Resumen de Scaling window for mean-field percolation of averages
For a complete graph of size n, assign each edge an i.i.d. exponential variable with mean n. For λ>0, consider the length of the longest path whose average weight is at most λ. It was shown by Aldous [Combin. Probab. Comput. 7 (1998) 1–10] that the length is of order logn for λ<1/e and of order n for λ>1/e. Aldous [Open problems (2003) Preprint] posed the question on detailed behavior at and near criticality 1/e. In particular, Aldous asked whether there exist scaling exponents μ, ν such that for λ within 1/e of order n−μ, the length for the longest path of average weight at most λ has order nν.
We answer this question by showing that the critical behavior is far richer: For λ around 1/e within a window of α(logn)−2 with a small absolute constant α>0, the longest path is of order (logn)3. Furthermore, for λ≥1/e+β(logn)−2 with β a large absolute constant, the longest path is at least of length a polynomial in n. An interesting consequence of our result is the existence of a second transition point in 1/e+[α(logn)−2,β(logn)−2]. In addition, we demonstrate a smooth transition from subcritical to critical regime. Our results were not known before even in a heuristic sense.
