Percolation Models

Department: 
Math
Course Number: 
8803-DAM
Hours - Lecture: 
3
Hours - Lab: 
0
Hours - Recitation: 
0
Hours - Total Credit: 
3
Typical Scheduling: 
Not regularly scheduled

Special Topics course offered in Fall 2020 by Michael Damron.

Prerequisites: 

MATH 6241 , MATH 6242 

 

Course Text: 

We will likely use different sources, but probably the main two references will be:

Auffinger-Damron-Hanson: "50 years of first-passage percolation" (the arXiv version has tons of typos and missing material)

Grimmett: "Percolation"
 

Topic Outline: 

These may change as the semester goes on. Most of these topics are from "50 years," but we may include more topics from standard
percolation (Grimmett's book for example).

Some typical random growth models:
     first-passage percolation (FPP)
     last-passage percolation (LPP)
     diffusional limited aggregation
Basics of FPP:
     existence of time constant
     shape theorems for the random ball at the origin
     conjectured properties of limit shape
Some basics of percolation theory in general
     existence of phase transition
     sharpness of phase transition
     gluing methods in 2D
Fluctuation bounds for FPP:
     variance bounds
     concentration inequalities
     large deviations
     cases where Gaussian fluctuations occur, critical FPP
Geodesics in FPP:
     finite geodesics and their sizes
     one-sided infinite geodesics
     directions of infinite geodesics
     doubly-infinite geodesics and relation to spin models
Busemann functions
     duality between Busemann functions and limit shape
     existence of Busemann functions on the half-plane
     use of Busemann functions to direct geodesics
     Busemann gradient fields
Growth and competition models
     Richardson's model
     Eden model for cell growth
     FPP competition interface
Relation between LPP and random matrices