Anisotropic Growth of Random Surfaces in 2 + 1 Dimensions

2008

167 citations recorded by the source corpus.

We construct a family of stochastic growth models in 2 + 1 dimensions, that belong to the anisotropic KPZ class. Appropriate projections of these models yield 1 + 1 dimensional growth models in the KPZ class and random tiling models. We show that correlation functions associated to our models have determinantal structure, and we study large time asymptotics for one of the models.The main asymptotic results are: (1) The growing surface has a limit shape that consists of facets interpolated by a curved piece. (2) The one-point fluctuations of the height function in the curved part are asymptotically normal with variance of order ln(t) for time $${t \gg 1}$$t≫1. (3) There is a map of the (2 + 1)-dimensional space-time to the upper half-plane $${\mathbb{H}}$$H such that on space-like submanifolds the multi-point fluctuations of the height function are asymptotically equal to those of the pullback of the Gaussian free (massless) field on $${\mathbb{H}}$$H.

Subfields: Geometry and Topology, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/d492c9191a855337cc526efc478c9679ed2ba74e

Citations in the corpus

• Height Fluctuations for the Stationary KPZ Equation (2014, unknown): “These statistical properties agreed between the models, but depended non-trivially on the type of initial data or geometry for the growth models, such as curved [6, 10, 21, 22, 60, 61, 68], flat [10, 21, 23–25, 43, 48, 73] or stationary [7–9, 47, 49, 57, 69].”
• Stochastic six-vertex model (2014, unknown): “ics models with height functions. Indeed, in many related models the asymptotic uctuations of the height function are Gaussian and, more precisely, are governed by the Gaussian Free Field, cf. [Ken], [BF2], [P], [Bor], [BG]. Theorem 1.2 can probably be better understood from the point of view of int”