Longest common substring for random subshifts of finite type

In this paper, we study the behaviour of the longest common substring for random subshifts of finite type (for dynamicists) or of the longest common substring for random sequences in random environments (for probabilists). We prove that, under some exponential mixing assumptions, this behaviour is linked to the R'enyi entropy of the stationary measure. We emphasize that what we establish is a quenched result.