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people:romik:osp [2019/09/09 22:31] 99.33.86.8 [7 Jeu de taquin and the joint distribution of finishing time pairs $V_n(k), V_n(k+1)$] |
people:romik:osp [2019/09/09 22:36] (current) 99.33.86.8 |
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| The next thing to note is that this TASEP with initial condition $\eta_0$ can be described in terms of a last passage percolation process with precisely the weight array $\boldsymbol{\tau}^{(n,k)}$ using the standard correspondence between TASEPs and LPP, since setting the corner weight $X_{1,1}'$ achieves the goal of causing the standard step initial configuration $(1,\ldots,1,1,0,0,\ldots,0)$ to jump immediately to $\eta_0$ at time $t=0$; from there the evolution proceeds according to the usual TASEP rules. | The next thing to note is that this TASEP with initial condition $\eta_0$ can be described in terms of a last passage percolation process with precisely the weight array $\boldsymbol{\tau}^{(n,k)}$ using the standard correspondence between TASEPs and LPP, since setting the corner weight $X_{1,1}'$ achieves the goal of causing the standard step initial configuration $(1,\ldots,1,1,0,0,\ldots,0)$ to jump immediately to $\eta_0$ at time $t=0$; from there the evolution proceeds according to the usual TASEP rules. | ||
| - | Finally, one needs to check that under this way of realizing the random vector $(V^n_k, V^n_{k+1})$, the vector maps precisely to $(T_{k, n+1-k}'', T_{k+1, n-k}'')$. This is because //* * *I will add those details later* * *//. | + | Finally, one needs to check that under this way of realizing the random vector $(V^n_k, V^n_{k+1})$, the vector maps precisely to $(T_{k, n+1-k}'', T_{k+1, n-k}'')$. This is related to the fact, also explained in the Romik-Sniady paper, that the jeu de taquin path precisely encodes the path of the second class particle (vertical steps of the jeu de taquin path correspond to the second-class particle moving to the left, and horizontal steps correspond to the second-class particle moving to the right). One can now separate into two cases depending on whether the jeu de taquin path ever makes it to the last column of the array; this division into cases corresponds precisely to the question of which of the two random variables $V^n_k$, $V^n_{k+1}$ is smaller in value. //Need to explain this better - I will add more details later.// |
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