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 and 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), 1129-1156. doi: 10.1239/aap/1231340167.

[2]

R. J. Baxter, Corner transfer matrices, Physica A, 106 (1981), 18-27. doi: 10.1016/0378-4371(81)90203-X.

[3]

R. J. Baxter, "Exactly Solved Models in Statistical Mechanics," Academic Press, London, 1982.

[4]

J. E. Besag, Spatial interaction and statistical analysis of lattice systems (with discussion), J. Roy. Statist. Soc., Series B, 36 (1974), 192-236.

[5]

L. Bogachev, G. Derfel, S. Molchanov and J. Ockendon, On bounded solutions of the balanced generalized pantograph equation, in "Topics in Stochastic Analysis and Nonparametric Estimation" IMA Vol. Math. Appl., 145 29-49, Springer, New York, 2008.

[6]

F. Champagnat, J. Idier and Y. Goussard, Stationary Markov random fields on a finite rectangular lattice, IEEE Trans. Inform. Theory, 44 (1998), 2901-2916. doi: 10.1109/18.737521.

[7]

F. Champagnat and J. Idier, On the correlation structure of unilateral AR processes on the plane, Adv. Appl. Prob., 32 (2000), 408-425. doi: 10.1239/aap/1013540171.

[8]

N. Cressie and J. L. Davidson, Image analysis with partially ordered Markov models, Comput. Stat. Data Anal., 29 (1998), 1-26.

[9]

T. M. Cover and J. A. Thomas, "Elements of Information Theory," 2nd ed., Wiley, Hoboken, New Jersey, 2006.

[10]

G. A. Derfel, Probabilistic method for a class of functional-differential equations, Ukr. Math. J., 41 (1989), 1137-1141.

[11]

V. Elser, Solution of the dimer problem on a hexagonal lattice with boundary, J. Phys. A: Math. Gen., 17 (1984), 1509-1513.

[12]

M. E. Fisher, Statistical mechanics of dimers on a plane lattice, Phys. Rev., 124 (1961), 1664-1672.

[13]

M. Fisher and H. Temperley, The dimer problem in statistical mechanics - an exact result, Phil. Mag., 6 (1961), 1061-1063.

[14]

U. Frisch, "Turbulence: the Legacy of A. N. Kolmogorov," Cambridge University Press, Cambridge, 1995.

[15]

I. I. Gikhman and A. V. Skorokhod, "The Theory of Stochastic Processes," Springer, Berlin, 2004.

[16]

J. K. Goutsias, Mutually compatible Gibbs random fields, IEEE Trans. Inform. Theory, 35 (1989), 1233-1249.

[17]

R. M. Gray, "Entropy and Information Theory," Springer-Verlag, New York, 1990.

[18]

R. Hayn and V. N. Plechko, Grassmann variable analysis for dimer problems in two dimensions, J. Phys. A: Math. Gen., 27 (1994), 4753-4760.

[19]

R. A. Horn and C. R. Johnson, "Matrix Analysis," Cambridge University Press, New York, 2007.

[20]

J. M. Hammersley and V. V. Menon, A lower bound for the monomer-dimer problem, IMA J. Appl. Maths., 6 (1970), 341-364.

[21]

K. Huang, "Statistical Mechanics," 2nd ed., John Wiley & Sons, New York, 1987.

[22]

J. Idier and Y. Goussard, "Champs de Pickard tridimensionnels," Tech. Rep., IGB/GPI-LSS, 1999, http://www.irccyn.ec-nantes.fr/~idier/pub/idier99d.pdf.

[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), 134-139.

[24]

S. Janson, "Gaussian Hilbert Spaces,'' Cambridge University Press, Cambridge, 1997.

[25]

P. W. Kasteleyn, The statistics of dimers on a lattice I. The number of dimer arrangements on a quadratic lattice, Physica, 27 (1961), 1209-1225. doi: 10.1016/0031-8914(61)90063-5.

[26]

T. Kato and J. B. McLeod, The functional-differential equation $y'(x) = ay(\lambda x) + by(x)$, Bull. Amer. Math. Soc., 77 (1971), 891-937.

[27]

C. Kenyon, D. Randall and A. Sinclair, Approximating the number of monomer-dimer coverings of a lattice, J. Stat. Phys., 83 (1996), 637-659. doi: 10.1007/BF02183743.

[28]

V. Kozyakin, N. Kuznetsov, A. Pokrovskii and I. Vladimirov, Some problems in analysis of discretizations of continuous dynamical systems, Nonlin. Anal., Theor. Meth. Appl., 30 (1997), 767-778.

[29]

E. H. Lieb, Solution of the dimer problem by the transfer matrix method, J. Math. Phys., 8 (1967), 2339-2341.

[30]

M. Loebl, On the dimer problem and the Ising problem in finite 3-dimensional lattices, Electr. J. Combinator., 9 (2002), 1-25.

[31]

K. Mahler, On a special functional equation, J. London Math. Soc., 15 (1940), 115-123. doi: 10.1112/jlms/s1-15.2.115.

[32]

P. Malliavin, "Integration and Probability," Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4612-4202-4.

[33]

P. Malliavin, "Stochastic Analysis," Springer, Berlin, 1997.

[34]

N. F. G. Martin and J. W. England, "Mathematical Theory of Entropy," Addison-Wesley, Reading, Mass., 1981.

[35]

P.-A. Meyer, "Quantum Probability for Probabilists," 2nd ed., Springer, Berlin, 1995.

[36]

U. U. Müller, A. Schick and W. Wefelmeyer, Inference for alternating time series, in: "Recent Advances in Stochastic Modeling and Data Analysis'' (ed. C. H. Skiadas), World Scientific, Singapore, (2007), 589-596.

[37]

D. K. Pickard, A curious binary lattice process, J. Appl. Prob., 14 (1977), 717-731. doi: 10.2307/3213345.

[38]

D. K. Pickard, Unilateral Markov fields, Adv. Appl. Prob., 12 (1980), 655-671. doi: 10.2307/1426425.

[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), 2465-2487. doi: 10.1098/rspa.2000.0621.

[40]

H. Rue and L. Held, "Gaussian Markov Random Fields," Chapman & Hall, 2006.

[41]

D. Ruelle, "Thermodynamic Formalism," 2nd ed., Cambridge University Press, Cambridge, 2004.

[42]

J. J. Sakurai, "Modern Quantum Mechanics," Addison-Wesley, Reading, Mass., 1994.

[43]

A. N. Shiryaev, "Probability," 2nd ed., Springer, New York, 1995.

[44]

V. Spiridonov, Universal superpositions of coherent states and self-similar potentials, Phys. Rev. A, 52 (1995), 1909-1935. doi: 10.1103/PhysRevA.52.1909.

[45]

E. M. Tory and D. K. Pickard, Unilateral Gaussian fields, Adv. Appl. Prob., 24 (1992), 95-112. doi: 10.2307/1427731.

[46]

I. Vladimirov, "Quantized Linear Systems on Integer Lattices: Frequency-Based Approach," Center for Applied Dynamical Systems and Environmental Modeling, Deakin University, Geelong, Victoria, Australia, CADSEM Reports 96-032 (1996), 1-37; 96-033 (1996), 1-50.

[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), 251-272. doi: 10.1016/S0378-4754(00)00154-3.

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), 1129-1156. doi: 10.1239/aap/1231340167.

[2]

R. J. Baxter, Corner transfer matrices, Physica A, 106 (1981), 18-27. doi: 10.1016/0378-4371(81)90203-X.

[3]

R. J. Baxter, "Exactly Solved Models in Statistical Mechanics," Academic Press, London, 1982.

[4]

J. E. Besag, Spatial interaction and statistical analysis of lattice systems (with discussion), J. Roy. Statist. Soc., Series B, 36 (1974), 192-236.

[5]

L. Bogachev, G. Derfel, S. Molchanov and J. Ockendon, On bounded solutions of the balanced generalized pantograph equation, in "Topics in Stochastic Analysis and Nonparametric Estimation" IMA Vol. Math. Appl., 145 29-49, Springer, New York, 2008.

[6]

F. Champagnat, J. Idier and Y. Goussard, Stationary Markov random fields on a finite rectangular lattice, IEEE Trans. Inform. Theory, 44 (1998), 2901-2916. doi: 10.1109/18.737521.

[7]

F. Champagnat and J. Idier, On the correlation structure of unilateral AR processes on the plane, Adv. Appl. Prob., 32 (2000), 408-425. doi: 10.1239/aap/1013540171.

[8]

N. Cressie and J. L. Davidson, Image analysis with partially ordered Markov models, Comput. Stat. Data Anal., 29 (1998), 1-26.

[9]

T. M. Cover and J. A. Thomas, "Elements of Information Theory," 2nd ed., Wiley, Hoboken, New Jersey, 2006.

[10]

G. A. Derfel, Probabilistic method for a class of functional-differential equations, Ukr. Math. J., 41 (1989), 1137-1141.

[11]

V. Elser, Solution of the dimer problem on a hexagonal lattice with boundary, J. Phys. A: Math. Gen., 17 (1984), 1509-1513.

[12]

M. E. Fisher, Statistical mechanics of dimers on a plane lattice, Phys. Rev., 124 (1961), 1664-1672.

[13]

M. Fisher and H. Temperley, The dimer problem in statistical mechanics - an exact result, Phil. Mag., 6 (1961), 1061-1063.

[14]

U. Frisch, "Turbulence: the Legacy of A. N. Kolmogorov," Cambridge University Press, Cambridge, 1995.

[15]

I. I. Gikhman and A. V. Skorokhod, "The Theory of Stochastic Processes," Springer, Berlin, 2004.

[16]

J. K. Goutsias, Mutually compatible Gibbs random fields, IEEE Trans. Inform. Theory, 35 (1989), 1233-1249.

[17]

R. M. Gray, "Entropy and Information Theory," Springer-Verlag, New York, 1990.

[18]

R. Hayn and V. N. Plechko, Grassmann variable analysis for dimer problems in two dimensions, J. Phys. A: Math. Gen., 27 (1994), 4753-4760.

[19]

R. A. Horn and C. R. Johnson, "Matrix Analysis," Cambridge University Press, New York, 2007.

[20]

J. M. Hammersley and V. V. Menon, A lower bound for the monomer-dimer problem, IMA J. Appl. Maths., 6 (1970), 341-364.

[21]

K. Huang, "Statistical Mechanics," 2nd ed., John Wiley & Sons, New York, 1987.

[22]

J. Idier and Y. Goussard, "Champs de Pickard tridimensionnels," Tech. Rep., IGB/GPI-LSS, 1999, http://www.irccyn.ec-nantes.fr/~idier/pub/idier99d.pdf.

[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), 134-139.

[24]

S. Janson, "Gaussian Hilbert Spaces,'' Cambridge University Press, Cambridge, 1997.

[25]

P. W. Kasteleyn, The statistics of dimers on a lattice I. The number of dimer arrangements on a quadratic lattice, Physica, 27 (1961), 1209-1225. doi: 10.1016/0031-8914(61)90063-5.

[26]

T. Kato and J. B. McLeod, The functional-differential equation $y'(x) = ay(\lambda x) + by(x)$, Bull. Amer. Math. Soc., 77 (1971), 891-937.

[27]

C. Kenyon, D. Randall and A. Sinclair, Approximating the number of monomer-dimer coverings of a lattice, J. Stat. Phys., 83 (1996), 637-659. doi: 10.1007/BF02183743.

[28]

V. Kozyakin, N. Kuznetsov, A. Pokrovskii and I. Vladimirov, Some problems in analysis of discretizations of continuous dynamical systems, Nonlin. Anal., Theor. Meth. Appl., 30 (1997), 767-778.

[29]

E. H. Lieb, Solution of the dimer problem by the transfer matrix method, J. Math. Phys., 8 (1967), 2339-2341.

[30]

M. Loebl, On the dimer problem and the Ising problem in finite 3-dimensional lattices, Electr. J. Combinator., 9 (2002), 1-25.

[31]

K. Mahler, On a special functional equation, J. London Math. Soc., 15 (1940), 115-123. doi: 10.1112/jlms/s1-15.2.115.

[32]

P. Malliavin, "Integration and Probability," Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4612-4202-4.

[33]

P. Malliavin, "Stochastic Analysis," Springer, Berlin, 1997.

[34]

N. F. G. Martin and J. W. England, "Mathematical Theory of Entropy," Addison-Wesley, Reading, Mass., 1981.

[35]

P.-A. Meyer, "Quantum Probability for Probabilists," 2nd ed., Springer, Berlin, 1995.

[36]

U. U. Müller, A. Schick and W. Wefelmeyer, Inference for alternating time series, in: "Recent Advances in Stochastic Modeling and Data Analysis'' (ed. C. H. Skiadas), World Scientific, Singapore, (2007), 589-596.

[37]

D. K. Pickard, A curious binary lattice process, J. Appl. Prob., 14 (1977), 717-731. doi: 10.2307/3213345.

[38]

D. K. Pickard, Unilateral Markov fields, Adv. Appl. Prob., 12 (1980), 655-671. doi: 10.2307/1426425.

[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), 2465-2487. doi: 10.1098/rspa.2000.0621.

[40]

H. Rue and L. Held, "Gaussian Markov Random Fields," Chapman & Hall, 2006.

[41]

D. Ruelle, "Thermodynamic Formalism," 2nd ed., Cambridge University Press, Cambridge, 2004.

[42]

J. J. Sakurai, "Modern Quantum Mechanics," Addison-Wesley, Reading, Mass., 1994.

[43]

A. N. Shiryaev, "Probability," 2nd ed., Springer, New York, 1995.

[44]

V. Spiridonov, Universal superpositions of coherent states and self-similar potentials, Phys. Rev. A, 52 (1995), 1909-1935. doi: 10.1103/PhysRevA.52.1909.

[45]

E. M. Tory and D. K. Pickard, Unilateral Gaussian fields, Adv. Appl. Prob., 24 (1992), 95-112. doi: 10.2307/1427731.

[46]

I. Vladimirov, "Quantized Linear Systems on Integer Lattices: Frequency-Based Approach," Center for Applied Dynamical Systems and Environmental Modeling, Deakin University, Geelong, Victoria, Australia, CADSEM Reports 96-032 (1996), 1-37; 96-033 (1996), 1-50.

[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), 251-272. doi: 10.1016/S0378-4754(00)00154-3.

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