Characterizing Optimal Sampling of Binary Contingency Tables via the Configuration Model

A binary contingency table is an m×n array of binary entries with row sums r = (r1, . . . , rm) and column sums c = (c1, . . . , cn). The configuration model generates a contingency table by considering ri tokens of type 1 for each row i and cj tokens of type 2 for each column j, and then taking a uniformly random pairing between type-1 and type-2 tokens. We give a necessary and sufficient condition so that the probability that the configuration model outputs a binary contingency table remains bounded away from 0 as N = ∑m i=1 ri = ∑n j=1 cj goes to ∞. Our finding shows surprising differences from recent results for binary symmetric contingency tables.