How Do Surfaces Grow?
Le Chen
le.chen@auburn.eduDepartment of Mathematics and Statistics
Auburn University
Auburn Summer Science Institute (AU-SSI)
Auburn, AL, June 6, 2025
DMS-Probability: No. 2246850
(2023-2026)
No. 959981
(2022-2027)
Two graduate students who have been working on this project throughout
Math 7820/30: Applied Stochastic Processes (2023/24):
Sand pile at the beach vs Snow pile in winter
Image created by OpenAI's DALL-E
Simulation by Tetris-Ballistic package
Central Limit Theorem (CLT)
Galton Box
Video from by Wikipeidia
Video created by 3Blue1Brown
Population Height Distribution
(by Demographics)
Video created by 3Blue1Brown
Experiments with weighted dices
Video created by 3Blue1Brown
Mean and standard deviation
Video created by 3Blue1Brown
General ideas of CLT
Video created by 3Blue1Brown
Large N
Independent of underlying details
Video created by 3Blue1Brown
Assumptions for CLT
Video created by 3Blue1Brown
Complete video by 3Blue1Brown
Quizzes
What does the Central Limit Theorem say?
- A. The mean of a sample is always equal to the mean of the population
- B. The shape of any population is always normal
- C. The distribution of sample means tends to be normal as sample size increases
- D. All distributions have the same standard deviation
Show Answer
Correct Answer: C
Which of the following is a requirement for the CLT to work well?
- A. The original data must be normal
- B. The sample size must be large enough
- C. You must only sample once
- D. All data must be integers
Show Answer
Correct Answer: B
Which kind of population shape can the CLT handle?
- A. Skewed
- B. Bumpy
- C. Uniform
- D. All of the above
Show Answer
Correct Answer: D
Which of the following is likely not independent?
- A. Rolling a die 50 times
- B. Measuring height of 50 random people
- C. Recording daily temperatures for 50 days in a row
- D. Tossing a coin
Show Answer
Correct Answer: C
Simulation by Tetris-Ballistic package
Nonsticky
Sticky
Growing interfaces in a thin film
Other examples
Paper inking
Paper burning
Snow pile in one dimension
Why doesn’t the Central Limit Theorem work for surface growth models?
- A. Because the surface never grows
- B. Because particles always fall in the same place
- C. Because the heights at different locations are not independent
- D. Because the average height is always zero
Show Answer
Correct Answer: C
What makes surface growth models more delicate than standard random sampling?
- A. They are just harder to simulate
- B. The interactions between sites introduce dependencies
- C. The surface must be flat
- D. We can only drop one particle at a time
Show Answer
Correct Answer: B
Tetris Pieces?
Any crossover of Gaussian and KPZ?
5% nonsticky + 95% sticky
50% nonsticky + 50% sticky
90% nonsticky + 10% sticky
95% nonsticky + 5% sticky
98% nonsticky + 2% sticky
99% nonsticky + 1% sticky
Install
PyPI: Tetris Ballistic
Source code
GitHub: Simulations on Some Surface Growth Models
Diffusion-limited aggregation (DLA)
Simulation by markstock/dla-nd