# Graduate student reading seminar

Time and place: Friday 2:30PM-4PM, B211

## Contents

## Electrical networks

Random Walks and Electric Networks by Doyle and Snell

Probability on Trees and Networks by Russell Lyons with Yuval Peres

February 3: Review 2.1 and 2.2 from the Lyons-Peres book and read 2.3

February 10: Read 2.4 and start reading 2.5

February 17: Read 2.5

February 24: Read 2.6

HW problem: Let [math] B_N=[-N,N]^2[/math] and let [math]E_N=\{(x,y): x=N, |y|\le N\}[/math] the east side of this box. Consider a simple RW started at (0,0) on the lattice where the jump probabilities are [math]1/4-\epsilon, 1/4, 1/4+\epsilon, 1/4[/math] for the W, N, E and S directions ([math]\epsilon \gt 0[/math] is fixed). Let [math]\tau_N[/math] be the hitting time of the boundary of [math]B_N[/math]. Show using the machinery of electrical networks that [math]P(X_{\tau_N}\in E_N)\to 1[/math]. How can you change the aspect ratio of the box so the result stays true?

## Stochastic Calculus

March 2: Read the first section of the lecture notes. March 9: Read the second section. March 16: 3rd section (up to Ito integrals)

## Fall 2011

## Determinantal point processes

Determinantal point processes: Chapters 4 and 6

Determinantal random point fields by Alexander Soshnikov

Terry Tao's blog entry on determinantal point processes

Random matrices and determinantal processes by K. Johansson

Determinantal point processes by A. Borodin

Determinantal probability measures by R. Lyons

September 13: start reading the HKPV book (Chapter 4). You can also have a look at the other survey articles listed above.

September 20: finish Section 4.2 and go through the first example in 4.3 (non-intersecting random walks)

September 27: Corollary 4.3.3, the rest of the examples in 4.3 and 4.4 (how to generate determinantal processes)

October 4: there is no reading seminar (you should go to the Probability Seminar instead)

October 11: start reading Section 4.5

October 18: existence and the necessary and sufficient condition (4.5)

October 25: there is no reading seminar this week

November 1: simultaneously observable subsets (end of 4.5), 4.6-4.8

November 8: High powers of complex polynomial processes (4.8), uniform spanning trees (6.1)

November 15: Uniform spanning trees cont. (6.1)

November 22: Ginibre ensemble, circular ensemble (.2, 6.4)

## Electrical networks

Random Walks and Electric Networks by Doyle and Snell

Probability on Trees and Networks by Russell Lyons with Yuval Peres

December 6: Electrical networks. Start reading Chapter 2 of the Lyons-Peres book.

December 13: Continue reading Chapter 2