Distribution of dynamical quantities in the contact process, random walks, and quantum spin chains in random environments

We study the distribution of dynamical quantities in various one-dimensional, disordered models the critical behavior of which is described by an infinite randomness fixed point. In the {\it disordered contact process}, the quenched survival probability $\mathcal{P}(t)$ defined in fixed random environments is found to show multi-scaling in the critical point, meaning that $\mathcal{P}(t)=t^{-\delta}$, where the (environment and time-dependent) exponent $\delta$ has a universal limit distribution when $t\to\infty$. The limit distribution is determined by the strong disorder renormalization group method analytically in the end point of a semi-infinite lattice, where it is found to be exponential, while, in the infinite system, conjectures on its limiting behaviors for small and large $\delta$, which are based on numerical results, are formulated. By the same method, the quenched survival probability in the problem of {\it random walks in random environments} is also shown to exhibit multi-scaling with an exponential limit distribution. In addition to this, the (imaginary-time) spin-spin autocorrelation function of the {\it random transverse-field Ising chain} is found to have a form similar to that of survival probability of the contact process at the level of the renormalization approach. Consequently, a relationship between the corresponding limit distributions in the two problems can be established. Finally, the distribution of the spontaneous magnetization in this model is also discussed.