Abstract
We consider the multi-point equal time height fluctuations of the one-dimensional polynuclear growth model in half-space. For special values of the nucleation rate at the origin, the multi-layer version of the model is reduced to a process with a determinantal weight, for which the asymptotics can be analyzed. In the scaling limit, the fluctuations near the origin are shown to be equivalent to those of the largest eigenvalue of the orthogonal/symplectic to unitary transition ensemble at soft edge in random matrix theory.
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REFERENCES
J. Krug and H. Spohn, Kinetic roughening of growing interfaces, in Solids Far from Equilibrium: Growth, Morphology, and Defects, C. Godrèche, ed. (1992), pp. 479–582.
P. Meakin, Fractals, Scaling, and Growth Far from Equilibrium (Cambridge, 1998).
M. Kardar, G. Parisi, and Y. C. Zhang, Dynamic scaling of growing interfaces, Phys. Rev. Lett. 56:889–892 (1986).
L.-H. Gwa and H. Spohn, Six-vertex model, roughened surfaces, and an asymmetric spin Hamiltonian, Phys. Rev. Lett. 68:725–728 (1992).
D. Kim, Bethe ansatz solution for crossover scaling functions of the asymmetric XXZ chain and the Kardar-Parisi-Zhang-type growth model, Phys. Rev. E 52:3512–3524 (1995).
K. Johansson, Shape fluctuations and random matrices, Commun. Math. Phys. 209: 437–476 (2000).
M. Prähofer and H. Spohn, Universal distributions for growth processes in 1+1 dimensions and random matrices, Phys. Rev. Lett 84:4882–4885 (2000).
M. Prähofer and H. Spohn, Statistical self-similarity of one-dimensional growth processes, Physica A 279:342–352 (2000).
J. Baik and E. M. Rains, Limiting distributions for a polynuclear growth model with external sources, J. Stat. Phys 100:523–541 (2000).
J. Gravner, C. A. Tracy, and H. Widom, Limit theorems for height fluctuations in a class of discrete space and time growth models, J. Stat. Phys. 102:1085–1132 (2001).
J. Gravner, C. A. Tracy, and H. Widom, A growth model in a random environment, Ann. Probab. 30:1340–1369 (2002).
J. Gravner, C. A. Tracy, and H. Widom, Fluctuations in the composite regime of a disordered growth model, Commun. Math. Phys. 229:433–458 (2002).
M. Prähofer and H. Spohn, Current fluctuations for the totally asymmetric simple exclusion process, in In and Out of Equilibrium, V. Sidoravicius, ed., Progress in Probability, Vol. 51 (2002), pp. 185–204.
M. Prähofer and H. Spohn, Scale invariance of the PNG droplet and the Airy process, J. Stat. Phys. 108:1071–1106 (2002).
M. Prähofer and H. Spohn, Exact scaling functions for one-dimensional stationary KPZ growth, cond-mat/0212519.
J. Baik, P. A. Deift, and K. Johansson, On the distribution of the length of the longest increasing subsequence in a random permutation, J. Amer. Math. Soc. 12:1119–1178 (1999).
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).
K. Johansson, Non-intersecting paths, random tilings, and random matrices, Probab. Theory Related Fields 123:225–280 (2002).
K. Johansson, Discrete polynuclear growth and determinantal processes, Commun. Math. Phys. 242:277–329 (2003).
C. A. Tracy and H. Widom, Level-spacing distributions and the Airy kernel, Commun. Math. Phys. 159:151–174 (1994).
M. L. Mehta, Random Matrices, 2nd ed. (Academic, 1991).
C. A. Tracy and H. Widom, On orthogonal and symplectic matrix ensembles, Commun. Math. Phys. 177:727–754 (1996).
J. Baik and E. M. Rains, Algebraic aspects of increasing subsequences, Duke Math. J. 109:1–65 (2001).
J. Baik and E. M. Rains, The asymptotics of monotone subsequences of involutions, Duke Math. J. 109:205–281 (2001).
J. Baik and E. M. Rains, Symmetrized random permutations, in Random Matrix Models and Their Applications, P. M. Bleher and A. R. Its, eds. (2001), pp. 1–29.
F. J. Dyson, A Brownian-motion model for the eigenvalues of a random matrix, J. Math. Phys 3:1191–1198 (1962).
K. Johansson, Discrete orthogonal polynomial ensembles and the Plancherel measure, Ann. Math. 153:259–296 (2001).
P. J. Forrester, T. Nagao, and G. Honner, Correlations for the orthogonal-unitary and symplectic transitions at the hard and soft edges, Nucl. Phys. B 553:601–643 (1999).
B. Lindström, On the vector representations of induced matroids, Bull. London Math. Soc. 5:85–90 (1973).
I. Gessel and G. Viennot, Binomial determinants, paths, and hook length formulae, Adv. Math. 58:300–321 (1985).
I. Gessel and G. Viennot, Determinants, paths, and plane partitions, preprint (1989).
J. R. Stembridge, Nonintersecting paths, pfaffians, and plane partitions, Adv. Math. 83: 96–131 (1990).
S. Karlin and L. McGregor, Coincidence properties of birth and death processes, Pacific J. 9:1109–1140 (1959).
S. Karlin and L. McGregor, Coincidence probabilities, Pacific J. 9:1141–1164 (1959).
J. Baik, Random vicious walks and random matrices, Comm. Pure Appl. Math. 53: 1385–1410 (2000).
T. Nagao and P. J. Forrester, Vicious random walkers and a discretization of Gaussian random matrix ensembles, Nucl. Phys. B 620:551–565 (2002).
T. Nagao, M. Katori, and H. Tanemura, Dynamical correlations among vicious random walkers, Phys. Lett. A 307:29–35 (2003).
C. A. Tracy and H. Widom, Correlation functions, cluster functions, and spacing distributions for random matrices, J. Stat. Phys. 92:809–835 (1998).
N. G. de Bruijn, On some multiple integrals involving determinants, J. Indian Math. Soc. 19:133–151 (1955).
B. M. McCoy and T. T. Wu, The Two-Dimensional Ising Model (Harvard University Press, 1973).
H. Rost, Non-equilibrium behavior of a many particle process: Density profile and local equilibria, Z. Warsch. Verw. Gebiete 58:41–53 (1981).
A. M. S. Macâedo, Universal parametric correlations at the soft edge of spectrum of random matrix ensembles, Europhys. Lett. 26:641–646 (1994).
T. Nagao and P. J. Forrester, Multilevel dynamical correlation functions for Dyson's Brownian motion model of random matrices, Phys. Lett. A 247:42–46 (1998).
C. A. Tracy and H. Widom, System of differential equations for the Airy process, Electron. Comm. Probab. 8:93–98 (2003).
C. A. Tracy and H. Widom, Differential equations for Dyson processes, math.PR/ 0309082.
M. Adler and P. van Moerbeke, A PDE for the joint distributions of the Airy process, math.PR/0302329.
M. Katori and T. Tanemura, in preparation.
T. Nagao and M. Wadati, Correlation functions of random matrix ensembles related to classical orthogonal polynomials, J. Phys. Soc. Japan 60:3298–3322 (1991).
T. Nagao and M. Wadati, Correlation functions of random matrix ensembles related to classical orthogonal polynomials II, J. Phys. Soc. Japan 61:78–88 (1992).
T. Nagao and M. Wadati, Correlation functions of random matrix ensembles related to classical orthogonal polynomials III, J. Phys. Soc. Japan 61:1910–1918 (1992).
W. H. Burge, Four correspondences between graphs and generalized Young tableaux, J. Combin. Theory Ser. A 17:12–30 (1974).
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Sasamoto, T., Imamura, T. Fluctuations of the One-Dimensional Polynuclear Growth Model in Half-Space. Journal of Statistical Physics 115, 749–803 (2004). https://doi.org/10.1023/B:JOSS.0000022374.73462.85
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DOI: https://doi.org/10.1023/B:JOSS.0000022374.73462.85