Suggested texts
| Date | Topics | Relevant text | Suggested exercises |
|---|---|---|---|
| Feb 13 | Random walk basics: LLN, CLT, LCLT | D Theorem 3.5.2 | (HE) Exercises 1.8, 1.9; (RW) Exercises 1.4, 1.7, 2.7, 2.9 |
| Feb 20 | LCLT, Ornstein coupling, RW proof of Liouville theorem | D Theorem 3.5.2 | Show that two simple random walks in \(\mathbb{Z}^d\) can be coupled so that they are eventually equal. (Hint: use the Ornstein coupling idea in each dimension separately.) And: (RW2) Exercises 2.4, 2.6 |
| Feb 27 | Green's function (transient case) | I Chapter 2, RW Chapters 4, 6 | Let \(z = (1,0, \ldots, 0) \in \mathbb{Z}^d\). Show that the probability that a SRW \(S\) ever visits \(z\) is \(p = 1-G(0,0)^{-1}\). Let \(T\) be the first return time of \(S\) to the set \(\{0, z\}\). Show that \(\mathbb{P}(S_T = 0) = \mathbb{P}(S_T = z) = \frac{p}{1+p}\). And: (RW) Exercises 4.1, 4.6, 4.7 |
| Mar 6 | WLLN for Range of 2D SRW; Midpoint of 1D SRW Range is Uniform | Erdos & Dvoretzky (1950), Sections 2-3, Discussion of Hot Potato puzzle | Let \(M_n, m_n\) denote (respectively) the maximum/minimum of the range \(R_n\) of 1D SRW at time \(n\). Prove or disprove: \(\frac{M_n-m_n}{\sqrt{n}}\) converges in distribution as \(n \to \infty\). If it converges, identify the limit. |
| Mar 13 | Last exit decomposition, capacity, Wiener's test, potential kernel in \(\mathbb{Z}^2\) | I Chapter 2, RW Chapters 4, 6 | Use the potential kernel in \(\mathbb{Z}^2\) to compute the asymptotic for the expected number of visits to site \(x\) by a simple random walk started at 0 and stopped on hitting 0 (again) as \(|x| \to \infty\). |
| Mar 20 | Bernoulli Percolation in \(\mathbb{Z}^d\), Peirels argument | DC Chapter 2 | (DC) Exercises 5, 8, 10 |
| Mar 27 | Basic tools: Increasing coupling, FKG, Russo | DC Chapter 2 | (DC) Exercises 11, 14 |
| April 3 | \(p_c = \frac{1}{2}\) for Bernoulli bond percolation in \(\mathbb{Z}^2\), via influences | Sharp threshold phenomena in statistical physics |