**Class Timings: **Friday 9am to 1pm at ISI

**Mode: **Online

**Course Description: **i) Review of discrete probability, First and Second Moment methods, Chernoff bounds and some applications. ii) Percolation on lattices: Phase-transition phenomena, subcritical and supercriti- cal phases, Uniqueness. iii) Random graphs: Phase transition, Influences, Russoï¿½s formula and Sharp thresh- olds. Noise Sensitivity and Stability. iv) Introduction to Markov chains and Martingales. Branching processes. Random walks and electrical networks, Uniform spanning trees.

**References: **

(a) C. Garban and J. Steif: Noise Sensitivity of Boolean Functions and Percolation.

(b) N. Lanchier: Stochastic Modelling.

(c) Sebastien Roch: Modern Discrete Probability: A toolkit. (Notes).

(d) R. Lyons and Y. Peres: Probability on trees and networks.

(e) M. Barlow: Random walks and heat kernel on Graphs.

**Prerequisites: **Basic Probability Theory, Analysis, Markov Chains discrete time (not necessary will help), Measure theory (not necessary will help),R-Programming (not necessary will help).

**Evaluation: **TBA

**Outcome: **Students will be able to understand Random walks, Harmonic functions and other aspects of discrete Probability models at the end of the course.

- Teacher: Siva Athreya