## Osion kuvaus

• ### Yleinen

Course description:

The central topic of this course is the following question. Let 0<p<1, and consider the d-dimensional square lattice Z^d. For each edge of Z^d, independently delete the edge with probability 1-p, and keep it with probability p. What remains is a random subgraph of Z^d. What is the probability that this random subgraph contains an infinitely large connected component? And how does this depend on p (and on d)? We will answer these questions and more.

You can get a further feel for the topic by watching the second part of this video: https://www.youtube.com/watch?v=a-767WnbaCQ&t=97s

Overview:

Definitions, some measure theory. Basic properties of percolation: monotonicity, existence of phase transition, ergodicity, unique infinite cluster. Covariance formula, sharpness, FKG and BK inequalities. Percolation in Z^2: RSW estimates, p_c=1/2, conformal invariance.

We will roughly follow the lecture notes of Hugo Duminil-Copin, https://www.ihes.fr/~duminil/publi/2017percolation.pdf. One can do further reading in Grimmett's book: https://www.statslab.cam.ac.uk/~grg/papers/perc/perc.html.

One does not need to have taken a measure theory course - we will introduce the measure theory that we need as we go.

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Assessment:

There will be one exercise sheet each week. To pass the course, you need to complete a solution to at least one problem per week. You will then either present your solution on the board at the exercise class, or hand in a written solution at the start of the exercise class (we will randomly select those who present). Based on numbers, you can expect to present 2 or 3 times out of the six weeks. Grading will just be pass/fail.

To help select who will present, before the exercise class please fill in which exercises you've completed: https://forms.gle/xGTo6gVGxtvLcDyF9

Exercise sheets will appear here on the MyCourses page.

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Lecture notes will appear here on the MyCourses page.