Abstract
We establish that the static height fluctuations of a particular growth model, the PNG droplet, converges upon proper rescaling to a limit process, which we call the Airy process A(y). The Airy process is stationary, it has continuous sample paths, its single “time” (fixed y) distribution is the Tracy–Widom distribution of the largest eigenvalue of a GUE random matrix, and the Airy process has a slow decay of correlations as y −2. Roughly the Airy process describes the last line of Dyson's Brownian motion model for random matrices. Our construction uses a multi-layer version of the PNG model, which can be analyzed through fermionic techniques. Specializing our result to a fixed value of y, one reobtains the celebrated result of Baik, Deift, and Johansson on the length of the longest increasing subsequence of a random permutation.
We’re sorry, something doesn't seem to be working properly.
Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.
REFERENCES
P. Meakin, Fractals, Scaling, and Growth Far From Equilibrium (Cambridge University Press, Cambridge, 1998).
M. Prähofer and H. Spohn, Universal distributions for growth processes in 1+1 dimensions and random matrices, Phys. Rev. Lett. 84:4882-4885 (2000).
J. Baik, P. Deift, and K. Johansson, On the distribution of the length of the longest increasing subsequence in a random permutation, J. Amer. Math. Soc. 12:1189-1178 (1999).
C. A. Tracy and H. Widom, Level-spacing distributions and the Airy kernel, Comm. Math. Phys. 159:151-174 (1994).
M. Prähofer and H. Spohn, Statistical self-similarity of one-dimensional growth processes, Physica A 279:342-352 (2000).
K. Johansson, Non-intersecting paths, random tilings and random matrices, preprint, arXiv: math.PR/0011250.
D. J. Gates and M. Westcott, Stationary states of crystal growth in three dimensions, J. Stat. Phys. 81:681-715 (1995).
M. Prähofer and H. Spohn, An exactly solved model of three-dimensional surface growth in the anisotropic KPZ regime, J. Stat. Phys. 88:999-1012 (1997).
G. Viennot, Une forme géométrique de la correspondence de Robinson-Schensted, in Combinatoire et Représentation du Groupe Symétrique, D. Foata, ed., Lecture Notes in Mathematics, Vol. 579 (Springer-Verlag, Berlin, 1977), pp. 29-58.
H. Helfgott, Edge Effects on Local Statistics in Lattice Dimers: A Study of the Aztec Diamond, B.A. thesis, Brandeis University, 1998. arXiv: math.CO/0007136.
M. Reed and B. Simon, Methods of Modern Mathematical Physics Vol. IV: Analysis of Operators (Academic Press, New York, 1978).
A. Soshnikov, Determinantal random point fields, Russ. Math. Surv. 55:923-975 (2000). arXiv: math.PR/0002099.
H. Spohn, Interacting Brownian particles: A study of Dyson's model, in Hydrodynamic Behavior and Interacting Particle Systems, G. Papanicolaou, ed. (Springer-Verlag, New York, 1987).
M. Abramowitz and I. A. Stegun (eds.), Pocketbook of Mathematical Functions (Verlag Harri Deutsch, Thun; Frankfurt/Main, 1984).
O. Bratteli and D. W. Robinson, Operator Algebras and Quantum Statistical Mechanics 2 (Springer-Verlag, New York, 1997).
K. Johansson, Discrete orthogonal polynomial ensembles and the Plancherel measure, Ann. Math. 153:259-296 (2001).
L. J. Landau, Bessel functions: Monotonicity and bounds, J. London Math. Society 61:197-215 (2000).
A. Borodin, A. Okounkov, and G. Olshanski, Asymptotics of Plancherel measures for symmetric groups, J. Amer. Math. Soc. 13:481-515 (2000).
P. J. Forrester, The spectrum edge of random matrix ensembles, Nucl. Phys. B 402:709-728 (1993).
D. C. Mattis and E. H. Lieb, Exact solution of a many-fermion system and its associated Boson field, J. Math. Phys. 6:304-312 (1965).
M. Salmhofer, Renormalization, an Introduction, Text and Monographs in Physics (Springer, Berlin, 1999).
J. Baik and E. Rains, Symmetrized random permutations, in Random Matrix Models and Their Applications, P. Bleher and A. Its, eds., MSRI Publications, Vol. 40 (Cambridge University Press, Cambridge, 2001), pp. 1-19.
O. Kallenberg, Foundations of Modern Probability (Springer-Verlag, New York, 1997).
J. Kerstan, Infinitely Divisible Point Processes (Wiley, New York, 1978).
A. Okounkov and N. Reshetikhin, Correlation function of Schur process with application to local geometry of a random 3-dimensional Young diagram, preprint, math.CO/ 0107056.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Prähofer, M., Spohn, H. Scale Invariance of the PNG Droplet and the Airy Process. Journal of Statistical Physics 108, 1071–1106 (2002). https://doi.org/10.1023/A:1019791415147
Issue Date:
DOI: https://doi.org/10.1023/A:1019791415147