abstract: With respect to various notions of randomness, one can ask what topological and geometric behaviors of a 3-manifold obtain `generically'. In this talk I will discuss a proof of a conjecture of N. Dunfield and W. Thurston that the volume of a random Heegaard splitting grows linearly in the word-length of the gluing map as an element of the mapping class group. This talk represents joint work with Igor Rivin and Juan Souto.