Single-species reactions on a random catalytic chain

We present an exact solution for a catalytically activated annihilation A + A --> 0 reaction taking place on a one-dimensional chain in which some segments (placed at random, with mean concentration p) possess special, catalytic properties. An annihilation reaction takes place as soon as any two A particles land from the reservoir onto two vacant sites at the extremities of the catalytic segment, or when any A particle lands onto a vacant site on a catalytic segment while the site at the other extremity of this segment is already occupied by another A particle. We find that the disorder-average pressure P-(quen) per site of such a chain is given by P-(quen) = P-(Lan) + beta(-1) F, where P-(Lan) = beta(-1) ln(l + z) is the Langmuir adsorption pressure, (z being the activity and beta(-1) the temperature), while beta(-1) F is the reaction-induced contribution, which can be expressed, under appropriate change of notation, as the Lyapunov exponent for the product of 2 x 2 random matrices, obtained exactly by Derrida and Hilhorst (1983 J. Phys. A:Math. Gen. 16 2641). Explicit asymptotic formulae for the particle mean density and the compressibility are also presented.