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We consider two independent and stationary measures over $\chi^\mathbb{N}$,where $\chi$ finite or countable alphabet. For each pair of $n$-strings in theproduct space we define $T_n^{(2)}$ as the length of the shortest pathconnecting one string to the other where the paths are generating by theunderlying dynamics of the measure. For ergodic measures with positive entropywe prove that, for almost every pair of realizations $(x,y)$, $T^{(2)}_n/n$concentrates in one, as $n$ diverges. Under mild extra conditions we prove alarge deviation principle. This principle is linked to a quantity that computethe similarity between the two measures that we also introduce. We furtherprove its existence and other properties. We also show that the fluctuations of$T_n^{(2)}$ converge (only) in distribution to a non-degenerated distribution.Several examples are provided for all results.

From the divergence between two measures to the shortest path between two observables