sharp versus coarse thresholds

Once we know a monotone property has a threshold p*, the next question is how abruptly the transition happens. A threshold is sharp if the probability of the property jumps from near 0 to near 1 over a window of p that is much narrower than p* itself — formally, if for every fixed epsilon > 0 the property goes from probability epsilon to probability 1 - epsilon as p moves through an interval of length o(p*). A threshold is coarse if the transition is spread out over a window comparable in size to p* (a constant factor of it). The distinction is the difference between a sudden switch and a gradual fade.

The deep fact, made precise by Friedgut and Kalai and then by Friedgut's sharp-threshold theorem, is that whether a property is sharp or coarse is governed by symmetry and 'locality'. Coarse thresholds are caused by local obstructions: a property has a coarse threshold essentially only when it can be approximated by the appearance of a fixed bounded-size subgraph. Containing a triangle is coarse precisely because it is the local event 'somewhere there are three mutually joined vertices', and near the threshold the number of triangles is approximately Poisson, so P(at least one triangle) tends to 1 - e^(-mu) which moves smoothly from 0 to 1 as the mean mu = (np)^3/6 sweeps from 0 to infinity over a constant-factor window of p. Global, symmetric properties that cannot be reduced to a local witness — connectivity, Hamiltonicity, k-colourability, having no isolated vertex — have sharp thresholds; the Friedgut criterion says a sharp threshold fails only in the presence of such a local cause.

This matters because sharpness tells you whether a system behaves like a clean phase transition (water to ice) or a soft crossover. For sharp thresholds one can often pin down the exact critical window and even the limiting probability inside it; for connectivity, for instance, the limiting probability is e^(-e^(-c)) when p = (log n + c)/n, the same double-exponential law as the count of isolated vertices becoming Poisson. A common misconception is that all interesting thresholds are sharp; the honest picture is that subgraph-appearance thresholds are coarse and Poisson-governed, while most 'global' monotone properties are sharp — and proving sharpness in a given case can require the heavy machinery of discrete Fourier analysis and influence inequalities.