Threshold Intervals under Group Symmetries

Summary

This article contains a brief description of new results on threshold phenomena for monotone properties of random systems. These results sharpen recent estimates of Talagrand, Russo and Margulis. In particular, for isomorphism invariant properties of random graphs, we get a threshold whose length is only of order l/(log n)^2-ϵ, instead of previous estimates of the order l/log n. The new ingredients are delicate inequalities in the spirit of harmonic analysis on the Cantor group.

If A is monotone, then μ_p(A) is clearly an increasing function of p. Considering A as a “property”, one observes in many cases a threshold phenomenon, in the sense that μ_p(A) jumps from near to near 1 in a short interval when n→∞. Well known examples of these phase transitions appear for instance in the theory of random graphs. A general understanding of such threshold effects has been pursued by various authors (see for instance Margulis [M] and Russo [RD. It turns out that this phenomenon occurs as soon as A depends little on each individual coordinate (Russo's zero-one law). A precise statement was given by Talagrand [T] in the form of the following inequality.

Define for i = 1, … , n where U_i(x) is obtained by replacement of the i-th coordinate X_i by 1 - X_i and leaving the other coordinates unchanged. The number μ_p(A_i) is the influence of the i-th coordinate (with respect to μ_p).