Geodesic
Transient dynamics of two interacting random strings
Fundamentalʹnaâ i prikladnaâ matematika, Tome 2 (1996) no. 4, pp. 1029-1043.

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A finite string is just a sequence of symbols from finite alphabet. We consider a Markov chain with the state space equal to the set of all pairs of strings. Transition probabilities depend only on $d$ leftmost symbols in each string. Besides that, the jumps of the chain are bounded: the lengths of strings at subsequent moments of time cannot differ by more than some $d$. We consider the case when dynamics of Markov chain is transient, i.e. as $t\to\infty$ the lengths of both strings tend to infinity with probability 1. In this situation we prove stabilization law: the distribution of symbols close to left ends of strings tends to those of some random process. 
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     author = {A. A. Zamyatin and A. A. Yambartsev},
     title = {Transient dynamics of two interacting random strings},
     journal = {Fundamentalʹna\^a i prikladna\^a matematika},
     pages = {1029--1043},
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     volume = {2},
     number = {4},
     year = {1996},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FPM_1996_2_4_a5/}
}
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A. A. Zamyatin; A. A. Yambartsev. Transient dynamics of two interacting random strings. Fundamentalʹnaâ i prikladnaâ matematika, Tome 2 (1996) no. 4, pp. 1029-1043. http://geodesic.mathdoc.fr/item/FPM_1996_2_4_a5/