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Surface Growth Models with Random Tetris Pieces

Le Chen

le.chen@auburn.edu
Department of Mathematics and Statistics
Auburn University

AMS Special Session on Topics in Stochastic Analysis/Rough Paths/SPDE and Applications in Machine Learning
Tallahassee, FL, March 23-24, 2024

Two graduate students who have been working on this project throughout

Math 7820/30: Applied Stochastic Processes (2023/24):

Mauricio Montes

mauricio.montes@auburn.edu

Ian Ruau

ian.ruau@auburn.edu

DMS-Probability: No. 2246850
(2023-2026)

No. 959981
(2022-2027)

1980's

Nonsticky

Sticky

Nonsticky

Sticky

Family-Vicsek scaling relation

\begin{align*} \sigma_L(M) ∼ L^β f\left(M/L^γ\right), \quad β = 1/2, \quad γ = 3/2. \end{align*} \begin{align*} f(x) \sim \begin{cases} x^ν, & x ≪ 1, \\ 1, & x ≫ 1, \end{cases} \quad ν = 0.30 ± 0.02. \end{align*}

[1] F. Family and T. Vicsek. Scaling of the active zone in the eden process on percolation networks and the ballistic deposition model. Journal of Physics A: Mathematical and General, 18(2):L75, feb 1985. [ bib | DOI | http ]

How to explain growth rate 1/3

Kardar-Parisi-Zhang Equation

\begin{align*} ∂_t h = ν ∇^2 h + λ(∇h)^2 + η, \end{align*}

[2] M. Kardar, G. Parisi, and Y.-C. Zhang. Dynamic scaling of growing interfaces. Phys. Rev. Lett., 56(9):889, 1986. [ bib | DOI | http ]

\begin{align*} δ h = \sqrt{(v δ t)^2 + (v δ t ∇ h)^2 } ≈ (v δ t) (1 + (∇ h)^2/2) \end{align*}