Graduate Probability Seminar

We meet every Thursday 2 - 2:50 pm, in CIF 3038.

This semester, we are planning to learn about random walks. We will use two references: Random Walk: A Modern Introduction, by G. Lawler and V. Limic (link), and Modern Discrete Probability, by S. Roch (link). We plan to focus on random walks from the first book, while exploring some broader methods and connections from the second, including hitting times, Green’s functions, potential theory, coupling methods, and applications in electrical networks.

If you want to give a talk, please contact one of the organizers.

• September 3: Rui Yu, Liyuan Xu
  Title: Introduction to Random Walks and Organization Meeting
  Abstract: N/A

• September 10: Rui Yu
  Title:  Green’s function of random walks
  Abstract: This talk will be based primarily on Chapter 4 of Lawler and Limic’s Random Walk: A Modern Introduction. We give an overview of Chapter 4, introducing the Green’s function and its main properties. Particular attention will be given to Section 4.6, which studies the Green function for a random walk killed upon leaving a domain and relates expected occupation times to hitting probabilities. Finally, we will discuss some applications of Green function asymptotics, including the dimension-dependent behavior of intersections between independent random walks.

• September 17: Liyuan Xu
  Title: Dirichlet Problem and Lyapunov functions
  Abstract: A very classic topic appeared in potential theory is the Dirichlet problem. Today I will talk about how to construct a solution to the Dirichlet problem on a finite subset of Z^d under the framework of random walks. I will also introduce Lyapunov functions that can be used to estimate hitting times of random walks, followed by some intuitive examples.

• September 24: Taegu Kang
  Title: Random Walks and Electrical Networks
  Abstract: Electrical networks provide a useful framework for studying reversible Markov chains. In this talk, we introduce the correspondence between random walks and electrical networks, interpreting voltages as hitting probabilities and currents as expected net edge crossings. We then relate effective resistance to escape probabilities and show that a random walk is transient precisely when the effective resistance to infinity is finite. Examples on paths and trees illustrate how network reduction simplifies probabilistic calculations.

• October 1: Rui Yu
  Title: Variational Principles and Recurrence of Random Walks
  Abstract: Electrical networks provide useful tools for studying recurrence and transience of random walks. In this talk, we introduce Thomson’s and Dirichlet’s variational principles, which characterize effective resistance and conductance through energy minimization. We then discuss the Nash–Williams inequality and the characterization of transience by finite-energy flows. As an application, we give an electrical network proof of Pólya’s theorem.

• October 8: TBA
  Title: TBA
  Abstract: TBA

• October 15: Cancelled (Midwest Probability Colloquium)
  Title: N/A
  Abstract: N/A

• October 22: TBA
  Title: TBA
  Abstract: TBA

• October 29: TBA
  Title: TBA
  Abstract: TBA

• November 5: TBA
  Title: TBA
  Abstract: TBA

• November 12: TBA
  Title: TBA
  Abstract: TBA

• November 19: TBA
  Title: TBA
  Abstract: TBA

• December 3: TBA
  Title: TBA
  Abstract: TBA