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

Igor G. Vladimirov

doi:10.3934/dcdsb.2013.18.575

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2013, Volume 18, Issue 2: 575-600. Doi: 10.3934/dcdsb.2013.18.575

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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

Received: September 30, 2011

Published: February 2013

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Abstract

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.

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Mathematics Subject Classification: Primary: 60G60, 60G15, 60C05, 60J57, 37H15; Secondary: 37D35, 82B20, 82B31, 60J22.

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