The oriented swap process and last passage percolation
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
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2020
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| author | Bisi, Elia Cunden, Fabio Deelan Gibbons, Shane Romik, Dan |
| author_facet | Bisi, Elia Cunden, Fabio Deelan Gibbons, Shane Romik, Dan |
| contents | We present new probabilistic and combinatorial identities relating three random processes: the oriented swap process on $n$ particles, the corner growth process, and the last passage percolation model. We prove one of the probabilistic identities, relating a random vector of last passage percolation times to its dual, using the duality between the Robinson-Schensted-Knuth and Burge correspondences. A second probabilistic identity, relating those two vectors to a vector of 'last swap times' in the oriented swap process, is conjectural. We give a computer-assisted proof of this identity for $n\le 6$ after first reformulating it as a purely combinatorial identity, and discuss its relation to the Edelman-Greene correspondence. The conjectural identity provides precise finite-$n$ and asymptotic predictions on the distribution of the absorbing time of the oriented swap process, thus conditionally solving an open problem posed by Angel, Holroyd and Romik. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2005_02043 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | The oriented swap process and last passage percolation Bisi, Elia Cunden, Fabio Deelan Gibbons, Shane Romik, Dan Probability Combinatorics We present new probabilistic and combinatorial identities relating three random processes: the oriented swap process on $n$ particles, the corner growth process, and the last passage percolation model. We prove one of the probabilistic identities, relating a random vector of last passage percolation times to its dual, using the duality between the Robinson-Schensted-Knuth and Burge correspondences. A second probabilistic identity, relating those two vectors to a vector of 'last swap times' in the oriented swap process, is conjectural. We give a computer-assisted proof of this identity for $n\le 6$ after first reformulating it as a purely combinatorial identity, and discuss its relation to the Edelman-Greene correspondence. The conjectural identity provides precise finite-$n$ and asymptotic predictions on the distribution of the absorbing time of the oriented swap process, thus conditionally solving an open problem posed by Angel, Holroyd and Romik. |
| title | The oriented swap process and last passage percolation |
| topic | Probability Combinatorics |
| url | https://arxiv.org/abs/2005.02043 |