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 df. The spectral dimension dₛ of such structures is always dₛ=1. The kinetics of the diffusion-limited reaction in such systems is shown to depend only on the spectral dimension dₛ, whereas the spatial structure of the clusters formed in the course of the reaction depends strongly on df. We analyze explicitly the cluster structure for Peano curves filling a Sierpinski gasket or the interior of a plane triangle.