ON RANDOM REPRESENTABLE MATROIDS.
Results are obtained on the likely connectivity properties and sizes of circuits in the column dependence matroid of a random r multiplied by n matrix over a finite field, for large r and n. In a sense made precise in the paper, it is shown to be highly probable that when n is less than r such a matroid is the free matroid on n points, while if n exceeds r it is a connected matroid of rank r. Moreover, the connectivity can be strengthened under additional hypotheses on the growth of n and r, using the notion of vertical connectivity; and the values of k for which circuits of size k exist can be determined in terms of n and r.
Publication Source (Journal or Book title)
Studies in Applied Mathematics
Kelly, D., & Oxley, J. (1984). ON RANDOM REPRESENTABLE MATROIDS.. Studies in Applied Mathematics, 71 (3), 181-205. https://doi.org/10.1002/sapm1984713181