Description

How to sample diverse subsets? How to find optimal measurement locations for a signal? How to approximate an integral using from function values? How to approximate the spectrum of an integral operator? This book looks at all these questions (and many more) from the viewpoint of Determinantal Point Processes, an important class of point processes that feature repulsion. The first part of the book is a didactical introduction to DPPs in the discrete case, more advanced topics are covered in the second part. This draft is not fully complete, I am planning to add some more material on theory and applications. If you find any errors or typos, please let me know!

The draft is available here

Update Jan 20, 2026

Many thanks to Li-Fu Chen (U Utah) and an anonymous reviewer. Various typos have been fixed in chapters 0 to 7.

More significant errors that are now fixed:

  • In ch. 2, the computation of the error rate for stratified sampling in 1D integrals was incorrect (the result was stated as $\mathcal{O}(n^{-1})$, it is actually $\mathcal{O}(n^{-3/2})$ )
  • In ch. 3, there was a hole in the proof of the Macchi-Soshnikov theorem, identified by an anonymous reviewer. The section has been largely rewritten to use the moment-generating function, simplifying some of the proofs. A similar hole was present in the continuous version of the theorem (ch 7) as well, a correct proof is now sketched (I’ll write a full proof later, in the meanwhile the reader can have a look at Soshnikov’s proof).
  • In ch. 7, there was an error, identified by Li-Fu Chen, in an equation for $\tilde{c}(x,y)$ (now on page 195), in the case of non-uniform weights.

Things that still need to be fixed:

  • Some confusing notation in ch. 5
  • Ch. 6 is unclear in places. The next version will be more explicit about reference measures, symmetric functions and maybe Palm measures if I work up the courage.

Table of Contents

  • Chapter 1: Determinantal Point Processes in the Wild
  • Chapter 2: Generalities on Discrete Point Processes
  • Chapter 3: Determinantal Point Processes in the Discrete Case
  • Chapter 4: Applications of discrete DPPS
  • Chapter 5: Sampling, Design, Quadrature, etc.
  • Chapter 6: Spatial Point Processes
  • Chapter 7: Determinantal Point Processes in the Continuous Case
  • Chapter 8: Orthogonal Polynomial Ensembles

Citation

Simon Barthelmé (2024). An Introduction to Determinantal Point Processes and Related Topics. Draft.

@book{IntroDPP,
author = {Simon Barthelmé},
year = {2024},
title = {An Introduction to Determinantal Point Processes and Related Topics},
publisher = {DRAFT},
url = {}}