Abstract: Construct a random m × n matrix by independently setting each entry to 1 with probability p and to 0 otherwise. We study the joint distribution of the row sums s = (s1, ..., sm) and column sums t = (t1, ..., tn). Clearly s and t have the same sum, but otherwise their dependencies are complicated. We prove that under certain conditions the distribution of (s, t) is accurately modelled by (S1, ..., Sm, T1, ..., Tn), where each Sj has the binomial distribution Binom(n, p'), each Tk has the binomial distribution Binom(m, p'), p' is drawn from a truncated normal distribution, and S1, ..., Sm, T1, ..., Tn are independent apart from satisfying Σj=1,...,m Sj = Σk=1,...,n Tk. We also consider the case of random 0-1 matrices where only the number of 1s is specified, and also the distribution of s when t is specified. In the seminar I will include details of one of the bounding arguments used in this last case. This bounding argument is an application of the generalised Doob's martingale process.
These results can also be expressed in the language of random bipartite graphs.
Joint work with Brendan McKay.