Russo's formula and pivotal edges
/ Russo: ROO-soh; Margulis: mar-GOO-lis /
How fast does the probability of a connectivity event change as we turn up p? Russo's formula gives the exact derivative, and in doing so identifies WHICH edges are responsible for the change: the pivotal edges, those whose flipping from closed to open would tip the event from not-occurring to occurring. It converts a question about a derivative in the parameter into a geometric question about pivotal locations, and it is the engine behind almost every sharp-threshold and critical-exponent argument.
An edge e is pivotal for an increasing event A in a configuration omega if A occurs when e is forced open and fails when e is forced closed (holding all other edges fixed). Russo's formula (also Margulis-Russo) states that for product measure P_p and an increasing event A depending on finitely many edges, d/dp P_p(A) = sum over edges e of P_p(e is pivotal for A). The probabilistic content: increasing p slightly raises P_p(A) at a rate equal to the expected number of pivotal edges — the more 'edges on the knife's edge', the more sensitive A is to p. There is a sharper differential-inequality version due to the same circle of ideas and to the BKR/Talagrand machinery: the derivative is controlled by the total influence of the edges, and when many edges each have small influence (the situation near criticality), the threshold is sharp — P_p(A) jumps from near 0 to near 1 over a parameter window that shrinks as the system grows.
Russo's formula is the bridge from correlation inequalities to phase-transition sharpness. Combined with FKG and BK it yields the OSSS / influence inequalities that Duminil-Copin, Raoufi and Tassion used to give short modern proofs that the percolation transition is sharp and that magnetization in the Ising model has exponential decay off-critical. The thing to watch: pivotality is defined per configuration and per event, and edges that are pivotal are typically RARE and special — near the critical point a single pivotal edge can decide a macroscopic crossing, which is exactly why critical percolation is so delicate. Also note the formula needs A to be increasing (or one handles general A by splitting into positive/negative influence); for non-monotone events the clean identity must be stated with care.
Let A be 'there is an open left-to-right crossing of an n-by-n box'. An edge e is pivotal exactly when removing it would break the only available crossing and adding it would complete one — geometrically, when there is an open path to e from the left, an open path from the right, and closed (dual) arms separating top and bottom. Russo's formula then gives d/dp P_p(A) as the expected number of such edges, which is large precisely near p = 1/2.
A pivotal edge for a box-crossing: four arms (two open, two dual-closed) meet at e.
Russo's formula needs the event to be increasing and to depend on finitely many edges; for infinite-volume or non-monotone events one works with influences and differential inequalities. Pivotal edges are typically rare, yet near criticality a single one can decide a macroscopic event.