## A discrete Hammersley process as an extreme case of oriented percolation flows

Consider the finite square lattice \( \{0,1, \ldots, n-1\} \times \{0,1, \ldots, n-1\} \)
and \(0 < p < 1\) and the usual oriented percolation model: edges are independently *open* with probability
\( p \) or else *closed*, and a *path* consists of edges oriented either *up* or *right*.
Standard theory includes existence of a critical value \(p_*\) such that, in the infinite lattice,
infinite open paths exist for \( p > p_* \) but not for \( p < p_* \).
We consider flows, as follows. Imagine open edges have capacity \(1 \) and closed edges have capacity \( 0 \).
Imagine sources at each vertex on the *left* and *bottom* sides of the square; what volume of flow can be transported to the
top and right sides?
In other words, we consider the random variable

\(V(n,p) := \) maximum number of edge-disjoint open paths starting from some vertex in
\( \{ (0,0), (0,1), \ldots (0,n-2), (1,0), (2,0), \ldots (n-2,0) \} \)
and ending at some vertex in
\( \{ (0,n-1), (1, n-1), \ldots , (n-1,n-1), (n-1, n-2),\ldots (n-1,0) \} \)

Straightforward arguments show that there is a deterministic limit function \(v(p)\) such that
$$ \frac{V(n,p)}{2n} \to v(p) \mbox{ in } L^1 \mbox{ as } n \to \infty $$
and that
$$ v(p) = 0, \ p < p_*; \quad v(p) > 0, \ p > p_*; \quad v(p) \to 1 \mbox{ as } p \to 1 . $$
Studying scaling properties as \( p \downarrow p_* \)
would be a classical style of hard statistical physics project.
Instead we consider \( p \uparrow 1\) behavior.
By considering complements we see that (up to negligible terms)
$$ 1 - v(p) = \lim_n \frac{U(n,p)}{2n} $$
where
\(U(n,p) := \) minimum number of edge-disjoint paths, starting
and ending as above, which cover all the closed edges.

We now start heuristic arguments.
For \( p \) near \( 1 \), the closed edges are approximately a Poisson point process of rate
\( 2(1-p) \) per unit area in the continuum square \( [0,n]^2 \)
and so \(U(n,p) \) should be approximately \( U^*(n, \delta) \), the minimum number of oriented paths in
the continuum square \( [0,n]^2 \) needed to cover all the points of a Poisson (rate \( \delta = 2(1-p) \)) process in the square.
By scaling \( U^*(n, \delta) \) has the same distribution as \( U^*(\delta^{-1/2}n, 1) \)
and by the theory surrounding the Hammersley process
(see e.g. Figure 3 of Aldous-Diaconis (1995); I'm not sure if this is written explicitly anywhere)
$$ U^*(m,1) = (1 + o(1)) m \mbox{ as } m \to \infty . $$
Putting this all together leads to
**Conjecture.**
$$ 1 - v(p) \sim \sqrt{2(1-p)} \mbox{ as } p \uparrow 1 $$

Thinking more carefully, the heuristic argument above involves an "interchange of limits" which seems rather difficult to justify.

**History.**
The conjecture was formulated in work with Jason Lenderman in 2004 but we were unable to prove it.
Simulations by Jason support the conjecture.

Back to **Open Problem list**