  The affine Toda field theory is studied as a 2+1-dimensional system. The
third dimension appears as the discrete space dimension, corresponding to the
simple roots in the $A_N$ affine root system, enumerated according to the
cyclic order on the $A_N$ affine Dynkin diagram. We show that there exists a
natural discretization of the affine Toda theory, where the equations of motion
are invariant with respect to permutations of all discrete coordinates. The
discrete evolution operator is constructed explicitly. The thermodynamic Bethe
ansatz of the affine Toda system is studied in the limit $L,N\to\infty$. Some
conjectures about the structure of the spectrum of the corresponding discrete
models are stated.
