Transfer-matrix DMRG for stochastic models: The Domany-Kinzel cellular automaton

Abstract

We apply the transfer-matrix DMRG (TMRG) to a stochastic model, the Domany-Kinzel cellular automaton, which exhibits a non-equilibrium phase transition in the directed percolation universality class. Estimates for the stochastic time evolution, phase boundaries and critical exponents can be obtained with high precision. This is possible using only modest numerical effort since the thermodynamic limit can be taken analytically in our approach. We also point out further advantages of the TMRG over other numerical approaches, such as classical DMRG or Monte-Carlo simulations.

9 pages, 9 figures, uses IOP style

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• Dynamical simulations of classical stochastic systems using matrix product states
• Ageing phenomena without detailed balance: the contact process
• Non-Hermitian Luttinger liquids and flux line pinning in planar superconductors
• On the choice of the density matrix in the stochastic TMRG
• Stochastic Light-Cone CTMRG: a new DMRG approach to stochastic models
• Construction of a matrix product stationary state from solutions of finite size system
• Quantum Scaling Approach to Nonequilibrium Models
• Quasi-stationary distributions for the Domany-Kinzel stochastic cellular automaton
• Thermodynamic properties and thermal correlation lengths of a Hubbard model with bond-charge interaction
• Tensor-network algorithm for nonequilibrium relaxation in the thermodynamic limit
• The inactive-active phase transition in the noisy additive (exclusive-or) probabilistic cellular automaton
• Biorthonormal Matrix-Product-State Analysis for Non-Hermitian Transfer-Matrix Renormalization-Group in the Thermodynamic Limit
• Tensor-Network Approaches to Counting Statistics for the Current in a Boundary-Driven Diffusive System
• Dualities for Multi-State Probabilistic Cellular Automata
• Plaquette Models, Cellular Automata, and Measurement-induced Criticality