What is the probability that the number of triangles in an Erdős–Rényi random graph exceeds its mean by a constant factor? In this talk, I will discuss some recent progress on this problem.
Already the order in the exponent of the tail probability was a long standing open problem until several years ago when it was solved by DeMarco and Kahn, and independently by Chatterjee. We now wish to determine the exponential rate of the tail probability. Thanks for the works of Chatterjee--Varadhan (dense setting) and Chatterjee--Dembo (sparse setting), this large deviations problem reduces to a natural variational problem. We solve this variational problem asymptotically, thereby determining the large deviation rate, which is valid at least for p > 1/n^c for some c > 0.
Based on joint work with Bhaswar Bhattacharya, Shirshendu Ganguly, and Eyal Lubetzky.
- Combinatorial Theory Seminar