This is a rather soft question, but I would like to know how physicists would approach a problem which seems to be hard from the mathematical prospective.

The Grimmett percolation model is defined as follows. Consider the $\mathbb{Z}^2$ lattice and randomly assign an orientation for every bond of the lattice in the following way. Let $p$ be the percolation parameter and for every horizontal bond orient it rightward with probability $p$ and leftwards otherwise. The same rule applies for vertical bonds: a bond is oriented upwards with probility $p$ and downwards otherwise. Thus we get an oriented graph supported on the whole lattice $\mathbb{Z}^2$.

Grimmett conjectures that for all $p \neq 1/2$ there is an infinite directed path from the origin to infinity. By coupling with classical bond percolation it is not hard to show that at $1/2$ the system does not percolate - so presumably $1/2$ is a critical point. On the other hand, the conjecture is quite old and seems to be very far from resolution mainly because all the known methods used for other percolation models fail to work.

I am wondering if this model has any meaning in statistical mechanics and if so how a physicist would approach Grimmett's conjecture?

  • $\begingroup$ May you provide a link or reference for this particular percolation model? $\endgroup$
    – Quillo
    May 30, 2022 at 7:15


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