A chord diagram is a perfect matching on 2n points arranged on a circle, equivalently a partition of {1,...,2n} into n two-element chords. Suppose a chord diagram of size n is chosen uniformly at random from the (2n-1)!! possibilities. Any subset of k chords induces one of (2k-1)!! possible patterns, and we study the joint distribution of the number of occurrences of each pattern for large n.

---

We show that this vector of pattern counts exhibits diverse scaling behaviors with n in different directions, and we characterize the corresponding subspaces of different orders of magnitude, using tools from the representation theory of the symmetric group and algebras of words. To leading order, the distribution of pattern counts is multivariate normal and supported on a k(2k-3)-dimensional subspace. We explicitly describe and diagonalize its covariance matrix for every k. We also relate this spectral decomposition to the spectrum of a dynamical process on k-chord diagrams in which, at each step, one endpoint of a chord is relocated to a uniformly random position.

---

MSc advisor: Chaim Even-Zohar