zbMATH — the first resource for mathematics

Sorting networks, staircase Young tableaux, and last passage percolation. (English) Zbl 1447.05218
Summary: We present new combinatorial and probabilistic 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.
05E10 Combinatorial aspects of representation theory
60C05 Combinatorial probability
Full Text: Link
[1] O. Angel, D. Dauvergne, A. E. Holroyd, and B. Virág. “The local limit of random sorting networks”.Ann. Inst. H. Poincaré Probab. Stat.55.1 (2019), pp. 412-440.Link. · Zbl 1455.60016
[2] O. Angel, A. E. Holroyd, and D. Romik. “The oriented swap process”.Ann. Probab.37.5 (2009), pp. 1970-1998. · Zbl 1180.82125
[3] O. Angel, A. E. Holroyd, D. Romik, and B. Virág. “Random sorting networks”.Adv. Math. 215.2 (2007), pp. 839-868.Link. · Zbl 1132.60008
[4] E. Bisi, F. D. Cunden, S. Gibbons, and D. Romik.OrientedSwaps: a Mathematica package. https://www.math.ucdavis.edu/ romik/orientedswaps/. 2019.
[5] E. Bisi, N. O’Connell, and N. Zygouras. “The geometric Burge correspondence and the partition function of polymer replicas”. 2020.arXiv:2001.09145.
[6] E. Bisi and N. Zygouras.GOE andAiry2→1marginal distribution via symplectic Schur functions. Ed. by P. Friz et al. Berlin: Springer, 2019. · Zbl 1429.60015
[7] D. Dauvergne. “The Archimedean limit of random sorting networks”. Preprint. 2018.Link.
[8] D. Dauvergne and B. Virág. “Circular support in random sorting networks”.Trans. Amer. Math. Soc.373(2020), pp. 1529-1553.Link. · Zbl 1455.60020
[9] P. Edelman and C. Greene. “Balanced tableaux”.Adv. Math.63.1 (1987), pp. 42-99.Link. · Zbl 0616.05005
[10] W. Fulton.Young Tableaux: With Applications to Representation Theory and Geometry. London Mathematical Society Student Texts. Cambridge University Press, 1997.Link. · Zbl 0878.14034
[11] C. Greene. “An extension of Schensted’s theorem”.Adv. Math.14.2 (1974), pp. 254-265. Link. · Zbl 0303.05006
[12] C. Krattenthaler. “Growth diagrams, and increasing and decreasing chains in fillings of Ferrers shapes”.Adv. Appl. Math.37.3 (2006), pp. 404-431.Link. · Zbl 1108.05095
[13] D. Romik.The Surprising Mathematics of Longest Increasing Subsequences. Cambridge University Press, 2015.Link. · Zbl 1345.05003
[14] R.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.