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Limit Theorems for Height Fluctuations in a Class of Discrete Space and Time Growth Models

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Abstract

We introduce a class of one-dimensional discrete space-discrete time stochastic growth models described by a height function ht(x) with corner initialization. We prove, with one exception, that the limiting distribution function of ht(x) (suitably centered and normalized) equals a Fredholm determinant previously encountered in random matrix theory. In particular, in the universal regime of large x and large t the limiting distribution is the Fredholm determinant with Airy kernel. In the exceptional case, called the critical regime, the limiting distribution seems not to have previously occurred. The proofs use the dual RSK algorithm, Gessel's theorem, the Borodin–Okounkov identity and a novel, rigorous saddle point analysis. In the fixed x, large t regime, we find a Brownian motion representation. This model is equilvalent to the Seppäläinen–Johansson model. Hence some of our results are not new, but the proofs are.

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REFERENCES

  1. M. Adler and P. van Moerbeke, Integrals over classical groups, random permutations, Toda and Toeplitz lattices, preprint (arXiv: math. CO/9912143).

  2. D. Aldous and P. Diaconis, Hammersley's interacting particle process and longest increasing subsequences, Probab. Theory Related Fields 103:199-213 (1995).

    Google Scholar 

  3. D. Aldous and P. Diaconis, Longest increasing subsequences: From patience sorting to the Baik-Deift-Johansson theorem, Bull. Amer. Math. Soc. 36:413-432 (1999).

    Google Scholar 

  4. Z. D. Bai and Y. Q. Yin, Necessary and sufficient conditions for almost sure convergence of the largest eigenvalue of a Wigner matrix, Ann. Prob. 16:1729-1741 (1988).

    Google Scholar 

  5. 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. 12:1119-1178 (1999).

    Google Scholar 

  6. J. Baik and E. M. Rains, The asymptotics of monotone subsequences of involutions, preprint (arXiv: math. CO/9905084).

  7. J. Baik and E. M. Rains, Symmetrized random permutations, preprint (arXiv: math. CO/9910019).

  8. J. Baik and E. Rains, Limiting distributions for a polynuclear growth model with external sources, J. Stat. Phys. 100:523-541 (2000).

    Google Scholar 

  9. E. Basor and H. Widom, On a Toeplitz determinant identity of Borodin and Okounkov, Int. Eqns. Oper. Th. 37:397-401 (2000).

    Google Scholar 

  10. A. Borodin and A. Okounkov, A Fredholm determinant formula for Toeplitz determinants, Int. Eqns. Oper. Th. 37:386-396 (2000).

    Google Scholar 

  11. A. Borodin, A. Okounkov, and G. Olshanski, Asymptotics of Plancherel measures for symmetric groups, J. Amer. Math. Soc. 13:481-515 (2000).

    Google Scholar 

  12. C. Chester, B. Friedman, and F. Ursell, An extension of the method of steepest descents, Proc. Cambridge Philos. Soc. 53:599-611 (1957).

    Google Scholar 

  13. J. T. Cox, A. Gandolfi, P. S. Griffin, and H. Kesten, Greedy lattice animals I: Upper bounds, Ann. Appl. Prob. 3:1151-1169 (1993).

    Google Scholar 

  14. P. A. Deift, Integrable operators, in Differential Operators and Spectral Theory: M. Sh. Birman's 70th Anniversary Collection, V. Buslaev, M. Solomyak, and D. Yafaev, eds., American Mathematical Society Translations, Ser. 2, v. 189 (AMS, Providence, RI, 1999).

    Google Scholar 

  15. P. A. Deift, A. R. Its, and X. Zhou, A Riemann-Hilbert approach to asymtptotic problems arising in the theory of random matrix models, and also in the theory of integrable statistical mechanics, Ann. of Math. 146:149-235 (1997).

    Google Scholar 

  16. P. A. Deift and X. Zhou, A steepest descent method for oscillatory Riemann-Hilbert problems: Asymptotics for the MKdV equation, Ann. Math. 137:295-368 (1993).

    Google Scholar 

  17. R. Durrett, Probability: Theory and Examples, 2nd ed. (Wadsworth Publ. Co., Belmont, 1996).

    Google Scholar 

  18. I. M. Gessel, Symmetric functions and P-recursiveness, J. Comb. Theory, Ser. A 53: 257-285 (1990).

    Google Scholar 

  19. D. J. Grabiner, Brownian motion in a Weyl chamber, non-colliding particles, and random matrices, Ann. Inst. H. Poincaré Probab. Statist. 35:177-204 (1999).

    Google Scholar 

  20. J. Gravner, Cellular automata models of ring dynamics, Int. J. Mod. Phys. C 7:863-871 (1996).

    Google Scholar 

  21. J. Gravner, Recurrent ring dynamics in two-dimensional excitable cellular automata, J. Appl. Prob. 36:1-20 (1999).

    Google Scholar 

  22. J. Gravner and D. Griffeath, Cellular automaton growth on Z2: Theorems, examples, and problems, Adv. Appl. Math. 21:241-304 (1998).

    Google Scholar 

  23. D. Griffeath, Self-organization of random cellular automata: four snapshots, in Probability and Phase Transitions, NATO Adv. Sci. Inst. Ser. C: Math. and Phys. Sci., Vol. 420, G. Grimmett, ed. (Kluwer Academic Publishers, Dordrecht, 1994), pp. 49-67.

    Google Scholar 

  24. D. Griffeath, Primordial Soup Kitchen, http://psoup.math.wisc.edu/kitchen.html.

  25. P.W. Glynn and W. Whitt, Departure from many queues in series, Ann. Appl. Prob. 1:546-572 (1991).

    Google Scholar 

  26. A. Its, C. A. Tracy, and H. Widom, Random words, Toeplitz determinants and integrable systems. I, preprint (arXiv: math. CO/9909169).

  27. A. Its, C. A. Tracy, and H. Widom, Random words, Toeplitz determinants and integrable systems. II, preprint (arXiv: nlin.SI/0004018).

  28. K. Johansson, Shape fluctuations and random matrices, Commun. Math. Phys. 209: 437-476 (2000).

    Google Scholar 

  29. K. Johansson, Discrete orthogonal polynomial ensembles and the Plancherel measure, preprint (arXiv: math. CO/9906120).

  30. D.E. Knuth, Permutations, matrices and generalized Young tableaux, Pacific J. Math. 34:709-727 (1970).

    Google Scholar 

  31. G. Kuperberg, Random words, quantum statistics, central limits, random matrices, preprint (arXiv: math. PR/9909104).

  32. M. Lässig, On growth, disorder and field theory, J. Phys.: Condens. Matter 10:9905-9950 (1998).

    Google Scholar 

  33. M. L. Mehta, Random Matrices, 2nd ed. (Academic Press, San Diego, 1991).

    Google Scholar 

  34. A. Okounkov, Random matrices and random permutations, preprint (arXiv: math.CO/9903176).

  35. M. Prähofer and H. Spohn, Statistical self-similarity of one-dimensional growth processes, Physica A 279:342-352 (2000).

    Google Scholar 

  36. M. Prähofer and H. Spohn, Universal distributions for growth processes in 1+1 dimensions and random matrices, Phys. Rev. Lett. 84:4882-4885 (2000).

    Google Scholar 

  37. T. Seppäläinen, A scaling limit for queues in series, Ann. Appl. Prob. 7:855-872 (1997).

    Google Scholar 

  38. T. Seppäläinen, Exact limiting shape for a simplified model of first-passage percolation in the plane, Ann. Prob. 26:1232-1250 (1998).

    Google Scholar 

  39. R. P. Stanley, Enumerative Combinatorics, Vol. 2 (Cambridge University Press, 1999).

  40. C. A. Tracy and H. Widom, Level-spacing distributions and the Airy kernel, Commun. Math. Phys. 159:151-174 (1994).

    Google Scholar 

  41. C. A. Tracy and H. Widom, On orthogonal and symplectic ensembles, Commun. Math. Phys. 177:727-754 (1996).

    Google Scholar 

  42. C. A. Tracy and H. Widom, Correlation functions, cluster functions and spacing distributions for random matrices, J. Stat. Phys. 92:809-835 (1998).

    Google Scholar 

  43. C. A. Tracy and H. Widom, Random unitary matrices, permutations and Painlevé, Commun. Math. Phys. 207:665-685 (1999).

    Google Scholar 

  44. C. A. Tracy and H. Widom, On the distributions of the lengths of the longest monotone subsequences in random words, preprint (arXiv: math. CO/9904042).

  45. P. van Moerbeke, Integrable lattices: random matrices and random permutations, preprint (arXiv: math. CO/0010135).

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Authors and Affiliations

  1. Department of Mathematics, University of California, Davis, California, 95616

    Janko Gravner

  2. Department of Mathematics, Institute of Theoretical Dynamics, University of California, Davis, California, 95616

    Craig A. Tracy

  3. Department of Mathematics, University of California, Santa Cruz, California, 95064

    Harold Widom

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  1. Janko Gravner
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  2. Craig A. Tracy
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Gravner, J., Tracy, C.A. & Widom, H. Limit Theorems for Height Fluctuations in a Class of Discrete Space and Time Growth Models. Journal of Statistical Physics 102, 1085–1132 (2001). https://doi.org/10.1023/A:1004879725949

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