American Institute of Mathematical Sciences

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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

Igor G. Vladimirov 1,

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.
Keywords: pantograph equation., Gaussian random field, Monomer-dimer problem, Pickard random field, product moments, partition function, moment Lyapunov exponent.
Mathematics Subject Classification: Primary: 60G60, 60G15, 60C05, 60J57, 37H15; Secondary: 37D35, 82B20, 82B31, 60J2.
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), 1129-1156. doi: 10.1239/aap/1231340167.  Google Scholar

[2]

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

[3]

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

[4]

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

[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.  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), 2901-2916. 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), 408-425. 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), 1-26.  Google Scholar

[9]

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

[10]

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

[11]

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

[12]

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

[13]

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

[14]

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

[15]

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

[16]

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

[17]

R. M. Gray, "Entropy and Information Theory," Springer-Verlag, New York, 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), 4753-4760.  Google Scholar

[19]

R. A. Horn and C. R. Johnson, "Matrix Analysis," Cambridge University Press, New York, 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), 341-364.  Google Scholar

[21]

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

[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. 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), 134-139. Google Scholar

[24]

S. Janson, "Gaussian Hilbert Spaces,'' Cambridge University Press, Cambridge, 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), 1209-1225. 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), 891-937.  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), 637-659. 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., Theor. Meth. Appl., 30 (1997), 767-778.  Google Scholar

[29]

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

[30]

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

[31]

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

[32]

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

[33]

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

[34]

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

[35]

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

[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. Google Scholar

[37]

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

[38]

D. K. Pickard, Unilateral Markov fields, Adv. Appl. Prob., 12 (1980), 655-671. 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), 2465-2487. 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., Cambridge University Press, Cambridge, 2004.  Google Scholar

[42]

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

[43]

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

[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.  Google Scholar

[45]

E. M. Tory and D. K. Pickard, Unilateral Gaussian fields, Adv. Appl. Prob., 24 (1992), 95-112. 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, Deakin University, Geelong, Victoria, Australia, CADSEM Reports 96-032 (1996), 1-37; 96-033 (1996), 1-50. 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), 251-272. 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), 1129-1156. doi: 10.1239/aap/1231340167.  Google Scholar

[2]

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

[3]

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

[4]

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

[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.  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), 2901-2916. 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), 408-425. 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), 1-26.  Google Scholar

[9]

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

[10]

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

[11]

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

[12]

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

[13]

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

[14]

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

[15]

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

[16]

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

[17]

R. M. Gray, "Entropy and Information Theory," Springer-Verlag, New York, 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), 4753-4760.  Google Scholar

[19]

R. A. Horn and C. R. Johnson, "Matrix Analysis," Cambridge University Press, New York, 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), 341-364.  Google Scholar

[21]

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

[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. 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), 134-139. Google Scholar

[24]

S. Janson, "Gaussian Hilbert Spaces,'' Cambridge University Press, Cambridge, 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), 1209-1225. 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), 891-937.  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), 637-659. 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., Theor. Meth. Appl., 30 (1997), 767-778.  Google Scholar

[29]

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

[30]

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

[31]

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

[32]

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

[33]

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

[34]

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

[35]

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

[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. Google Scholar

[37]

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

[38]

D. K. Pickard, Unilateral Markov fields, Adv. Appl. Prob., 12 (1980), 655-671. 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), 2465-2487. 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., Cambridge University Press, Cambridge, 2004.  Google Scholar

[42]

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

[43]

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

[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.  Google Scholar

[45]

E. M. Tory and D. K. Pickard, Unilateral Gaussian fields, Adv. Appl. Prob., 24 (1992), 95-112. 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, Deakin University, Geelong, Victoria, Australia, CADSEM Reports 96-032 (1996), 1-37; 96-033 (1996), 1-50. 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), 251-272. doi: 10.1016/S0378-4754(00)00154-3.  Google Scholar

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