Abstract of my PhD thesis "Shape fluctuation of crystal facets and surface growth in one dimension"
In this thesis we consider two models, the first belonging to non-equilibrium and the second one to equilibrium statistical mechanics. The two models are connected the mathematical methods used to their analysis.
The first model analyzed is the polynuclear growth model (PNG) in one dimension, which belongs to the KPZ (Kardar-Parisi-Zhang) universality class. For growth processes, when the growth time t is large, the statistical properties of the surface are expected to depend only on qualitative properties of the dynamics and on symmetries, but not on the details of the models. In the case of the PNG, for large growth time t the surface height fluctuations scale as t1/3 and the spatial correlation length as t2/3. For boundary conditions inducing a droplet shaped surface, it was shown by Prhofer and Spohn that the statistics of the surface is described by the Airy process. This result was obtained by extending the surface line to a multi-layer model. In this thesis we consider the space-translation invariant case and determine the limit point process of the multi-layer model at fixed position. The process coincides with the edge scaling of eigenvalues of the Gaussian orthogonal ensemble (GOE) of random matrices.
The second model we study is the 3D-Ising corner at zero temperature. The corner of the crystal is composed by three facets (flat pieces) and a rounded piece interpolating between them. We analyze the border line between the rounded and a flat piece. When the corner defect size is large, say of linear length L, the fluctuations of the border line are of order L1/3 and the spatial correlation length scales as L2/3. We prove that the (properly rescaled) border line is well described by the Airy process. This is also the case for the terrace-ledge-kink (TLK) model, a simple model used to describe surfaces close to the high symmetry ones. We expect that the Airy process describes the border of the facets in the class of surface models with short range interactions.
Although the two models describe physically very different systems, the mathematical methods employed for their investigation are similar. Both models can be mapped into some non-intersecting line ensembles, which can also be viewed as trajectories of fermions. One can associate some point processes to the line ensembles. For the 3D-Ising corner it is an extended determinantal point process, whose kernel converges to the extended Airy kernel. The Airy kernel appears also in the edge scaling of Dyson's Brownian motion for GUE random matrices. The process for the PNG is a Pfaffian point process (at fixed position) and the 2x2 matrix kernel converges to the one of GOE random matrices. In the thesis we also discuss the connection with some other models: the longest increasing subsequence problem, directed polymers, last passage percolation, totally asymmetric exclusion process, random tiling, 3D-Young diagrams, and indirectly, Gaussian ensembles of random matrices.