Tomaž Prosen, Reversible cellular automata: Minimal models of solvable statistical mechanics

I will discuss reversible cellular automata in 1+1 dimensions which describe deterministic interacting particle systems, and present several nontrivial examples that show certain aspects of integrability. Most notable is the reversible Rule 54, for which one can obtain exact matrix product expressions for probability state vectors in various setups, ranging from nonequilibrium steady state of the system coupled to markovian stochastic boundaries (and completely diagonalizing the corresponding markov chain) to time-dependent statistical ensembles.

I will present also two deformations which turn the model into either quantum or stochastic cellular automaton. Both deformations, being parametrixed by an arbitrary unitary or stochastic 2x2 matrix, respectively, exhibit some features of integrability. The later (stochastic version) can be interpeted as a lattice discretization of deformed polynuclear growth model. Depending on the remaining time, I will discuss the construction of non-trivial conservation laws of the quantum deformation and/or the matrix product form of the steady state of the boundary driven stochastic deformation.