Hi Williams,
you ask whether photons of a smaller wavelength would be more impeded by the atoms of space.
I wish I were already there!
The general question behind this, I guess, would be: what are, really, particles in a causal set, intended as a discrete model of spacetime?
A general answer would be: since the only tool we have for building the universe is a discrete spacetime -- a directed graph made of nodes (events without attributes) and causal relations among them -- a particle is a trajectory, a worldline, a periodic pattern made of events.
But talking about the SPEED of a particle in a causal set is already quite difficult, since we need a reference frame, and that's not easy to define either, since we cannot enjoy the advantages of a continuous setting, such as Minkowski spacetime. One way to proceed would be to identify the origin of a reference frame with a particle, as defined above, so that you end up with a system of particles that observe each other and detect their mutual speeds... But I have not jet investigated this line of thought.
Defining what a photon is in a causal set is indeed particularly challenging, for the following reason.
While the definitions of time-like and space-like sets of events are immediately available, via the notions of chain and anti-chain for partial orders, the definition of light-like path is problematic, since you do not have a continuous background where you can take limits.
The difficulty is as follows. Consider the set of nodes forming the photon's trajectory.
If these are in time-like relation, we have something for which Lorentz distance progresses (Lorentz distance between two events in a causal set is the length of the longest directed path between them - this works fine, as the people in the Causal Set Programme know well), but then we would have a progression of the photon proper time, which contradicts its living on a null spacetime cone.
If the points are in a space-like relation, no information can be carried by the photon, violating the idea that this particle is the fastest causality carrier.
The attitude one assumes with the type of research I have described is to set up computational experiments and, basically, see what happens, without preconceived expectations. I find this approach justified in light of the highly creative power exhibited by emergence in simple models of computation. Of course the idea is then to establish relations between what emerges and familiar physical phenomena, as I suggested, for example, with the entanglement-like effect in Figure 3 of my essay.
At the moment, photons and null cones appear to be still at large, in my causets.
