Strong law of large numbers for the interface in ballistic deposition |
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Authors: | Timo Sepplinen |
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Institution: | Department of Mathematics, Iowa State University, Ames, IA 50011, USA |
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Abstract: | We prove a hydrodynamic limit for ballistic deposition on a multidimensional integer lattice. In this growth model particles rain down at random and stick to the growing cluster at the first point of contact. The theorem is that if the initial random interface converges to a deterministic macroscopic function, then at later times the height of the scaled interface converges to the viscosity solution of a Hamilton–Jacobi equation. The proof idea is to decompose the interface into the shapes that grow from individual seeds of the initial interface. This decomposition converges to a variational formula that defines viscosity solutions of the macrosopic equation. The technical side of the proof involves subadditive methods and large deviation bounds for related first-passage percolation processes. |
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Keywords: | Ballistic deposition Interacting particle system Hydrodynamic limit Interface model Viscosity solution Hamilton– Jacobi equation Hopf– Lax formula Subadditive ergodic theorem First-passage percolation |
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