Introduction

This Python package offers a suite of tools designed for simulating various random surface growth models. The foundational concepts and methodologies are significantly influenced by the work presented in the book by Barabási and Stanley [BarabasiS95] (1995) on fractal concepts in surface growth.

It is known that the fluctuations in the surface growth models fall into a few universality classes. The most famous one is the Kardar-Parisi-Zhang (KPZ) equation Kardar et al. [KPZ86]

(1)\[ \frac{\partial h(t,x)}{\partial t} = \nu \nabla^2 h(t,x) + \frac{\lambda}{2} (\nabla h(t,x))^2 + \eta(t,x), \quad t>0, x \in \mathbb{R},\]

where \(\eta\) is a centered Gaussian noise, which is white in both space and time. The KPZ equation (1) is a stochastic partial differential equation (SPDE) [Wal86].

KPZ equation (1) was solved by Hairer [Hai13] in 2013. The fluctuations of the KPZ equation (1) are in the KPZ universality class; see Amir et al. [ACQ11]. Numerous discrete models report the same universality class as the KPZ universality class, including

Theoretical results include the fluctuations of:

  • the longest increasing subsequence of a random permutation: [BDJ99].

  • the largest eigenvalue of a random matrix: [TW93] and [TW94].

  • the asymmetric simple exclusion process (ASEP): [Joh00] and [TW09].

History of the Package

The foundational elements of this package, which include the random deposition, random deposition with surface relaxation, the ballistic deposition models, and a visualization component, were initially crafted by the first author. These components, forming the basis of the package, were part of a graduate student seminar in October 2023, as documented in Chen [Che23a]. The seminar’s slides provide an excellent introduction to these foundational aspects of the package.

Building upon this foundation, the package has been enriched with simulations for growth models with Tetris pieces. This feature is to test the belief of the universality of the growth models. This feature was developed as a final course project for the course “Math 7820: Applied Stochastic Processes I,” undertaken by the second and third authors in Fall 2023. This is still an ongoing project. More feature and functionalities will be added in the future. Some simulation experiments will be carried out to test the universality of the growth models.

Acknowledgments

The references throughout this document have been meticulously compiled and are available in a comprehensive bibliography bank [Che23b].

This work is partially supported by the National Science Foundation (NSF) under Grant No. No. 2246850 (2023–2026) and the collaboration/travel award from the Simons foundation under Award No. 959981 (2022–2027).

Bibliography

[ACQ11]

Gideon Amir, Ivan Corwin, and Jeremy Quastel. Probability distribution of the free energy of the continuum directed random polymer in $1+1$ dimensions. Comm. Pure Appl. Math., 64(4):466–537, 2011. URL: https://doi.org/10.1002/cpa.20347, doi:10.1002/cpa.20347.

[BDJ99]

Jinho Baik, Percy Deift, and Kurt Johansson. On the distribution of the length of the longest increasing subsequence of random permutations. J. Amer. Math. Soc., 12(4):1119–1178, 1999. URL: https://doi.org/10.1090/S0894-0347-99-00307-0, doi:10.1090/S0894-0347-99-00307-0.

[BKR+88]

R. Baiod, D. Kessler, P. Ramanlal, L. Sander, and R. Savit. Dynamical scaling of the surface of finite-density ballistic aggregation. Phys. Rev. A, 38:3672–3679, Oct 1988. URL: https://link.aps.org/doi/10.1103/PhysRevA.38.3672, doi:10.1103/PhysRevA.38.3672.

[BarabasiS95]

Albert-László Barabási and H. Eugene Stanley. Fractal concepts in surface growth. Cambridge University Press, Cambridge, 1995. ISBN 0-521-48318-2. URL: https://doi.org/10.1017/CBO9780511599798, doi:10.1017/CBO9780511599798.

[Che23a]

Le Chen. Graduate student seminars by Le Chen. nov 2023. URL: https://doi.org/10.5281/zenodo.10206966, doi:10.5281/zenodo.10206966.

[Che23b]

Le Chen. SPDEs-Bib: a comprehensive bibliography of stochastic partial differential equations and related topics. nov 2023. URL: https://doi.org/10.5281/zenodo.10143431, doi:10.5281/zenodo.10143431.

[FV85] (1,2)

Fereydoon Family and Tamás 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. URL: https://dx.doi.org/10.1088/0305-4470/18/2/005, doi:10.1088/0305-4470/18/2/005.

[Hai13]

Martin Hairer. Solving the KPZ equation. Ann. of Math. (2), 178(2):559–664, 2013. URL: https://doi.org/10.4007/annals.2013.178.2.4, doi:10.4007/annals.2013.178.2.4.

[Joh00]

Kurt Johansson. Shape fluctuations and random matrices. Comm. Math. Phys., 209(2):437–476, 2000. URL: https://doi.org/10.1007/s002200050027, doi:10.1007/s002200050027.

[JB85]

R Jullien and R Botet. Scaling properties of the surface of the eden model in d=2, 3, 4. Journal of Physics A: Mathematical and General, 18(12):2279, aug 1985. URL: https://dx.doi.org/10.1088/0305-4470/18/12/026, doi:10.1088/0305-4470/18/12/026.

[KPZ86]

Mehran Kardar, Giorgio Parisi, and Yi-Cheng Zhang. Dynamic scaling of growing interfaces. Phys. Rev. Lett., 56(9):889, 1986. URL: https://doi.org/10.1103/PhysRevLett.56.889, doi:10.1103/PhysRevLett.56.889.

[MJB86]

P. Meakin, R. Jullien, and R. Botet. Large-scale numerical investigation of the surface of eden clusters. Europhysics Letters, 1(12):609, jun 1986. URL: https://dx.doi.org/10.1209/0295-5075/1/12/001, doi:10.1209/0295-5075/1/12/001.

[MRSB86]

Paul Meakin, P. Ramanlal, L. M. Sander, and R. C. Ball. Ballistic deposition on surfaces. Phys. Rev. A, 34:5091–5103, Dec 1986. URL: https://link.aps.org/doi/10.1103/PhysRevA.34.5091, doi:10.1103/PhysRevA.34.5091.

[PRacz85]

Michael Plischke and Zoltán Rácz. Dynamic scaling and the surface structure of eden clusters. Phys. Rev. A, 32:3825–3828, Dec 1985. URL: https://link.aps.org/doi/10.1103/PhysRevA.32.3825, doi:10.1103/PhysRevA.32.3825.

[TW93]

Craig A. Tracy and Harold Widom. Level-spacing distributions and the Airy kernel. Phys. Lett. B, 305(1-2):115–118, 1993. URL: https://doi.org/10.1016/0370-2693(93)91114-3, doi:10.1016/0370-2693(93)91114-3.

[TW94]

Craig A. Tracy and Harold Widom. Level spacing distributions and the Bessel kernel. Comm. Math. Phys., 161(2):289–309, 1994. URL: http://projecteuclid.org/euclid.cmp/1104269903.

[TW09]

Craig A. Tracy and Harold Widom. Asymptotics in ASEP with step initial condition. Comm. Math. Phys., 290(1):129–154, 2009. URL: https://doi.org/10.1007/s00220-009-0761-0, doi:10.1007/s00220-009-0761-0.

[Wal86]

John B. Walsh. An introduction to stochastic partial differential equations. In École d'été de probabilités de Saint-Flour, XIV—1984, volume 1180 of Lecture Notes in Math., pages 265–439. Springer, Berlin, 1986. URL: https://doi.org/10.1007/BFb0074920, doi:10.1007/BFb0074920.

[ZS86]

J. G. Zabolitzky and D. Stauffer. Simulation of large eden clusters. Phys. Rev. A, 34:1523–1530, Aug 1986. URL: https://link.aps.org/doi/10.1103/PhysRevA.34.1523, doi:10.1103/PhysRevA.34.1523.