Reaction diffusion models in one dimension with disorder

We study a large class of one-dimensional reaction diffusion models with quenched disorder using a real space renormalization group method (RSRG) which yields exact results at large time. Particles (e.g., of several species) undergo diffusion with random local bias (Sinai model) and may react upon meeting. We obtain a detailed description of the asymptotic states (i.e., attractive fixed points of the RSRG), such as the large time decay of the density of each specie, their associated universal amplitudes, and the spatial distribution of particles. We also derive the spectrum of nontrivial exponents which characterize the convergence towards the asymptotic states. For reactions which lead to several possible asymptotic states separated by unstable fixed points, we analyze the dynamical phase diagram and obtain the critical exponents characterizing the transitions. We also obtain a detailed characterization of the persistence properties for single particles as well as more complex patterns. We compute the decay exponents for the probability of no crossing of a given point by, respectively, the single particle trajectories $\left(\theta \right)$ or the thermally averaged packets $\left(\overline{\theta }\right).\mathrm{}$ The generalized persistence exponents associated to $n$ crossings are also obtained. Specifying to the process $A+\overrightarrow{A}\varnothing$ or $A$ with probabilities $(r,1\mathrm{}-r),$ we compute exactly the exponents $\delta \mathrm{}\left(r\right)\mathrm{}$ and $\psi \mathrm{}\left(r\right)\mathrm{}$ characterizing the survival up to time $t$ of a domain without any merging or with mergings, respectively, and the exponents ${\delta }_A\mathrm{}\left(r\right)\mathrm{}$ and ${\psi }_A\mathrm{}\left(r\right)\mathrm{}$ characterizing the survival up to time $t$ of a particle $A$ without any coalescence or with coalescences, respectively. $\overline{\theta },$ ψ, and $\delta$ obey hypergeometric equations and are numerically surprisingly close to pure system exponents (though associated to a completely different diffusion length). The effect of additional disorder in the reaction rates, as well as some open questions, are also discussed.