Exponents appearing in the zero temperature dynamics of one dimensional spin models.
One of the simplest problems of stochastic automata is the Glauber dynamics of ferromagnetic spin models such as Ising or Potts models. At zero temperature, if the initial condition is random, one observes a pattern of growing domains with a characteristic size which increases with time like $t^{1/2}$. In this self similar regime, one can show that the fraction of spins which never move up to time $t$ decreases like $t^{-\theta}$ where the exponent $\theta$ is non trivial and depends both on the number of states of the Potts model and on the dimension of space. This exponent can be calculated exactly in one dimension, confirming the values previously determined by numerical methods. Similar non-trivial exponents are also present in even simpler models of coarsening, where the dynamical rule is deterministic.