Dynamical barriers for the random ferromagnetic Ising model on the Cayley tree: traveling-wave solution of the real space renormalization flow

We consider the stochastic dynamics near zero temperature of the random ferromagnetic Ising model on a Cayley tree of branching ratio K. We apply the boundary real space renormalization procedure introduced in our previous work (Monthus and Garel 2013 J. Stat. Mech. P02037) in order to derive the renormalization rule for dynamical barriers. We obtain that the probability distribution P_n(B) of a dynamical barrier for a sub-tree of n generations converges for large n towards some traveling-wave P_n(B) ≃ P*(B - nv), i.e. the width of the probability distribution remains finite around an average value that grows linearly with the number n of generations. We present numerical results for the branching ratios K = 2 and K = 3. We also compute the weak-disorder expansion of the velocity v for K = 2.