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

TL;DR

This paper connects the monomer-dimer problem in statistical mechanics to the moment Lyapunov exponents of Gaussian random fields, providing new computational methods and insights for elastic dimer-monomer systems.

Contribution

It introduces an elastic version of the monomer-dimer problem and relates its free energy to the MLE of Gaussian random fields, extending classical models and proposing recursive computation approaches.

Findings

Reduced free energy to MLE of Gaussian fields

Computed MLE for 1D Manhattan EDM systems as eigenvalues

Linked the problem to quantum harmonic oscillator operators

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 l1-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.