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 disordered contact process, the survival probability P(t) is found to show multiscaling in the critical point, meaning that P(t)=t-δ, where the (environment and time-dependent) exponent δ has a universal limit distribution when t→∞. 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 δ, which are based on numerical results, are formulated. By the same method, the survival probability in the problem of random walks in random environments is also shown to exhibit multiscaling with an exponential limit distribution. In addition to this, the (imaginary-time) spin-spin autocorrelation function of the 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.