Doubly geometric corner growth model

Axel Saenz


The model [1]

The model depends on two parameters $\beta>0$ and $-\beta\le \nu<1$. The evolution of a height function $H_T(N)$ is as follows. At each discrete time step, add a box to a place where we can with probability $\beta/(1+\beta)$. If a box is added, simultaneously add a random number of boxes according to a truncated geometric random variable with parameter $(\nu+1)/(\beta+1)$, i.e., such that if an overhang occurs, we truncate the resulting added boxes. The special case $\nu=-\beta$ reduces the model to the classical geometric corner growth whose fluctuations were studied in [2].

The next picture indicates one possible time step in DGCG (doubly geometric corner growth).

DGCG

Sampling algorithm

The simulation is a simple forward sampling of the discrete time Markov chain.

Data file format

There is no data for this simulation. The simulation pictures are shown in “Russian notation”, i.e., rotated by 45 degrees. The random height function is given in blue, and there is also a theoretical limit shape (a cubic curve) obtained in [1] superimposed onto it.


code • (Main GitHub repo)

(note: parameters in the code might differ from the ones in simulation results below)

simulation results

  1. Geometric (symmetric) corner growth case • (image: 24 KB)
    Discrete (unscaled) time $T=500$, $\beta=\frac14$, $\nu=-\frac14$
    Geometric (symmetric) corner growth case
  2. DGCG, almost symmetric • (image: 24 KB)
    Discrete (unscaled) time $T=500$, $\beta=\frac14$, $\nu=0$
    DGCG, almost symmetric
  3. DGCG, more asymmetric • (image: 24 KB)
    Discrete (unscaled) time $T=500$, $\beta=\frac14$, $\nu=\frac14$
    DGCG, more asymmetric
  4. DGCG, even more asymmetric • (image: 24 KB)
    Discrete (unscaled) time $T=500$, $\beta=\frac14$, $\nu=\frac12$
    DGCG, even more asymmetric

references

  1. A. Knizel, L. Petrov, A. Saenz. In preparation (2018)
  2. K. Johansson. Shape fluctuations and random matrices. Commun. Math. Phys. 209 (2000) no. 2, 437--476 • https://arxiv.org/abs/math/9903134https://link.springer.com/article/10.1007/s002200050027