# Topics in high-dimensional probability

## Summary

This is a theoretical course about probability in high dimensions. We will look at some mathematical phenomena appearing as the number of random variables grows large - e.g. concentration of measure or universality.

## Content

There are several interesting properties that become visible in large random systems like large random graphs, or random walks or random matrices etc...

For example, in such systems one observes:

- Averaging and concentration: the simplest example is the Law of Large numbers, where the average of i.i.d. random variables converges to its mean. In fact, more generally functions of many independent random variables will often be close to their expectation and one can quantitatively bound the flucatuations.

- Universality: in different large systems microscopic properties might lose their importance and some universal properties appear. The simplest example is the Central limit theorem - if you add up i.i.d. random variables with finite variance and normalize properly, the Gaussian law always appears and the fine details of initial laws of the random variables don't matter. There are much richer examples of such phenomena.

We look into these two topics and also some others.

## Learning Prerequisites

## Required courses

Mathematics Bachelor's level knowledge of analysis, linear algebra and probability (for example, the Bloc "Science de Base" in EPFL Mathematics Bachelor's program). I will assume working knowledge of measure-theoretical probability / vocabulary, even though it won't be central.

## Assessment methods

Oral exam

## Resources

## Bibliography

In parts I will probably follow Ramon van Handel "Probability in high dimension" available on his webpage.

## Moodle Link

## In the programs

**Semester:**Fall**Exam form:**Oral (winter session)**Subject examined:**Topics in high-dimensional probability**Lecture:**2 Hour(s) per week x 14 weeks**Exercises:**2 Hour(s) per week x 14 weeks**Type:**optional

**Semester:**Fall**Exam form:**Oral (winter session)**Subject examined:**Topics in high-dimensional probability**Lecture:**2 Hour(s) per week x 14 weeks**Exercises:**2 Hour(s) per week x 14 weeks**Type:**optional

**Semester:**Fall**Exam form:**Oral (winter session)**Subject examined:**Topics in high-dimensional probability**Lecture:**2 Hour(s) per week x 14 weeks**Exercises:**2 Hour(s) per week x 14 weeks**Type:**optional

**Semester:**Fall**Exam form:**Oral (winter session)**Subject examined:**Topics in high-dimensional probability**Lecture:**2 Hour(s) per week x 14 weeks**Exercises:**2 Hour(s) per week x 14 weeks**Type:**optional