The probabilistic method outside of discrete mathematics

Question

The probabilitic method is a genius idea in combinatorics, graph theory etc, where instead of constructing something by hand, you construct the thing randomly and show that there is a positive probability of whatever property you wanted to hold holding.

Q: Are there examples of this method being used to prove results in other areas of mathematics, e.g. geometry, topology, algebraic number theory,... ?

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For the purposes of this question, I want to avoid results which are "discrete mathematics in disguise", e.g. concerning structures which live on a triangulation of a topological space (but using discrete approximations of topological spaces to prove an original property of that space is OK). I would also like to avoid (non algebraic) number theory, because applications there are well known.

Edit: Finally, while there is a similar class of methods in algebraic geometry, e.g. where you show that the set of counterexamples has positive codimension, or restrict to generic points, I would like to avoid these examples because they are also fairly well known and have a slightly different flavour to the combinatorics probabilistic method

Answer 1

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.

Answer 2

The Wiener measure gives positive weight to those continuous functions $f:[0,1]\to\Bbb R$ that are nowhere differentiable. In particular, there do exist such functions.

Answer 3

The book Introduction to Banach spaces II by Daniel Li and Hervé Queffélec has a chapter on "the method of selectors" which seems to be a subgenre of the probabilistic method. Here is a sample of the applications given in the chapter:

• A theorem of Bourgain allowing on the extraction of "big" quasi-independent sets, with applications to theorems of Pisier and Drury about Sidon sets.

• A theorem of Bourgain on sums of sines: for any integer $N \geq 1$, there exists a subset $\Lambda \subseteq \mathbb N^*$, of cardinality $\lvert \Lambda \rvert = N$, such that $\left\| \sum_{k\in\Lambda} \sin(kx)\right\|_\infty \leq C_0\,N^{2/3}$, where $C_0$ is a numerical constant. This has an application to vectorial Hilbert tranforms.

• A theorem of Bourgain on the geometry of Banach spaces (a lower bound on the "K-convexity constant").

I don't know these subjects too well, but I feel that each of these examples live in a rich ecosystem of results, some of which also answer your question. For example, here are two other results on trigonometric polynomials which can also be obtained by probabilistic methods:

• Salem and Zygmund proved in 1954 that given amplitudes $\rho_1, \ldots, \rho_N$ and phases $\phi_1, \ldots, \phi_N$, one can find (random) signs such that $\left\| \sum_{k=1}^N \pm \rho_k \,\cos(kt+\phi_k) \right\|_\infty \leq C\,\sqrt{\sum_{k=1}^n \rho_k^2} \sqrt{\ln N}$, where $C$ is an absolute constant.

• Uchiyama proved in 1965 that there are subsets $\Lambda \subseteq [\![ 1, N ]\!]$ such that $\left\| \sum_{k \in \Lambda} e^{ikx} \right\|_1 \geq c\,\sqrt N$, where $c$ is an absolute constant.

I know I would love to see a user-friendly survey on this kind of constructions. Kahane's book is probably very enlightening, but it remains a bit formidable-looking to me.