One of the weakest estimates conjectured to hold for critical planar percolation models (and proved in many cases) is the so-called RSW estimate. RSW estimate is the statement that the probability of the existence of a crossing between the two opposite short sides of an $N\times 3N$ rectangle is bounded away from $0$ and $1$ as $N\to\infty$.

The RSW estimate was originally proved for bond percolation on the square lattice by Russo, Seymour, and Welsh, relying on the fact that this lattice is self-dual. The same self-duality argument applies to site percolation on the triangular lattice. Since the original work, RSW estimate has been proven for many other critical planar percolation models.

What is (currently) the most general setting in which RSW estimate is known for critical planar percolation?

(Of course, there may be no single result which encompasses all others, so the optimal answer may include a list of results).

As a side question (which may be answered at the same time as the main question) is: is there any model for critical planar percolation where RSW estimates are known but where the critical probability is not known?

(image credit: Random-Turn Hex and Other Selection Games by Yuval Peres, Oded Schramm, Scott Sheffield, and David B. Wilson, from http://arxiv.org/abs/math/0508580)

  • $\begingroup$ I would have written a better senior thesis if I had focused on percolation theory rather than its limit SLE. Here's a small bit of insight from Hugo Duminil-Coupin "In words, the property (RSW) can be interpreted as follows: a lower bound on the crossing probability for a rectangle of aspect ratio α implies a lower bound for a rectangle of larger aspect ratio β." which is kind of obvious but we need to prove it... Have you seen the Hex proof of the Brouwer fixed-point theorem? $\endgroup$ Aug 29 '17 at 20:21
  • $\begingroup$ I think if you want an RSW lattices other than square lattices of Hex lattices you are out of luck. Not that it can't be done but Google certainly isn't returning any. The percolation threshhold numbers are known for various lattices en.m.wikipedia.org/wiki/Percolation_threshold The best proof we have used Conformal Field Theory. I don't see how Percolation Theory doesn't have a more topological flavor. $\endgroup$ Aug 29 '17 at 21:27

This paper of Grimmett and Manolescu prove RSW for bond percolation on isoradial graphs with critical weights (see also this one). The critical weights are those for which the model satisfies the star-triangle relation, and they use this relation, essentially, to transform the graph into a regular grid.

Tassion recently proved the RSW bound for a large class of percolation models including Voronoi percolation, improving upon a weaker result of Bolobás and Riordan.

Then, of course, there's work on RSW for dependent percolation models (FK-percolation, nodal sets of Gaussian random fields).


The strongest known RSW result could be the exact formula:

The problem is that critical percolation (in the scaling limit) has a larger group of symmetries than the grid it lives on at any finite scale. Therefore we need some way to find "holomorphic" observables that works in $\displaystyle \lim_{\epsilon \to 0} \mathbb{Z}^2 \to \mathbb{C}$. We could define a "discrete holomorhpic function on the grid by making sure it behaves correctly under translations:

$$ f(z + dz ) = f(z) + f'(z) \, dz $$

This is really two equations, one for deformations $dx$ and another for $i \, dy$ and these should obey some type of Cauchy-Riemann equations: $$ \frac{f(z + dx) - f(z)}{dx} = \frac{f(z + i\, dy) - f(z)}{i\, dy} $$

Another relation that one might try to discretize is the Cauchy integral formula: $$ \oint f(z) \, dz = 0 $$ When we could draw a small loop around $z$ and make sure the average sums to zero: $$ f(z + dx ) \,(2 i \, dy) - f(z + i dy) \,(2\,dx) - f(z - dx) \,(2i\, dy) + f(z - i dy) \,(2\,dx) = 0$$ RSW is needed to establish Hölder continuity of certain observables in the scaling limit. We start to get concern the boundararies of these random regions in the plane, are simply too rough.

These resources may be slightly out of date, and are obviously not what you are looking for. Thank you Kostya for correcting me.


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