Biorthonormal transfer-matrix renormalization-group method for general Markov random fields

We seek numerical calculations of partition functions of general Markov random fields (MRFs) by using the biorthonormal transfer-matrix renormalization-group (BTMRG) method. The BTMRG is a modification of the conventional TMRG, 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 dimensions of the renormalized transfer matrix remain constant. We apply the BTMRG to the decimation of the fundamental transfer matrix for general MRFs. Four binary 2^nd order MRFs are selected for numerical simulations. Results of simulations show that all four models exhibit the phase transition phenomenon. This work shows that our BTMRG method is superior to the conventional TMRG in accuracy, computational speed, and in the possibility of treating a much larger system.