(Disclaimer: I'm not all that familiar with any of the two topics)
Consider the Ising model on some graph with, lets say, two heavily inter-connected components that are sparsely connected between each other.
This "bottleneck" would imply (or would be implied by?) that the Cheeger constant of the graph (and thus the first eigenvalue of the Laplacian matrix) is small and also that it would be very difficult for the spins in one component to interact with the spins in the other component.
My question then is: is there any result linking the Cheeger constant with the "magnetization" properties of the graph? Something like a bound on the temperature at which the phase transition occurs that depends on the Cheeger constant or vice versa?
Thanks in advance!