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

Juhász, Róbert (2014) Distribution of dynamical quantities in the contact process, random walks, and quantum spin chains in random environments. PHYSICAL REVIEW E - STATISTICAL, NONLINEAR AND SOFT MATTER PHYSICS (2001-2015), 89 (3). ISSN 1539-3755

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Abstract

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 P(t) defined in fixed random environments is found to show multi-scaling 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 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.

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