, in, 145 (2008), 29.   Google Scholar [6] F. Champagnat, J. Idier and Y. Goussard, Stationary Markov random fields on a finite rectangular lattice,, IEEE Trans. Inform. Theory, 44 (1998), 2901.  doi: 10.1109/18.737521.  Google Scholar [7] F. Champagnat and J. Idier, On the correlation structure of unilateral AR processes on the plane,, Adv. Appl. Prob., 32 (2000), 408.  doi: 10.1239/aap/1013540171.  Google Scholar [8] N. Cressie and J. L. Davidson, Image analysis with partially ordered Markov models,, Comput. Stat. Data Anal., 29 (1998), 1.   Google Scholar [9] T. M. Cover and J. A. Thomas, "Elements of Information Theory,", 2nd ed., (2006).   Google Scholar [10] G. A. Derfel, Probabilistic method for a class of functional-differential equations,, Ukr. Math. J., 41 (1989), 1137.   Google Scholar [11] V. Elser, Solution of the dimer problem on a hexagonal lattice with boundary,, J. Phys. A: Math. Gen., 17 (1984), 1509.   Google Scholar [12] M. E. Fisher, Statistical mechanics of dimers on a plane lattice,, Phys. Rev., 124 (1961), 1664.   Google Scholar [13] M. Fisher and H. Temperley, The dimer problem in statistical mechanics - an exact result,, Phil. Mag., 6 (1961), 1061.   Google Scholar [14] U. Frisch, "Turbulence: the Legacy of A. N. Kolmogorov,", Cambridge University Press, (1995).   Google Scholar [15] I. I. Gikhman and A. V. Skorokhod, "The Theory of Stochastic Processes,", Springer, (2004).   Google Scholar [16] J. K. Goutsias, Mutually compatible Gibbs random fields,, IEEE Trans. Inform. Theory, 35 (1989), 1233.   Google Scholar [17] R. M. Gray, "Entropy and Information Theory,", Springer-Verlag, (1990).   Google Scholar [18] R. Hayn and V. N. Plechko, Grassmann variable analysis for dimer problems in two dimensions,, J. Phys. A: Math. Gen., 27 (1994), 4753.   Google Scholar [19] R. A. Horn and C. R. Johnson, "Matrix Analysis,", Cambridge University Press, (2007).   Google Scholar [20] J. M. Hammersley and V. V. Menon, A lower bound for the monomer-dimer problem,, IMA J. Appl. Maths., 6 (1970), 341.   Google Scholar [21] K. Huang, "Statistical Mechanics,", 2nd ed., (1987).   Google Scholar [22] J. Idier and Y. Goussard, "Champs de Pickard tridimensionnels,", Tech. Rep., (1999).   Google Scholar [23] L. Isserlis, On a formula for the product-moment coefficient of any order of a normal frequency distribution in any number of variables,, Biometrika, 12 (1918), 134.   Google Scholar [24] S. Janson, "Gaussian Hilbert Spaces,'', Cambridge University Press, (1997).   Google Scholar [25] P. W. Kasteleyn, The statistics of dimers on a lattice I. The number of dimer arrangements on a quadratic lattice,, Physica, 27 (1961), 1209.  doi: 10.1016/0031-8914(61)90063-5.  Google Scholar [26] T. Kato and J. B. McLeod, The functional-differential equation $y'(x) = ay(\lambda x) + by(x)$,, Bull. Amer. Math. Soc., 77 (1971), 891.   Google Scholar [27] C. Kenyon, D. Randall and A. Sinclair, Approximating the number of monomer-dimer coverings of a lattice,, J. Stat. Phys., 83 (1996), 637.  doi: 10.1007/BF02183743.  Google Scholar [28] V. Kozyakin, N. Kuznetsov, A. Pokrovskii and I. Vladimirov, Some problems in analysis of discretizations of continuous dynamical systems,, Nonlin. Anal., 30 (1997), 767.   Google Scholar [29] E. H. Lieb, Solution of the dimer problem by the transfer matrix method,, J. Math. Phys., 8 (1967), 2339.   Google Scholar [30] M. Loebl, On the dimer problem and the Ising problem in finite 3-dimensional lattices,, Electr. J. Combinator., 9 (2002), 1.   Google Scholar [31] K. Mahler, On a special functional equation,, J. London Math. Soc., 15 (1940), 115.  doi: 10.1112/jlms/s1-15.2.115.  Google Scholar [32] P. Malliavin, "Integration and Probability,", Springer-Verlag, (1995).  doi: 10.1007/978-1-4612-4202-4.  Google Scholar [33] P. Malliavin, "Stochastic Analysis,", Springer, (1997).   Google Scholar [34] N. F. G. Martin and J. W. England, "Mathematical Theory of Entropy,", Addison-Wesley, (1981).   Google Scholar [35] P.-A. Meyer, "Quantum Probability for Probabilists,", 2nd ed., (1995).   Google Scholar [36] U. U. Müller, A. Schick and W. Wefelmeyer, Inference for alternating time series,, in:, (2007), 589.   Google Scholar [37] D. K. Pickard, A curious binary lattice process,, J. Appl. Prob., 14 (1977), 717.  doi: 10.2307/3213345.  Google Scholar [38] D. K. Pickard, Unilateral Markov fields,, Adv. Appl. Prob., 12 (1980), 655.  doi: 10.2307/1426425.  Google Scholar [39] A. V. Pokrovskii, A. J. Kent and J. G. McInerney, Mixed moments of random mappings and chaotic dynamical systems,, Proc. R. Soc. Lond. A, 456 (2000), 2465.  doi: 10.1098/rspa.2000.0621.  Google Scholar [40] H. Rue and L. Held, "Gaussian Markov Random Fields,", Chapman & Hall, (2006).   Google Scholar [41] D. Ruelle, "Thermodynamic Formalism,", 2nd ed., (2004).   Google Scholar [42] J. J. Sakurai, "Modern Quantum Mechanics,", Addison-Wesley, (1994).   Google Scholar [43] A. N. Shiryaev, "Probability,", 2nd ed., (1995).   Google Scholar [44] V. Spiridonov, Universal superpositions of coherent states and self-similar potentials,, Phys. Rev. A, 52 (1995), 1909.  doi: 10.1103/PhysRevA.52.1909.  Google Scholar [45] E. M. Tory and D. K. Pickard, Unilateral Gaussian fields,, Adv. Appl. Prob., 24 (1992), 95.  doi: 10.2307/1427731.  Google Scholar [46] I. Vladimirov, "Quantized Linear Systems on Integer Lattices: Frequency-Based Approach,", Center for Applied Dynamical Systems and Environmental Modeling, (1996), 96.   Google Scholar [47] I. Vladimirov, N. Kuznetsov and P. Diamond, Frequency measurability, algebras of quasiperiodic sets and spatial discretizations of smooth dynamical systems,, Math. Comp. Simul., 52 (2000), 251.  doi: 10.1016/S0378-4754(00)00154-3.  Google Scholar
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