Random semigroups à la Gromov

We introduce a model of random semigroups analogous to Gromov's density model for random groups. Fixing n >= 2 generators and a density $\delta \in (0,1)$, we impose $\lfloor{n^{2\delta l}}\rfloor$ uniformly chosen relations between distinct words of length at most l, and study the resulting semigroup as l tends to infinity. We show a phase transition at density 1/2: with probability tending to one, above this density the semigroup is trivial, while below it the semigroup contains a free subsemigroup of rank two and is neither left nor right amenable. We will also discuss small cancellation and residual finiteness at low densities.  Finally, we discuss the behavior at the critical density 1/2, highlighting some differences from the random group model.
Based on joint work with Be'eri Greenfeld and Margaret Shideler.