The Up–Down last-passage percolation distributional identity

Let $(X_{i,j})_{i,j\ge 1}$ be an array of independent, identically distributed exponential random variables of rate $1$. For each $n\ge 2$, let $\bm{V}_n$ denote the vector of last-passage percolation times

$$\bm{V}_n=(L(1,1;n-1,1),L(1,1;n-2,2),\ldots,L(1,1;1,n-1)),$$

where $L(a,b;c,d)$ is the maximum weight over directed lattice paths from $(a,b)$ to $(c,d)$, and let $\bm{U}_n$ be the corresponding vector arising from the up–down process on $n$ particles. The Up–Down LPP conjecture. For all $n\ge 2$,

$$\bm{U}_n\overset{D}{=}\bm{V}_n.$$

The conjecture would identify the joint distribution of the up–down process with a directed last-passage percolation model. Its one-coordinate marginal identities are known, and the full identity has been proved for $2\le n\le 6$; in general it remains open. It would imply the point-to-line LPP description and the corresponding asymptotic consequences for the total absorbing time of the up–down process.