One application of the probabilistic method in topology was found by Melanie Matchett Wood and myself:

Let $H$ be the finite group $(\mathbb Z/15) \rtimes Q_8$, where generators $i$ and $j$ of $Q_8$ act on $\mathbb Z/15$ by multiplication by -1 and 4 respectively.

Let $S$ be a finite set of primes including $2,3,$ and $5$. Then there exists a 3-manifold $M$ such that $H$ is the maximal quotient of $\pi_1(M)$ of order divisible only by primes in $S$.

However, $H$ is not itself the fundamental group of any 3-manifold.

In fact, $H$ is the smallest group with this property.

The existential part of this statement is proven using the probabilistic method, i.e. we prove that a *random* 3-manifold has a fundamental group of this form with positive probability. The notion of random 3-manifold we use was defined by Dunfield and Thurston, who also suggested using the probabilistic method to prove the existence of 3-manifolds with particular properties.

The relevant probability distribution is on a space of profinite groups, those most of the calculations reduce to sets of finite groups, which are discrete, so one could argue that this is discrete in disguise, but the set of isomorphism classes of finite groups certainly has a different flavor from what's usually considered in the probabilistic method in discrete mathematics. One could make the statement less discrete, though also less concrete, by saying we find a characterization of the closure of the set of fundamental groups of oriented 3-manifolds inside the space of isomorphism classes of profinite groups (with a natural topology) by the probabilistic method.

A reference is Proposition 8.17 of our paper [Finite quotients of 3-manifold groups][1]. 


  [1]: https://arxiv.org/pdf/2203.01140.pdf