Diffusion-controlled reactionA+B→0 on Peano curves

We investigate the diffusion-controlled A+B→0 reaction taking place on Peano curves, which are topologically one-dimensional unbranched chains, forming structures of arbitrary fractal dimension ${\mathit{d}}_{\mathit{f}}$. The spectral dimension ${\mathit{d}}_{\mathit{s}}$ of such structures is always ${\mathit{d}}_{\mathit{s}}$=1. The kinetics of the diffusion-limited reaction in such systems is shown to depend only on the spectral dimension ${\mathit{d}}_{\mathit{s}}$, whereas the spatial structure of the clusters formed in the course of the reaction depends strongly on ${\mathit{d}}_{\mathit{f}}$. We analyze explicitly the cluster structure for Peano curves filling a Sierpinski gasket or the interior of a plane triangle.