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

TL;DR

This paper studies the dynamics of the random ferromagnetic Ising model on a Cayley tree, revealing a traveling-wave behavior in the distribution of dynamical barriers and providing numerical and analytical insights into their properties.

Contribution

It introduces a boundary real space renormalization approach to analyze dynamical barriers, showing convergence to a traveling-wave distribution and calculating the velocity for specific cases.

Findings

Distribution of barriers converges to a traveling wave for large generations.

The average barrier grows linearly with the number of generations.

Numerical results are provided for branching ratios K=2 and K=3, with analytical velocity expansion for K=2.

Abstract

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 (C. Monthus and T. Garel, J. Stat. Mech. P02037 (2013)) in order to derive the renormalization rule for dynamical barriers. We obtain that the probability distribution $P_n(B)$ of dynamical barrier for a subtree of $n$ generations converges for large $n$ towards some traveling-wave $P_n(B) \simeq 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.