The largest components C_1,C_2,C_3,... of a G(n,p) random graph in the critical window np=1+t*n^{−1/3} are known to converge, in several concrete meanings, to a non-trivial random scaling limit. For example, the size of the largest component divided by n^(2/3) converges in distribution to a continuous random variable that was famously identified by Aldous in the 90’s. 

In this talk we will study the sensitivity of the limiting properties of these components under a noise operator that resamples every edge independently with probability epsilon. For instance, we ask: for which values of epsilon are the properties “|C_1| exceeds 2*n^(2/3)” or “the diameter of C_2 is smaller than its median” noise-sensitive?

Our main finding is a dichotomy for noise sensitivity that coincides with the critical window: If the noise parameter epsilon >> n^(−1/3) then such properties are noise-sensitive, while for epsilon << n^(−1/3) they are noise-stable.

If time permits, we will also present the very same dichotomy for the noise-sensitivity of metric properties of the random Minimum Spanning Tree of the complete n-vertex graph.

Based on joint works with Eyal Lubetzky and Omer Israeli.