Let be the n-letter word obtained by repeating a fixed word W, and let be a random n-letter word over the same alphabet. We show several results about the length of the longest common subsequence (LCS) between and ; in particular, we show that its expectation is for an efficiently-computable constant .
This is done by relating the problem to a new interacting particle system, which we dub “frog dynamics”. In this system, the particles (“frogs”) hop over one another in the order given by their labels. Stripped of the labeling, the frog dynamics reduces to a variant of the PushTASEP.
In the special case when all symbols of W are distinct, we obtain an explicit formula for the constant and a closed-form expression for the stationary distribution of the associated frog dynamics.
In addition, we propose new conjectures about the asymptotic of the LCS of a pair of random words. These conjectures are informed by computer experiments using a new heuristic algorithm to compute the LCS. Through our computations, we found periodic words that are more random-like than a random word, as measured by the LCS.
The first author was supported in part by Sloan Research Fellowship and by U.S. taxpayers through NSF CAREER Grant DMS-1555149.
The second author was supported in part by U.S. taxpayers through NSF CAREER Grant DMS-1555149.
We thank Tomasz Tkocz for discussions at the early stage of this research and for comments on a draft of this paper. We thank him additionally for the contribution of Proposition 40. We thank Alex Tiskin for pointing out the relevance of references  and . We owe the development of the frog metaphor used in this paper to a conversation with Laure Bukh. The frog symbol is from Froggy font by Vladimir Nikolic. The lily pad symbol is based on a drawing by FrauBieneMaja. We thank Zimu Xiang for pointing several typos, and two anonymous referees for valuable feedback on the earlier versions of the paper.
"Periodic words, common subsequences and frogs." Ann. Appl. Probab. 32 (2) 1295 - 1332, April 2022. https://doi.org/10.1214/21-AAP1709