Project Euclid

References

• [1] D. Aldous and P. Diaconis, Longest increasing subsequences: From patience sorting to the Baik-Deift-Johansson theorem, Bull. Amer. Math. Soc. (N.S.) 36 (1999), 413–432.
  Mathematical Reviews (MathSciNet): MR1694204
  Digital Object Identifier: doi:10.1090/S0273-0979-99-00796-X
  
• [2] J. Baik, P. Deift, and K. Johansson, On the distribution of the length of the longest increasing subsequence of random permutations, J. Amer. Math. Soc. 12 (1999), 1119–1178.
  Mathematical Reviews (MathSciNet): MR1682248
  Digital Object Identifier: doi:10.1090/S0894-0347-99-00307-0
  
• [3] M. Balázs, E. Cator, and T. Seppäläinen, Cube root fluctuations for the corner growth model associated to the exclusion process, Electron. J. Probab. 11 (2006), 1094–1132.
  Mathematical Reviews (MathSciNet): MR2268539
  
• [4] M. Balázs and T. Seppäläinen, Order of current variance and diffusivity in the asymmetric simple exclusion process, Ann. of Math. (2) 171 (2010), 1237–1265.
  Mathematical Reviews (MathSciNet): MR2630064
  Digital Object Identifier: doi:10.4007/annals.2010.171.1237
  
• [5] F. Baudoin and N. O’Connell, Exponential functionals of Brownian motion and class-one Whittaker functions, Ann. Inst. Henri Poincaré Probab. Stat. 47 (2011), 1096–1120.
  Mathematical Reviews (MathSciNet): MR2884226
  Digital Object Identifier: doi:10.1214/10-AIHP401
  Project Euclid: euclid.aihp/1317906503
  
• [6] A. Berenstein and D. Kazhdan, “Geometric and unipotent crystals” in GAFA 2000 (Tel Aviv, 1999), Geom. Funct. Anal. 2000, Birkhäuser, Basel, 2000, 188–236.
  Mathematical Reviews (MathSciNet): MR1826254
  
• [7] A. Berenstein and D. Kazhdan, “Lecture notes on geometric crystals and their combinatorial analogues” in Combinatorial Aspect of Integrable Systems, MSJ Mem. 17, Math. Soc. Japan, Tokyo, 2007.
  Mathematical Reviews (MathSciNet): MR2269125
  
• [8] A. Berenstein and A. N. Kirillov, The Robinson-Schensted-Knuth bijection, quantum matrices and piece-wise linear combinatorics, preprint, http://math.la.asu.edu/~fpsac01/PROGRAM/5.html (accessed 19 December 2013).
• [9] P. Biane, P. Bougerol, and N. O’Connell, Littelmann paths and Brownian paths, Duke Math. J. 130 (2005), 127–167.
  Mathematical Reviews (MathSciNet): MR2176549
  Digital Object Identifier: doi:10.1215/S0012-7094-05-13014-9
  Project Euclid: euclid.dmj/1131804021
  
• [10] Ph. Biane, Ph. Bougerol, N. O’Connell, Continuous crystals and Duistermaat-Heckman measure for Coxeter groups, Adv. Math. 221 (2009), 1522–1583.
  Mathematical Reviews (MathSciNet): MR2522427
  Digital Object Identifier: doi:10.1016/j.aim.2009.02.016
  
• [11] A. Borodin and I. Corwin, Macdonald processes, to appear in Probab. Theory Related Fields, preprint, arXiv:1111.4408v4 [math.PR].
  arXiv: 1111.4408v4
  Mathematical Reviews (MathSciNet): MR3152785
  Digital Object Identifier: doi:10.1007/s00440-013-0482-3
  
• [12] A. Borodin, I. Corwin, and D. Remenik, Log-Gamma polymer free energy fluctuations via a Fredholm determinant identity, Comm. Math. Phys. 324 (2013), 215–232.
  Mathematical Reviews (MathSciNet): MR3116323
  Digital Object Identifier: doi:10.1007/s00220-013-1750-x
  
• [13] A. Borodin and S. Péché, Airy kernel with two sets of parameters in directed percolation and random matrix theory, J. Stat. Phys. 132 (2008), 275–290.
  Mathematical Reviews (MathSciNet): MR2415103
  Digital Object Identifier: doi:10.1007/s10955-008-9553-8
  
• [14] P. Bougerol and T. Jeulin, Paths in Weyl chambers and random matrices, Probab. Theory Related Fields 124 (2002), 517–543.
  Mathematical Reviews (MathSciNet): MR1942321
  Digital Object Identifier: doi:10.1007/s004400200221
  
• [15] D. Bump, Automorphic Forms on $\mathrm{GL}(3,{\mathbb{R} })$, Lecture Notes in Math. 1083, Springer, Berlin, 1984.
  Mathematical Reviews (MathSciNet): MR765698
  
• [16] D. Bump, “The Rankin-Selberg method: A survey” in Number Theory, Trace Formulas, and Discrete Groups (Oslo, 1987), Academic Press, Boston, 1989, 49–109.
  Mathematical Reviews (MathSciNet): MR993311
  
• [17] E. Cator and P. Groeneboom, Second class particles and cube root asymptotics for Hammersley’s process, Ann. Probab. 34 (2006), 1273–1295.
  Mathematical Reviews (MathSciNet): MR2257647
  Digital Object Identifier: doi:10.1214/009117906000000089
  Project Euclid: euclid.aop/1158673319
  
• [18] I. Corwin, The Kardar-Parisi-Zhang equation and universality class, Random Matrices Theory Appl. 1 (2012), art. 1130001.
  Mathematical Reviews (MathSciNet): MR2930377
  Digital Object Identifier: doi:10.1142/S2010326311300014
  
• [19] I. Corwin and A. Hammond, The $H$-Brownian Gibbs property of the KPZ line ensemble, preprint, arXiv:1312.2600v1 [math.PR].
  arXiv: 1312.2600v1
  
• [20] V. Danilov and G. Koshevoy, The octahedron recurrence and RSK-correspondence, Sém. Lothar. Combin. 54A (2005/2007), Art. B54An.
  Mathematical Reviews (MathSciNet): MR2317686
  
• [21] M. Defosseux, Orbit measures, random matrix theory and interlaced determinantal processes, Ann. Inst. Henri Poincaré Probab. Stat. 46 (2010), 209–249.
  Mathematical Reviews (MathSciNet): MR2641777
  Digital Object Identifier: doi:10.1214/09-AIHP314
  Project Euclid: euclid.aihp/1267454115
  
• [22] A. B. Dieker and J. Warren, On the largest-eigenvalue process for generalized Wishart random matrices, ALEA Lat. Am. J. Probab. Math. Stat. 6 (2009), 369–376.
  Mathematical Reviews (MathSciNet): MR2557876
  
• [23] Y. Doumerc, “A note on representations of eigenvalues of classical Gaussian matrices” in Séminaire de Probabilités XXXVII, Lecture Notes in Math. 1832, Springer, Berlin, 2003, 370–384.
  Mathematical Reviews (MathSciNet): MR2053054
  
• [24] M. Draief, J. Mairesse, and N. O’Connell, Queues, stores, and tableaux, J. Appl. Probab. 42 (2005), 1145–1167.
  Mathematical Reviews (MathSciNet): MR2203829
  Digital Object Identifier: doi:10.1239/jap/1134587823
  Project Euclid: euclid.jap/1134587823
  
• [25] P. L. Ferrari and H. Spohn, “Random growth models” in The Oxford Handbook of Random Matrix Theory, Oxford Univ. Press, Oxford, 2011, 782–801.
  Mathematical Reviews (MathSciNet): MR2932658
  
• [26] P. J. Forrester and E. M. Rains, Jacobians and rank $1$ perturbations relating to unitary Hessenberg matrices, Int. Math. Res. Not. IMRN 2006, art. ID 48306.
  Mathematical Reviews (MathSciNet): MR2219210
  
• [27] A. Gerasimov, D. Lebedev, and S. Oblezin, Baxter operator and Archimedean Hecke algebra, Comm. Math. Phys. 284 (2008), 867–896.
  Mathematical Reviews (MathSciNet): MR2452597
  Digital Object Identifier: doi:10.1007/s00220-008-0547-9
  
• [28] A. Givental, “Stationary phase integrals, quantum Toda lattices, flag manifolds and the mirror conjecture” in Topics in Singularity Theory, Amer. Math. Soc. Transl. Ser. 2 180, Amer. Math. Soc., Providence, 1997, 103–115.
  Mathematical Reviews (MathSciNet): MR1767115
  
• [29] C. Greene, An extension of Schensted’s theorem, Adv. Math. 14 (1974), 254–265.
  Mathematical Reviews (MathSciNet): MR354395
  Digital Object Identifier: doi:10.1016/0001-8708(74)90031-0
  
• [30] K. Johansson, Shape fluctuations and random matrices, Comm. Math. Phys. 209 (2000), 437–476.
  Mathematical Reviews (MathSciNet): MR1737991
  Digital Object Identifier: doi:10.1007/s002200050027
  
• [31] K. Johansson, Discrete orthogonal polynomial ensembles and the Plancherel measure, Ann. of Math. (2) 153 (2001), 259–296.
  Mathematical Reviews (MathSciNet): MR1826414
  Digital Object Identifier: doi:10.2307/2661375
  
• [32] S. Kharchev and D. Lebedev, Integral representations for the eigenfunctions of quantum open and periodic Toda chains from the QISM formalism, J. Phys. A 34 (2001), 2247–2258.
  Mathematical Reviews (MathSciNet): MR1831292
  Digital Object Identifier: doi:10.1088/0305-4470/34/11/317
  
• [33] A. N. Kirillov, “Introduction to tropical combinatorics” in Physics and Combinatorics, 2000 (Nagoya, 2000), World Scientific, River Edge, N. J., 2001, 82–150.
  Mathematical Reviews (MathSciNet): MR1872253
  
• [34] W. König, N. O’Connell, and S. Roch, Non-colliding random walks, tandem queues, and discrete orthogonal polynomial ensembles, Electron. J. Probab. 7 (2002), 24 pp.
  Mathematical Reviews (MathSciNet): MR1887625
  
• [35] B. Kostant, “Quantisation and representation theory” in Representation Theory of Lie Groups (Oxford, 1977), London Math. Soc. Lecture Note Ser. 34, Cambridge Univ. Press, Cambridge, 1977, 287–316.
• [36] B. F. Logan and L. A. Shepp, A variational problem for random Young tableaux, Adv. Math. 26 (1977), 206–222.
  Mathematical Reviews (MathSciNet): MR1417317
  Digital Object Identifier: doi:10.1016/0001-8708(77)90030-5
  
• [37] I. Macdonald, Symmetric Functions and Hall Polynomials, 2nd ed., Oxford Math. Monogr., Oxford Univ. Press, New York, 1995.
  Mathematical Reviews (MathSciNet): MR1354144
  
• [38] H. Matsumoto and M. Yor, A version of Pitman’s $2M-X$ theorem for geometric Brownian motions, C. R. Math. Acad. Sci. Paris 328 (1999), 1067–1074.
  Mathematical Reviews (MathSciNet): MR1696208
  Digital Object Identifier: doi:10.1016/S0764-4442(99)80326-7
  
• [39] J. Moriarty and N. O’Connell, On the free energy of a directed polymer in a Brownian environment, Markov Process. Related Fields 13 (2007), 251–266.
  Mathematical Reviews (MathSciNet): MR2343849
  
• [40] M. Noumi and Y. Yamada, “Tropical Robinson-Schensted-Knuth correspondence and birational Weyl group actions” in Representation Theory of Algebraic Groups and Quantum Groups (Tokyo, 2001), Adv. Stud. Pure Math. 40, Math. Soc. Japan, Tokyo, 2004, 371–442.
  Mathematical Reviews (MathSciNet): MR2074600
  
• [41] N. O’Connell, “Random matrices, non-colliding processes and queues” in Séminaire de Probabilités XXXVI, Lecture Notes in Math. 1801, Springer, Berlin, 2002, 165–182.
  Mathematical Reviews (MathSciNet): MR1971584
  
• [42] N. O’Connell, Conditioned random walks and the RSK correspondence, J. Phys. A 36 (2003), 3049–3066.
  Mathematical Reviews (MathSciNet): MR1986407
  Digital Object Identifier: doi:10.1088/0305-4470/36/12/312
  
• [43] N. O’Connell, A path-transformation for random walks and the Robinson-Schensted correspondence, Trans. Amer. Math. Soc. 355, no. 9 (2003), 3669–3697.
  Mathematical Reviews (MathSciNet): MR1990168
  Digital Object Identifier: doi:10.1090/S0002-9947-03-03226-4
  
• [44] N. O’Connell, Directed polymers and the quantum Toda lattice, Ann. Probab. 40 (2012), 437–458.
  Mathematical Reviews (MathSciNet): MR2952082
  Digital Object Identifier: doi:10.1214/10-AOP632
  Project Euclid: euclid.aop/1332772711
  
• [45] N. O’Connell, T. Seppäläinen, and N. Zygouras, Geometric RSK correspondence, Whittaker functions and symmetrized random polymers, to appear in Invent. Math., preprint, arXiv:1210.5126v2 [math.PR].
  arXiv: 1210.5126v2
  
• [46] N. O’Connell and J. Warren, A multi-layer extension of the stochastic heat equation, preprint, arXiv:1104.3509v3 [math.PR].
  arXiv: 1104.3509v3
  
• [47] N. O’Connell and M. Yor, Brownian analogues of Burke’s theorem, Stochastic Process. Appl. 96 (2001), 285–304.
  Mathematical Reviews (MathSciNet): MR1865759
  Digital Object Identifier: doi:10.1016/S0304-4149(01)00119-3
  
• [48] N. O’Connell and M. Yor, A representation for non-colliding random walks, Electron. Commun. Probab. 7 (2002), 1–12.
  Mathematical Reviews (MathSciNet): MR1887169
  
• [49] A. Okounkov, Infinite wedge and random partitions, Selecta Math. (N.S.) 7 (2001), 57–81.
  Mathematical Reviews (MathSciNet): MR1856553
  Digital Object Identifier: doi:10.1007/PL00001398
  
• [50] K. Rietsch, A mirror construction for the totally nonnegative part of the Peterson variety, Nagoya Math. J. 183 (2006), 105–142.
  Mathematical Reviews (MathSciNet): MR2253887
  Project Euclid: euclid.nmj/1157490981
  
• [51] L. C. G. Rogers and J. W. Pitman, Markov functions, Ann. Probab. 9 (1981), 573–582.
  Mathematical Reviews (MathSciNet): MR624684
  Digital Object Identifier: doi:10.1214/aop/1176994363
  Project Euclid: euclid.aop/1176994363
  
• [52] T. Sasamoto and H. Spohn, The $1+1$-dimensional Kardar-Parisi-Zhang equation and its universality class, J. Stat. Mech. Theory Exp. 11 (2010), P11013 (electron.).
• [53] T. Seppäläinen, Hydrodynamic scaling, convex duality and asymptotic shapes of growth models, Markov Process. Related Fields 4 (1998), 1–26.
  Mathematical Reviews (MathSciNet): MR1625007
  
• [54] T. Seppäläinen, Exact limiting shape for a simplified model of first-passage percolation on the plane, Ann. Probab. 26 (1998), 1232–1250.
  Mathematical Reviews (MathSciNet): MR1640344
  Digital Object Identifier: doi:10.1214/aop/1022855751
  Project Euclid: euclid.aop/1022855751
  
• [55] T. Seppäläinen, Scaling for a one-dimensional directed polymer with boundary conditions, Ann. Probab. 40 (2012), 19–73.
  Mathematical Reviews (MathSciNet): MR2917766
  Digital Object Identifier: doi:10.1214/10-AOP617
  Project Euclid: euclid.aop/1325604996
  
• [56] T. Seppäläinen and B. Valko, Bounds for scaling exponents for a $1+1$ dimensional directed polymer in a Brownian environment, ALEA Lat. Am. J. Probab. Math. Stat. 7 (2010), 451–476.
  Mathematical Reviews (MathSciNet): MR2741194
  
• [57] E. Stade, Archimedean $L$-factors on $\mathrm{GL}(n)\times\mathrm{GL}(n)$ and generalized Barnes integrals, Israel J. Math. 127 (2002), 201–219.
  Mathematical Reviews (MathSciNet): MR1900699
  Digital Object Identifier: doi:10.1007/BF02784531
  
• [58] A. M. Vershik and S. Kerov, Asymptotic behavior of the Plancherel measure of the symmetric group and the limit form of Young tableaux (in Russian), Dokl. Akad. Nauk. SSSR 233, no. 6 (1977), 1024–1027; English translation in Soviet Math. Dokl. 233, no. 1-6 (1977), 527–531.
  Mathematical Reviews (MathSciNet): MR480398