Description
The colored discrete t-PNG model lives on the integer quadrant \(\{1,2,\dots\}^2\) with \(N\) colors (\(N =\) grid size).
It evolves by diagonals \(D = x + y\) (we avoid using the letter t for time to prevent confusion with a model parameter).
Boundaries are empty (no boundary emissions). Any new particle born at \((x,y)\) receives color \(x\).
State used for the update: for a target cell \(v=(x,y)\) on diagonal \(D+1\), define
\(s=(x-1,y)\) and \(s'=(x,y-1)\) on diagonal \(D\), and \(u=(x-1,y-1)\) on diagonal \(D-1\).
Values are \(0\) (empty) or a color in \(\{1,\dots,N\}\).
Parameters:
- b — birth probability when \(s=s'=0\) and \(u=0\).
- t — continuation probability when \(s=s'=0\) and \(u>0\) (not a crossing parameter).
- p — color crossing probability when \(s\neq s'\) and both are nonzero.
Update rules (exactly as implemented):
- Birth / propagation with empty neighbors:
- If \(s=s'=0\) and \(u=0\): \(v=0\) with prob \(1-b\); \(v=x\) with prob \(b\).
- If \(s=s'=0\) and \(u>0\): \(v=u\) with prob \(t\); \(v=0\) with prob \(1-t\).
- Same color on both sides: if \(s=s'=c>0\):
- If \(u=c\): \(v=0\).
- If \(u=0\): \(v=c\).
- If \(u\neq 0\) and \(u\neq c\): \(v=0\).
- Two different nonzero colors present: if \(s\neq s'\) with \(s>0\) and \(s'>0\):
- With prob \(p\) (crossing):
- If \(u=s\): \(v=s'\).
- If \(u=s'\): \(v=s\).
- If \(u=0\): \(v\) is chosen uniformly from \(\{s,s'\}\).
- If \(u\) is some other color: \(v=0\).
- With prob \(1-p\) (annihilation): \(v=0\).
- Exactly one of \(s, s'\) is zero: write \(\{s,s'\}=\{c,0\}\) with \(c>0\):
- If \(u=c\): \(v=0\).
- If \(u=0\): \(v=c\).
- Otherwise: \(v=0\).