  The time evolution of local operators in quantum compass models is
characterized by simplicity as it can be represented as expanding and
contracting strings of operators. Here we present an analytical solution to the
problem of growth of a local energy operator in a quantum compass model on a
Bethe lattice. We find a linear increase in time of the average operator length
and a diffusive spreading of the operator length distribution. By a moment
method we evaluate the local energy autocorrelation function that shows a
Lorentzian shape at low frequencies. Furthermore, by a stochastic method we
visualize the expansion of the string cloud.
