Exploring hypergraphs with martingales

Recently, in [Random Struct Algorithm 41 (2012), 441-450] we adapted exploration and martingale arguments of Nachmias and Peres [ALEA Lat Am J Probab Math Stat 3 (2007), 133-142], in turn based on ideas of Martin-Lof [J Appl Probab 23 (1986), 265-282], Karp [Random Struct Alg 1 (1990), 73-93] and Aldous [Ann Probab 25 (1997), 812-854], to prove asymptotic normality of the number L-1 of vertices in the largest component L1 of the random r-uniform hypergraph in the supercritical regime. In this paper we take these arguments further to prove two new results: strong tail bounds on the distribution of L-1, and joint asymptotic normality of L-1 and the number M-1 of edges of L1 in the sparsely supercritical case. These results are used in [Combin Probab Comput 25 (2016), 21-75], where we enumerate sparsely connected hypergraphs asymptotically. (c) 2016 Wiley Periodicals, Inc. Random Struct. Alg., 50, 325-352, 2017