(Georgia Institute of Technology, 2024-05-07)
Deslandes, Clement
This work considers some random words combinatorial problems and their applications. The starting point of this endeavor is the following question : given two random words, ”how much do they have in common” ? Even if this question has emerged independently in various fields, including computer science, biology, linguistics, it remains mostly unsolved. Firstly, we study the asymptotic distribution of the length of the longest common and increasing
subsequences. There we consider a totally ordered alphabet with an order, say 1,...,m, and the subsequences are simply made of a block of 1’s, followed
by a block of 2’s, ... and so on (such a subsequence is increasing, but not strictly). Secondly, we deal with
the problem of the variance of the LCS. By introducing a general framework going beyond this problem, partial results in this direction are presented, and various upper and lower variance bounds are revisited in diverse settings. Lastly, we consider the Longest
Increasing Subsequences (LIS) of one random word, and the surprising connection with quantum statistics.