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Consider last passage percolation model on $\mathbb{Z}^2$. I am interested to know if there is any known result for the stationary distribution of passage times, given some distribution for the weights. To be more precise, let us consider the passage times $G(R,U)$ from the origin to a point $(R,U)$

$$G(R,U)=\max_{p\in \Pi(R,U)}\sum_{(i,j)\in p} X(i,j)$$

where $\Pi(R,U)$ is an up-right path from the origin to $(R,U)$, and $X(i,j)$ is the weight at $(i,j)$, with as yet unspecified distribution.

When $i+j$ tends to infinity, but $i-j$ does not, we may expect that $G(i,j)-\mu(i,j)$ approaches a limiting distribution for semi-infinite paths, where $\mu(i,j)$ represents the mean (which obviously does grow).

My question. Are there any models where this limiting distribution is known?

Edit: I think the better way to define what I'm after is in terms of the Buseman functions

$$B(i,j|k,l)=G(i,j)-G(k,l)$$

which measure the difference of path lengths to two points. Now the question now concerns $B(i,j|k,l)$ for (say) $i+j=k+l\to \infty$. The simplification of considering both $(i,j)$ and $(k,l)$ on the same diagonal is that one is asking for a distribution of the $B(i,j|k,l)$ that is preserved under the recursion that propagates the $G(i,j)$ from one diagonal to the next.

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2 Answers 2

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Yes, when the weights are geometric or exponentials (and you need to rescale - the variance grows like $N^{2/3}$, and the limiting distribution is Tracy-Widom). This was proved by K. Johansson, see http://arxiv.org/pdf/math/9903134.pdf

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  • $\begingroup$ Thanks for the reply, Ofer. I'm actually looking for something a bit different from Johansson's result, which as you point out is not stationary in the sense that I'm after. On reflection the right way to phrase this is in terms of the Buseman functions $\endgroup$
    – Austen
    Commented Jun 26, 2015 at 14:23
  • $\begingroup$ This is a completely different question, and I don't know the answer. I would be interested to see an argument even for tightness when i=k+1. $\endgroup$
    – ofer zeitouni
    Commented Jun 26, 2015 at 16:25
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I know it's a bit late but I've just found this question completely randomly. We have a certain kind of stationarity detailed in http://projecteuclid.org/euclid.ejp/1464730577 (Lemma 4.2). One carefully prepares Exponential boundary weights with certain parameters for last passage and then $B(i,j|k,l)$ becomes the sum of independent Exponentials whenever $(i,j)$ and $(k,l)$ connects via a South-East or North-West path. $i+j=k+l$ seems to indicate that this is your case (talking about the first quadrant, right?), but I think for your original question of $G(i,j)-\mu(i,j)$ you also need North-East steps for which independence of increments is lost and the question becomes difficult.

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