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

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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

Abstract
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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 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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