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Solvable vertex models in statistical mechanics give rise to soliton cellular automata at q=0 in a ferromagnetic regime. By means of the crystal base theory we study a class of such automata associated with non-exceptional quantum affine algebras U′ $$_q$$ ( $$\widehat {\mathfrak{g}}$$ $$_n$$ ). Let B $$_l$$ be the crystal of the U′ $$_q$$ ( $$\widehat {\mathfrak{g}}$$ $$_n$$ )-module corresponding to the l-fold symmetric fusion of the vector representation. For any crystal of the form B = $$B_{l_1 }$$ ⊗ ...⊗ $$B_{l_N }$$ , we prove that the combinatorial R matrix B $$_M$$ ⊗B $$\widetilde \to$$ B⊗B $$_M$$ is factorized into a product of Weyl group operators in a certain domain if M is sufficiently large. It implies the factorization of certain transfer matrix at q=0, hence the time evolution in the associated cellular automata. The result generalizes the ball-moving algorithm in the box-ball systems.

We study, in the path realization, crystals for Demazure modules of affine Lie algebras of typesA(1)n,B(1)n,C(1)n,D(1)n,A(2)2n−1,A(2)2n, andD(2)n+1. We find a special sequence of affine Weyl group elements for the selected perfect crystal, and show that if the highest weight islΛ0, the Demazure crystal has a remarkably simple structure.

A new series of exactly solvable solid-on-solid models in 2D statistical mechanics is investigated. This series is a lattice analogue for the N=2 superconformal minimal models since the local state probability yields the characters of the N=2 superconformal algebra. This is the same correspondence as is observed in the Virasoro minimal series and the Andrews-Baxter-Forrester's RSOS model.

A generalization of the relation between the simple random walk on a regular lattice and the diffusion equation in a continuous space is described. In one dimension we consider a random walk of a walker with exponentially decreasing mobility with respect to time. It has an exact solution of the conditional probability, that is expressed in terms of the Gaussian polynomials, a generalization of binomial coefficients. Taking a suitable continuum limit we obtain the corresponding transport equation from the recursion relation of the discrete random walk process. The kernel of this differential equation is also directly obtained from that conditional probability by the same continuum limit.

The tropical (ultradiscrete) periodic Toda lattice is a dynamical system derived from a time-discretized version of the periodic Toda lattice through a limiting procedure called tropicalization. We propose a new formulation for this dynamical system with a representation by two-colored strips. Based on this formulation, a family of its commuting time evolutions is constructed.