First passage percolation is a natural way to put a random metric on a graph; instead of assigning each edge length 1 as in the standard graph distance, one assigns each individual edge a random length by independently sampling from a fixed probability distribution called the "weight measure." The distance between two vertices is then the minimal total length of an edge path between them.
Probabilists study the large-scale geometry of these random metric spaces; one classical theorem states that in many cases of interest, the random metric space obtained has a *non-random* scaling limit. Therefore, this construction gives a
map from probability measures on the nonnegative numbers ("weight measures") to metric spaces (scaling limits).
This map is hard to get one's hands on and in many ways quite mysterious. A classical paper of van den Berg and Kesten shows that in the case that our graph is the standard Cayley graph of Z^d, this map is in some sense strictly monotonic; under mild assumptions, if one of the weight measures is "strictly smaller" than the other, then the resulting scaling limit is also "strictly smaller". It turns out that this question is closely related to the question of what kinds of edges long geodesics in our random metric space end up using.
I will explain a recent paper of mine where I prove an extension of this classical result of van den Berg and Kesten to the case of graphs with a uniform upper and lower polynomial growth bound of the same degree. Perhaps surprisingly, the result does not require any quasi-transitivity assumption on the graph (although one gets nicer, stronger results under such an assumption). If time allows I will say something about a stronger form of monotonicity which is equivalent to fine-geometric condition on the graph, or I will say something about why "mild conditions" on our measures are necessary for strict monotonicity to hold---audience's choice.