Random plane partitions and corner distributions
Algebraic Combinatorics, Volume 4 (2021) no. 4, pp. 599-617.

We explore some probabilistic applications arising in connections with K-theoretic symmetric functions. For instance, we determine certain corner distributions of random lozenge tilings and plane partitions. We also introduce some distributions that are naturally related to the corner growth model. Our main tools are dual symmetric Grothendieck polynomials and normalized Schur functions.

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DOI: https://doi.org/10.5802/alco.171
Classification: 60K35,  60C05,  05E05
Keywords: Random plane partitions, lozenge tilings, dual Grothendieck polynomials.
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Yeliussizov, Damir. Random plane partitions and corner distributions. Algebraic Combinatorics, Volume 4 (2021) no. 4, pp. 599-617. doi : 10.5802/alco.171. https://alco.centre-mersenne.org/articles/10.5802/alco.171/

[1] Amanov, Alimzhan; Yeliussizov, Damir MacMahon’s statistics on higher-dimensional partitions (2020) (https://arxiv.org/abs/2009.00592)

[2] Baryshnikov, Yuliy GUEs and queues, Probab. Theory Related Fields, Volume 119 (2001) no. 2, pp. 256-274 | Article | MR 1818248 | Zbl 0980.60042

[3] Borodin, Alexei; Ferrari, Patrik L. Anisotropic growth of random surfaces in 2+1 dimensions, Comm. Math. Phys., Volume 325 (2014) no. 2, pp. 603-684 | Article | MR 3148098 | Zbl 1303.82015

[4] Borodin, Alexei; Okounkov, Andrei A Fredholm determinant formula for Toeplitz determinants, Integral Equations Operator Theory, Volume 37 (2000) no. 4, pp. 386-396 | Article | MR 1780118 | Zbl 0970.47014

[5] Borodin, Alexei; Petrov, Leonid Integrable probability: From representation theory to Macdonald processes, Probab. Surv., Volume 11 (2014), pp. 1-58 | Article | MR 3178541 | Zbl 1295.82023

[6] Borodin, Alexei; Petrov, Leonid Nearest neighbor Markov dynamics on Macdonald processes, Adv. Math., Volume 300 (2016), pp. 71-155 | Article | MR 3534830 | Zbl 1356.60161

[7] Buch, Anders S. A Littlewood–Richardson rule for the K-theory of Grassmannians, Acta Math., Volume 189 (2002) no. 1, pp. 37-78 | Article | MR 1946917 | Zbl 1090.14015

[8] Buch, Anders S. Combinatorial K-theory, Topics in cohomological studies of algebraic varieties (Trends Math.), Birkhäuser, Basel, 2005, pp. 87-103 | Article | MR 2143073

[9] Cohn, Henry; Kenyon, Richard; Propp, James A variational principle for domino tilings, J. Amer. Math. Soc., Volume 14 (2001) no. 2, pp. 297-346 | Article | MR 1815214 | Zbl 1037.82016

[10] Cohn, Henry; Larsen, Michael; Propp, James The Shape of a Typical Boxed Plane Partition, New York J. Math., Volume 4 (1998), pp. 137-165 | MR 1641839 | Zbl 0908.60083

[11] Glynn, Peter W.; Whitt, Ward Departures from many queues in series, Ann. Appl. Probab., Volume 1 (1991) no. 4, pp. 546-572 | MR 1129774 | Zbl 0749.60090

[12] Gorin, Vadim; Panova, Greta Asymptotics of symmetric polynomials with applications to statistical mechanics and representation theory, Ann. Probab., Volume 43 (2015) no. 6, pp. 3052-3132 | Article | MR 3433577 | Zbl 1390.05240

[13] Ikeda, Takeshi; Naruse, Hiroshi K-theoretic analogues of factorial Schur P- and Q-functions, Adv. Math., Volume 243 (2013), pp. 22-66 | Article | MR 3062739 | Zbl 1278.05240

[14] Johansson, Kurt Shape fluctuations and random matrices, Comm. Math. Phys., Volume 209 (2000) no. 2, pp. 437-476 | Article | MR 1737991 | Zbl 0969.15008

[15] Johansson, Kurt Random growth and random matrices, European Congress of Mathematics, Vol. I (Barcelona, 2000) (Progr. Math.), Volume 201 (2001), pp. 445-456 | Article | MR 1905334 | Zbl 1030.60094

[16] Johansson, Kurt Non-intersecting paths, random tilings and random matrices, Probab. Theory Related Fields, Volume 123 (2002) no. 2, pp. 225-280 | Article | MR 1900323 | Zbl 1008.60019

[17] Johansson, Kurt; Nordenstam, Eric Eigenvalues of GUE minors, Electron. J. Probab., Volume 11 (2006) no. 50, pp. 1342-1371 | Article | MR 2268547 | Zbl 1127.60047

[18] Kenyon, Richard Lectures on dimers, Statistical mechanics (IAS/Park City Math. Ser.), Volume 16 (2009), pp. 191-230 | Article | MR 2523460 | Zbl 1180.82001

[19] Lam, Thomas; Pylyavskyy, Pavlo Combinatorial Hopf algebras and K-homology of Grassmanians, Int. Math. Res. Not. IMRN (2007) no. 24, Paper no. rnm125, 48 pages | Article | MR 2377012 | Zbl 1134.16017

[20] Lenart, Cristian Combinatorial aspects of the K-theory of Grassmannians, Ann. Comb., Volume 4 (2000) no. 1, pp. 67-82 | Article | MR 1763950 | Zbl 0958.05128

[21] Macdonald, Ian G. Symmetric functions and Hall polynomials, Oxford University Press, 1998 | Zbl 0899.05068

[22] Motegi, Kohei; Sakai, Kazumitsu Vertex models, TASEP and Grothendieck polynomials, J. Phys. A, Volume 46 (2013) no. 35, Paper no. 355201, 26 pages | Article | MR 3100873 | Zbl 1278.82042

[23] Motegi, Kohei; Scrimshaw, Travis Refined dual Grothendieck polynomials, integrability, and the Schur measure (2020) (https://arxiv.org/abs/2012.15011)

[24] Okounkov, Andrei Infinite wedge and random partitions, Selecta Math. (N.S.), Volume 7 (2001) no. 1, pp. 57-81 | Article | MR 1856553 | Zbl 0986.05102

[25] Okounkov, Andrei; Reshetikhin, Nikolai Correlation function of Schur process with application to local geometry of a random 3-dimensional Young diagram, J. Amer. Math. Soc., Volume 16 (2003) no. 3, pp. 581-603 | Article | MR 1969205 | Zbl 1009.05134

[26] Petrov, Leonid Asymptotics of random lozenge tilings via Gelʼfand–Tsetlin schemes, Probab. Theory Related Fields, Volume 160 (2014) no. 3-4, pp. 429-487 | Article | MR 3278913 | Zbl 1315.60013

[27] Romik, Dan The surprising mathematics of longest increasing subsequences, Institute of Mathematical Statistics Textbooks, 4, Cambridge University Press, New York, 2015, xi+353 pages | MR 3468738 | Zbl 1345.05003

[28] Seppäläinen, Timo Lecture notes on the corner growth model (2009) (Unpublished notes)

[29] Shimozono, Mark; Zabrocki, Mike Stable Grothendieck symmetric functions and Ω-calculus (2003) (preprint)

[30] Soshnikov, Alexander Determinantal random point fields, Russ. Math. Surv., Volume 55 (2000) no. 5, pp. 923-975 | Article | MR 1799012 | Zbl 0991.60038

[31] Stanley, Richard P. Enumerative combinatorics. Vol. 2, Cambridge Studies in Advanced Mathematics, 62, Cambridge University Press, Cambridge, 1999, xii+581 pages | Article | MR 1676282 | Zbl 0928.05001

[32] Thomas, Hugh; Yong, Alexander Longest increasing subsequences, Plancherel-type measure and the Hecke insertion algorithm, Adv. in Appl. Math., Volume 46 (2011) no. 1-4, pp. 610-642 | Article | MR 2794040 | Zbl 1227.05262

[33] Tracy, Craig A.; Widom, Harold Level-spacing distributions and the Airy kernel, Comm. Math. Phys., Volume 159 (1994) no. 1, pp. 151-174 | Article | MR 1257246 | Zbl 0789.35152

[34] Vakil, Ravi A geometric Littlewood–Richardson rule, Ann. of Math. (2), Volume 164 (2006) no. 2, pp. 371-421 | Article | MR 2247964 | Zbl 1163.05337

[35] Yeliussizov, Damir Duality and deformations of stable Grothendieck polynomials, J. Algebraic Combin., Volume 45 (2017) no. 1, pp. 295-344 | Article | MR 3591379 | Zbl 1355.05263

[36] Yeliussizov, Damir Symmetric Grothendieck polynomials, skew Cauchy identities, and dual filtered Young graphs, J. Combin. Theory Ser. A, Volume 161 (2019), pp. 453-485 | Article | MR 3861787 | Zbl 1400.05264

[37] Yeliussizov, Damir Dual Grothendieck polynomials via last-passage percolation, C. R. Math. Acad. Sci. Paris, Volume 358 (2020) no. 4, pp. 497-503 | Article | MR 4134260 | Zbl 1444.05145

[38] Yeliussizov, Damir Positive specializations of symmetric Grothendieck polynomials, Adv. Math., Volume 363 (2020), p. 107000, 35 | Article | MR 4054055 | Zbl 1432.05119

[39] Yeliussizov, Damir Enumeration of plane partitions by descents, J. Combin. Theory Ser. A, Volume 178 (2021), Paper no. 105367, 18 pages | Article | MR 4179056 | Zbl 07304671

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