Question: How the maximal height grows for random Tetris like blocks falling down ? Numeric simulation (see below) shows leading term is linear with some constant depending on shapes of blocks allowed. Can the constant be calculated ? Can the subleading term like $h(T) = a T + b T^p + ...$ be found ( similar to "sticky disks" = "ballistic deposition" model where $p=1/3$) ?
Background: Considerable research interest attracted models like "Ballistic Deposition" where blocks randomly fall down and glue to neighbors. See MO Tetris-like falling sticky disks, youtube video from 36 second for illustration, Borodin's lectures, I. Corwin Notices 2016 Kardar-Parisi-Zhang Universality etc...
Numeric simulation 1. (Horizontal blocks) Consider horizontal blocks of fixed length $n$ falling down into the "box" (playing field in Tetris) of horizontal size $L$. The model has only two parameters (n,L) and n<=L. One uniformly generates random position (horizontal position x = 1... (L-n+1)) of the block and it falls down straight vertically. The heap of blocks grows - like in Tetris and we are interested in maximal height. In contrast to Tetris to simplify things let us forget about burning out the fully packed raws.
We are interested in height growth depending on number $T$ of fallen blocks: $$ h(T) = a T + ... $$ where constant $a$ depends on $(n,L)$. It is clear that for $n>L/2$, $a(n,L)=1$. Numeric estimation for $a(n,L)$ are given in table 1 below. It seems plausible that for $k=1$, $a(1,L) = 1/L$; and $a(n,L) = 2n/L + ...$.
It would be interesting to find subleading terms $h(T)$, however from numeric simulation it seems there is no reasonable term.
Numeric simulation 2. Young diagrams Consider Young diagrams for some fixed $n$ falling down into the "box" (playing field in Tetris) of horizontal size $L$. Let us use English notation for diagrams. The model has only two parameters (n,L) and n<=L. One uniformly generates random position of the block in horizontal direction and it falls down straight vertically (horizontal position is UNchanged).
Again we are interested how the height grows : $$ h(T) = a T + ... $$ where constant $a$ depends on $(n,L)$. Numeric similation results given in the table 2 below.
Table 1: Constant $a(n,L)$ for falling horizonal blocks of size $n$ and $L$ is horizontal size of "playing field".
Simulation number T= 200000
n\L 10.0000 20.0000 50.0000 80.0000 100.0000
1.0000 0.1014 0.0508 0.0207 0.0129 0.0106
2.0000 0.3808 0.1975 0.0803 0.0508 0.0408
3.0000 0.6100 0.3276 0.1349 0.0843 0.0679
4.0000 0.7999 0.4486 0.1875 0.1188 0.0952
5.0000 0.9510 0.5651 0.2404 0.1521 0.1227
10.0000 1.0000 0.9847 0.4923 0.3167 0.2562
25.0000 1.0000 1.0000 0.9972 0.7424 0.6181
40.0000 1.0000 1.0000 1.0000 0.9989 0.8956
50.0000 1.0000 1.0000 1.0000 1.0000 0.9993
Table 2: Constant $a(n,L)$ for falling Young diagrams (English notation), $n$ is Young diagrams's $n$ and $L$ is horizontal size of "playing field".
Simulation number T= 500000
n\L 10.0000 20.0000 50.0000 80.0000 100.0000
1.0000 0.1005 0.0508 0.0203 0.0130 0.0104
2.0000 0.3643 0.1876 0.0763 0.0478 0.0381
3.0000 0.6663 0.3491 0.1427 0.0895 0.0719
4.0000 0.8899 0.4728 0.1942 0.1226 0.0977
5.0000 1.3263 0.7220 0.2998 0.1891 0.1512
6.0000 1.6307 0.9046 0.3772 0.2387 0.1913
7.0000 2.0022 1.1360 0.4767 0.3015 0.2420
8.0000 2.3123 1.3349 0.5675 0.3591 0.2872
There are several other questions on Tetirs here and on MSE: MO Is there winning strategy in Tetris ? What if Young diagrams are falling?, MSE The Mathematics of Tetris, MSE Mathematics of Tetris 2.0, MSE An impossible sequence of Tetris pieces, but they seems to discuss different sides of the game.
It would be natural to ask what will happen with the height growth if Tetris rule: burning out the fully packed raws is introduced... It might be next question ...
