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IEICE Technical Report; IEICE Tech. Rep.

Calculation method for grid-structured markov random field using corner transfer matrix renormalization group

Yoshida Tomoharu, Kazuho Watanabe, Kyoji Umemura

Published Nov 2, 2017• 0 citations
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Theory and Application of the Density-Matrix Renormalization Group

Im Fachbereich Physik

•Jan 1, 2002•0 citations
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 2nd-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.
Critical behavior of the Ising model is investigated at the center of large scale finite size systems, where the lattice is represented as the tiling of pentagons. The system is on the hyperbolic plane, and the recursive structure of the lattice makes it possible to apply the corner transfer matrix renormalization group method. From the calculated nearest neighbor spin correlation function and the spontaneous magnetization, it is concluded that the phase transition of this model is mean-field like. One parameter deformation of the corner Hamiltonian on the hyperbolic plane is discussed.