Transfer-matrix renormalization group method for general Markov random fields

We seek the numerical calculation of partition functions of general Markov random fields (MRFs) on an infinitely long twisted cylindrical lattice by using the transfer-matrix renormalization group (TMRG) method. The TMRG is a variant of the density-matrix renormalization group (DMRG) which automatically truncates the Hilbert space so that the properties of large systems can be precisely calculated while the dimension of the renormalized Hilbert space remains constant. We apply the TMRG to the decimation of the fundamental transfer matrix that we have proposed previously for general MRFs. Instead of the standard S••E scheme for TMRG, we propose a new E•S•E scheme and propose an alternative method for selecting the renormalized basis. Specifically, the new E•S•E scheme keeps those singular-value decomposition (SVD) components of the fundamental transfer matrix that are relevant for the Perron state and truncates the other irrelevant ones. Results for the Ising model show that our method exhibits very impressive accuracy under a rather restricted computational resource. Simulations for another four general MRFs demonstrate that our TMRG method is superior to the classical Monte Carlo method in accuracy, computational speed, and in the possibility of treating a much larger system.