Abstract
We consider a discrete polynuclear growth (PNG) process and prove a functional limit theorem for its convergence to the Airy process. This generalizes previous results by Prähofer and Spohn. The result enables us to express the F 1 GOE Tracy- Widom distribution in terms of the Airy process. We also show some results, and give a conjecture, about the transversal fluctuations in a point to line last passage percolation problem. Furthermore we discuss a rather general class of measures given by products of determinants and show that these measures have determinantal correlation functions.
Similar content being viewed by others
References
Adler, M., van Moerbeke, P.: The spectrum of coupled random matrices. Ann. Math. 149, 921–976 (1999)
Baik, J., Deift, P.A., Johansson, K.: On the distribution of the length of the longest increasing subsequence in a random permutation. J. Am. Math. Soc. 12, 1119–1178 (1999)
Baik, J., Deift, P.A., McLaughlin, K., Miller, P., Zhou, X.: Optimal tail estimates for directed last passage site percolation with geometric random variables. Adv. Theor. Math. Phys. 5, 1207–1250 (2001)
Baik, J., Rains, E.: Symmetrized random permutations. In: Random Matrix Models and Their Applications, P.M. Bleher and A.R. Its, (eds.), MSRI Publications 40, Cambridge: Cambridgen Univ. Press, 2001
Baryshnikov, Yu.: GUES and QUEUES. Probab. Theory Relat. Fields 119, 256–274 (2001)
Billingsley, P.: Convergence of Probability measures. New York: John Wiley & Sons, 1968
Borodin, A.: Biorthogonal ensembles. Nuel. Phys. B 536, 704–732 (1999)
Böttcher, A., Silberman, B.: Introduction to large truncated Toeplitz Matrices. Berlin-Heidelberg-New York: Springer, 1999
Dyson, F.J.: A Brownian-Motion Model for the eigenvalues of a Random Matrix. J. Math. Phys. 3, 1191–1198 (1962)
Eynard, B., Mehta, M.L.: Matrices coupled in a chain I: Eigenvalue correlations. J. Phys. A 31, 4449–4456 (1998)
Fisher, M.E., Stephenson, J.: Statistical Mechanics of Dimers on a plane Lattice II: Dimer Correlations and Monomers. Phys. Rev. 132, 1411–1431 (1963)
Forrester, P.J.: Exact solution of the lock step model of vicious walkers. J. Phys. A: Math. Gen. 23, 1259–1273 (1990)
Forrester, P.J., Nagao, T., Honner, G.: Correlations for the orthogonal-unitary and symplectic-unitary transitions at the soft and hard edges. Nucl. Phys. B 553, 601–643 (1999)
Fulton, W.: Young Tableaux. London Mathematical Society, Student Texts 35, Cambridge: Cambridge Univ. Press, 1997
Horn, R.A., Johnson, C.R.: Matrix Analysis. Cambridge: Cambridge University Press, 1985
Johansson, K.: Shape fluctuations and random matrices. Commun. Math. Phys. 209, 437–476 (2000)
Johansson, K.: Transversal fluctuations for increasing subsequences on the plane. Probab. Theory Relat. Fields 116, 445–456 (2000)
Johansson, K.: Discrete orthogonal polynomial ensembles and the Plancherel measure. Ann. Math. 153, 259–296 (2001)
Johansson, K.: Random growth and Random matrices. In: European Congress of Mathematics, Barcelona, Vol. I, Baset-Bosten: Birkhäuser, 2001
Johansson, K.: Universality of the local spacing distribution in certain ensembles of hermitian Wigner matrices. Commun. Math. Phys. 215, 683–705 (2001)
Johansson, K.: Non-intersecting paths, random tilings and random matrices. Probab. Theory Relat. Fields 123, 225–280 (2002)
Johansson, K.: The arctic circle boundary and the Airy process. math.PR/0306216
Kenyon, R.: Local statistics of lattice dimers. Ann. Inst. H. Poincaré, Probabilités et Statistiques, 33, 591–618 (1997)
Krug, J., Spohn, H.: Kinetic Roughening of Growing Interfaces. In: Solids far from Equilibrium: Growth, Morphology and Defects, C. Godrèche, (ed.), Cambridge: Cambridge University Press, 1992, pp. 479–582
König, W., O’Connell, N., Roch, S.: Non-colliding random walks, tandem queues and discrete orthogonal polynomial ensembles. Electron. J. Probab. 7(5), (2002)
Macêdo, A.M.S.: Europhys. Lett. 26, 641 (1994)
Mehta, M.L.: Random Matrices. 2nd ed., San Diego: Academic Press, 1991
Nagle, J.F.: Yokoi, C.S.O., Bhattacharjee, S.M.: Dimer models on anisotropic lattices. In: Phase Transitions and Critical Phenomena, Vol. 13, C. Domb, J. L. Lebowitz, (eds.), London-New York: Academic Press, 1989
Okounkov, A.: Infinite wedge and random partitions. Selecta Math. (N.S.) 7, 57–81 (2001)
Okounkov, A., Reshetikhin, N.: Correlation function of Schur process with applications to local geometry with application to local geometry of a random 3-dimensional Young diagram. math.CO/0107056
Prähofer, M., Spohn, H.: Scale invariance of the PNG droplet and the Airy process. J. Stat. Phys. 108, 1076–1106 (2002)
Sagan, B.: The Symmetric Group. Monterey, CA: Brooks/Cole Publ. Comp. 1991
Simon, B.: Trace ideals and their applications. LMS Lecture Notes Series 35, Cambridge: Cambridge University Press, 1979
Soshnikov, A.: Determinantal random point fields. Russ. Math. Surv. 55, 923–975 (2000)
Stanley, R.P.: Enumerative Combinatorics. Vol. 2, Cambridge: Cambridge University Press, 1999
Tracy, C.A., Widom, H.: Level Spacing Distributions and the Airy Kernel. Commun. Math. Phys. 159, 151–174 (1994)
Tracy, C.A., Widom, H.: Correlation Functions, Cluster Functions, and Spacing Distributions for Random Matrices. J. Stat. Phys. 92, 809–835 (1998)
Viennot, G.: Une forme géométrique de la correspondance de Robinson-Schensted. Lecture Notes in Math. 579, Berlin: Springer, 1977, pp. 29–58
Widom, H.: On Convergence of Moments for Random Young Tableaux and a Random Growth Model. Int. Math. Res. Not. 9, 455–464 (2002)
Yokoi, C.S.O., Nagle, J.F., Salinas, S.R.: Dimer Pair Correlations on the Brick Lattice. J. Stat. Phys. 44, 729–747 (1986)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by H. Spohn
Rights and permissions
About this article
Cite this article
Johansson, K. Discrete Polynuclear Growth and Determinantal Processes. Commun. Math. Phys. 242, 277–329 (2003). https://doi.org/10.1007/s00220-003-0945-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-003-0945-y