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For a lattice in $\mathbb{R}^2$, if we include each edge independently with probability $p$ (i.e. bond percolation), it is well known that there is a critical probability $0 < p_c < 1$ depending on the lattice, such that if $ p > p_c$ then there is almost surely an infinite component (i.e. with probability $1$), and if $p < p_c$ then almost surely all components are finite.

The square lattice is special because it is self-dual. This allows one to guess that in this particular case $p_c = 1/2$, although the proof of this fact, originally due to Harry Kesten, is fairly subtle.

One wonders if there is an analogous fact for "two-dimensional" percolation in $\mathbb{R}^4$. In particular, consider the hypercubic lattice as a $1$-dimensional cubical complex, and then attach two-dimensional square "plaquettes" independently with probability $p$. Since the hypercubic lattice is self-dual, we might expect some kind of phase transition when $p=1/2$. My hope is that topology would give us the right language to talk about this.

One possibility, and perhaps the nicest answer I can imagine, was suggested to me by Russ Lyons: If $p > 1/2$ then there are almost surely embedded planes, and if $p< 1/2$ then there are almost surely not. Here the embedded plane should be the union of closed $2$-cells. It turns out that once $p > 1/2$ in $\mathbb{R}^2$, there are not only infinite components, which implies embedded half-lines, but embedded lines as well.

Another possibility is that once $p > 1/2$ there are almost surely bounded $1$-dimensional cycles in homology $H_1 ( - , \mathbb{Z} / 2\mathbb{Z})$, which are boundaries of unbounded two-dimensional complexes, and that when $p < 1/2$ there are almost surely not. (I believe that this is similar to the "plaquette percolation" studied by Jennifer Chayes in her Ph.D. thesis, but the work I know of is in $\mathbb{R}^3$.)

I am not an expert in percolation theory, and would really like to know if anyone knows about any previous work in this area, or any standing conjectures. (Or is there any obvious reason that either of the possibilities I suggested could be ruled out?)

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up vote 4 down vote accepted

The paper "PLAQUETTES, SPHERES, AND ENTANGLEMENT" by GEOFFREY R. GRIMMETT AND ALEXANDER E. HOLROYD does not deal with the self-dual problem, but nevertheless could be of interest for you. If shows that "The high-density plaquette percolation model in d dimensions contains a surface that is homeomorphic to the (d − 1)sphere and encloses the origin." Here's the link: http://www.statslab.cam.ac.uk/~grg/papers/sphere8.pdf

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A good way to see duality at p=1/2 is to consider p= 1/2 Voronoi percolation on H^4, the hyperbolic four space. That is, let each Voronoi cell of a Poisson processes on H^4 be colored white or back with probability 1/2, independently of the others. Now you will have either a black plane or white one connecting appropriate pieces of the boundary.

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