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arXiv · 1901.05235

Large deviations of the length of the longest increasing subsequence of random permutations and random walks

Jörn Börjes·Hendrik Schawe·Alexander K. Hartmann
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Abstract

We study numerically the distributions of the length $L$ of the longest increasing subsequence (LIS) for the two cases of random permutations and of one-dimensional random walks. Using sophisticated large-deviation algorithms, we are able to obtain very large parts of the distribution, especially also covering probabilities smaller than $P(L) = 10^{-1000}$. This enables us to verify for the length of the LIS of random permutations the analytically known asymptotics of the rate function and even the whole Tracy-Widom distribution, to which we observe a rather fast convergence in the larger than typical part. For the length $L$ of LIS of random walks, where no analytical results are known to us, we test a proposed scaling law and observe convergence of the tails into a collapse for increasing system size. Further, we obtain estimates for the leading order behavior of the rate functions of both tails.

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Jörn Börjes, Hendrik Schawe, Alexander K. Hartmann. 2019-01-16. Large deviations of the length of the longest increasing subsequence of random permutations and random walks. https://doi.org/10.1103/physreve.99.042104

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