Infrared bound and mean field theory in percolation theory I have seen various references to the phrases "infrared bound" and "mean field theory", together or separately in the context of various lattice models. (Percolation, Ising Model, Interacting Particle Systems, Self-avoiding walks...)  If it helps to be concrete, let's talk about percolation theory.
I was hoping that the meaning of these phrases could be clarified.  I could not find an explanation of the rationale behind this terminology that didn't assume I already knew a large part of the corresponding theories.
I am also wondering about "lace expansion."  For this one, I have a resource from which I could learn a lot of the theory, but again there appears to be no quick layman's definition that I found in the literature.
 A: Although it is a bit late, I hope the answer is still useful. I will do the discussion based on the Ising Model with nearest-neighbors interaction.
It is a bit unfortunate to talk about infrared bounds on Bernoulli percolation for instance, as, from as far as I know, the model doesn't have that property.
So, Infrared Bound refer to a couple of equivalent theorems that relate the Discrete Fourier Transform of the two point function $F_L(x)=\langle \sigma_0 \sigma_x \rangle_{\mathbb{T}_L}$ where $\mathbb{T}_L$ refers to the discrete torus of length $L$ and the Green function of the simple random walk.
Mean Field Theory is usually used to refer to a simplified model related to the one you are studying, but where you disregard the geometry of the model. For instance, the Curie-Weilss Model is a simplification of the Ising Model, where you no longer have that each spin only interact with its neighbors, instead, in the Curie-Weilss Model, each spin interact with all the other spins, no matter the distance, in the same manner. That leads to significant simplifications for computations.
I don't know an intuitive explanation for the Lace expansion, but it is a geometric representation of the Random Cluster Model that is very useful to study critical phenomena.
Lastly, I recommend the book Statistical Mechanics of Lattice Systems:
a Concrete Mathematical Introduction. In chapter two, it studies the Curie-Weilss Model, and in chapter ten, it proves a version of the infrared bound. 
