TITLE: Height growth for randomly falling Tetris like blocks ? What if Young diagrams are falling down?
QUESTION [5 upvotes]: 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 ...

REPLY [5 votes]: This model is known as "Random Walk in Heap Monoids", see Chapter 5 of Vincent Jugé's PhD thesis and the references therein. The law of large numbers, as usually, follows from Kingman's subadditive theorem. A calculation of the rate of growth amounts to finding the limit distribution on infinite heaps. However, I am not sure to what extent it can be made explicit (cf. Abbes-Mairesse). As for the sublinear term, I would still expect the central limit theorem to hold.