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Additional resources for Analysis and stochastics of growth processes and interface models
This means that a random Hermitian matrix is constructed by putting IID complex-valued Gaussian random variables above the diagonal, IID real-valued Gaussian random variables on the diagonal, and letting the Hermitian property determine the entries below the diagonal. Then as the matrix grows in size, the variances of the entries are scaled appropriately to obtain limits. The standard reference is (Mehta 2004). 3 and related results initially arose entirely outside probability theory (except for the statements themselves), involving the RSK correspondence and Gessel’s identity from combinatorics and techniques from integrable systems to analyse the asymptotics of the resulting determinants.
The dynamical ﬂuctuations are the universal ones described by the Tracy–Widom laws. 3. Further remarks. The polynuclear growth model (PNG) is another related (1+1)-dimensional growth model used by several authors for studies of Tracy– Widom ﬂuctuations and the Airy process in the KPZ scaling picture (Baik and Rains 2000; Ferrari 2004; Johansson 2003; Pr¨ ahofer and Spohn 2002, 2004). Like the Hammersley process, the graphical construction of the PNG utilizes a planar Poisson process, and in fact the same underlying last-passage model of increasing paths.
10(3), 525–47. Baik, J. (2005). Limiting distribution of last passage percolation models. In XIVth International Congress on Mathematical Physics, pp. 339–46. World Sci. , Hackensack, NJ. , Deift, P. and Johansson, K. (1999). On the distribution of the length of the longest increasing subsequence of random permutations. J. Amer. Math. Soc. 12(4), 1119–78. Baik, J. and Rains, E. (2000). Limiting distributions for a polynuclear growth model with external sources. J. Statist. Phys. 100(3–4), 523–41.