High-Dimensional Probability
Geometry, concentration, random matrices, and stochastic processes
High-dimensional probability studies random objects whose ambient dimension is large enough that ordinary geometric intuition becomes unreliable. Volume migrates toward boundaries, naïve discretizations become impossibly large, and yet randomness often produces remarkably stable behavior.
The organizing question. Which notions of size and complexity continue to be useful when dimension is large, and how do they control the behavior of random vectors, matrices, and processes?
These notes support MATH 598 / 784 at McGill University in Fall 2026.
Themes
High-dimensional geometry and basic probabilistic tools
The curse of dimensionality, convex hulls and sparse approximation, Gaussian width, and tails of random variables.
Concentration, embeddings, and random matrices
Orlicz norms, Bernstein’s inequality and dimension reduction, coordinate embeddings, and extreme singular values.
Convex applications and covariance estimation
Semidefinite relaxations, Grothendieck-type inequalities, Gaussian matrices, and empirical covariance operators.
Gaussian processes
Comparison inequalities, metric entropy, Gaussian concentration, and Gaussian width as a geometric invariant.
Chaining and extremal applications
Dudley’s entropy integral, generic chaining, concentration of energy, and smallest-singular-value problems.
Sources and acknowledgements
The selection and organization of the course owe a particular debt to three resources:
- Roman Vershynin, High-Dimensional Probability: An Introduction with Applications in Data Science (Vershynin 2026);
- Ramon van Handel, Probability in High Dimension (Handel 2016); and
- Shahar Mendelson, Probabilistic Methods in Analysis (Mendelson 2023).
The notes synthesize ideas from these sources with material developed in earlier versions of the course. Any errors in this presentation are, of course, mine.
Fall 2026
Course logistics live separately from the durable notes. See the course page, the date-by-date calendar, and the exercise sheets.
Download Exercise Sheet 1 (PDF). The sheet contains Exercises 4.2.5, 4.2.9, 4.2.10, and 4.2.16 from the first edition of Vershynin’s High-Dimensional Probability.