Near-one asymptotics for oriented-percolation flow

Definitions and mathematical statement

On the square vertex set $\{0,\ldots,n-1\}^2$, orient each nearest-neighbor edge upward or rightward. Independently give every edge capacity one with probability $p$ and zero otherwise. Let $V(n,p)$ be the maximum number of edge-disjoint directed open paths from the left or bottom boundary to the top or right boundary.

To specify corner conventions, the source set is \[ \{(0,j):0\le j\le n-2\}\cup\{(i,0):1\le i\le n-2\}, \] and the target set is the union of the top row and right column. Different paths may share vertices; the restriction concerns edges, each of which carries at most one unit of flow.

Let $v(p)$ denote the deterministic limit characterized by $V(n,p)/(2n)\to v(p)$ in $L^1$ as $n\to\infty$. Thus $\mathbb E|V(n,p)/(2n)-v(p)|\to0$. Establish the near-one asymptotic \[ \lim_{p\uparrow1}\frac{1-v(p)}{\sqrt{2(1-p)}}=1. \] The order of limits is part of the target: first define the macroscopic flow density for fixed $p$, then let the closed-edge density $1-p$ tend to zero.

This is a maximum-flow quantity, not the probability of one crossing or the maximum number of vertex-disjoint paths. The normalization allows roughly $2n$ units when every edge is available. The question asks for the leading constant and square-root exponent of the deficit, rather than only its convergence to zero.

Short English statement

Is the loss of oriented-percolation flow near full capacity asymptotic to the square root of twice the closed-edge density?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 9700042 (AMR-096-0042), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Aldous - Open problems, hammersley_flow.html. Original source indexed by AMR-096-0042. https://www.stat.berkeley.edu/~aldous/Research/OP/index.html.
• S3 D. Aldous, A discrete Hammersley process as an extreme case of oriented percolation flows. https://www.stat.berkeley.edu/~aldous/Research/OP/hammersley_flow.html.