Understanding the wrapping criterion in percolation theory

Question

Context:

When studying percolation in finite sized systems, there exist various definitions and criteria for determining when a given system is percolating, i.e., given a definition for connectivity, it contains a system-spanning cluster which mimics that of an infinite cluster in the limit of infinite system sizes. Examples of percolative systems might be physical, such as molecular systems, or more mathematical such as bond/site percolation in 2D lattice domains.

Two commonly used definitions for qualifying a cluster as percolating are

• The side-to-side spanning clusters, where a cluster is found to connect two opposing sides/walls of the system together. This is generally used when the system has free boundaries (no periodic conditions).
• The wrapping criterion is another one, where a cluster wraps around the system (box, domain, etc). This definition is used when the system domain is endowed with periodic boundary conditions. Wrapping is usually described as follows: all constituent bonds/sites in the wrapping cluster are connected by a contiguous path to their own periodic image.

More formally, below are two excerpts from Newman and Ziff 2001 (also relevant is Fig 7.):

Cluster spanning: In many calculations one would like to detect the onset of percolation in the system as sites or bonds are occupied. One way of doing this is to look for a cluster of occupied sites or bonds which spans the lattice from one side to the other...

Cluster wrapping: An alternative criterion for percolation is to use periodic boundary conditions and look for a cluster which wraps all the way around the lattice...

However, at least to me, the latter is still a very counter-intuitive image of what such cluster might look like, and how it differs from the more conventional spanning definition. And naively, on which level is having a cluster comprised of constituents that are connected to their respective periodic images equivalent to the wall-to-wall spanning definition of percolation.

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Questions:

1. Is there an intuitive way of seeing what the wrapping criterion entails? I am struggling in two particular senses: first, simply understanding what it means for a particle/bond/site to be connected to its periodic image, does that mean that if we draw neighbouring periodic images of our system, we see the cluster continuing in these images? (I have not found images that somehow visualise this idea).

2. And secondly, how does the wrapping definition relate to the more conventional spanning criterion used in percolation? For instance, is wrapping always a stronger condition? (namely, a wrapping cluster is also spanning in the usual sense if we were to remove periodic boundaries?...).

Any pictures/examples, or references where such questions might be tackled would be much appreciated. Unfortunately, I still cannot wrap my mind around the wrapping percolation criterion (no pun intended), so any conceptual or intuitive insights would definitely be very helpful.

Answer 1

Q1: Here is an image that shows a wrapping cluster [source]. So yes, the wrapping condition means that the cluster would extend out to infinity if the lattice is repeated periodically in all directions. Just imagine tiling the plane with the image, and you would find a band of colored sites extending from the lower left to the upper right.

Q2: This example also shows that wrapping is a weaker condition than spanning: the image does not have a cluster that connects opposite edges, so there is no spanning cluster while there is a wrapping cluster.