Asymptotics of the partition function of a general Markov random field on an infinite rectangular lattice

作者：Huang Yu Kun; Yu Sung Nien^*

来源：Physica A: Statistical Mechanics and Its Applications , 2010, 389(4): 736-746.

DOI：10.1016/j.physa.2009.10.016

• Markov random field
• Partition function
• Markov chain
• Non-negative matrix
• Perron root

We investigate theoretically and numerically the asymptotics of the partition function of a general Markov random field (MRF) on an infinite rectangular lattice. We first propose the general local energy LEF)-parameterized MRF. Then we prove that the thermodynamic limit of the free energy of the MRF can be exactly characterized by the Perron root of the fundamental transfer matrix of a particular Markov additive process (MAP). This matrix possesses a special structure and many interesting properties that enable parallel computation of the Perron root and may be beneficial for deriving an analytical form of the free energy. We also develop another transfer matrix for numerical computation of the desired Perron root. Specifically, the former is a site-to-site transfer matrix on a twisted cylindrical lattice, while the latter is the one associated with a row-to-row transition on a vertical strip. Numerical results show that our methods exhibit consistent finite-size scaling behavior even for small values of the lattice width. This study reveals that the fundamental transfer matrix is an alternative direction of research on the analysis of the partition function of general MRFs within the scope of matrix algebra.

• 出版日期2010-2-15