CGM

Definition: Corner Growth Model

Suppose that at time 0 we start with the wedge (the graph of $|x|$ ) as our shape. We have a clock that rings after a random exp(1) time and when it does so, a tile rotated by 45 degrees drops at the corner that the shape has at (0,0). Now, the new shape has two corners and each one of these corners is a candidate position for a tile to drop there and cover it. In each of these two corners we assign a independent clock that rings after an exp(1) time and when the clock of a corner rings, a new tile drops and covers this corner. On the new shape, we assign one independent exponential clock on each corner and we drop a tile at that corner when it rings. This is the corner growth model.

By discretising the system at set intervals, each available position now follows a geometric distribution. If that position is covered during a time step, that counts as a success. Otherwise it is a failure. By doing this, the time it takes to cover up to a certain point can be represented by converting the growth model to a matrix and defining a concept called the last passage percolation time.

Definition: Last Passage Percolation

Consider an $m \times n$ matrix $W=\left( w_{ij} \right)$ . If the set of all paths from (1,1) to (m,n) that only go down and to the right is denoted by $\Pi$ , then the LPP is: $\max_{\pi \in \Pi} \sum_{(i,j) \in \pi}w_{ij}$

If each entry of the matrix records the geometric times that it takes for the tile to be placed, then the LPP is the time it takes for the position (m,n) to be covered. Using some results of the RSK algorithm outlined in [CA15], it can be shown that the schur polynomials create a pushforward measure of the measure induced by the geometric distributions on W.

References

[CA15] Ahlbach, Connor. "The Robinson-Schensted-Knuth Correspondence: Properties, Applications, and Generalizations." (2015)