Softcover ISBN:  9781470467661 
Product Code:  STML/98 
List Price:  $59.00 
Individual Price:  $47.20 
eBook ISBN:  9781470472214 
Product Code:  STML/98.E 
List Price:  $59.00 
Individual Price:  $47.20 
Softcover ISBN:  9781470467661 
eBook: ISBN:  9781470472214 
Product Code:  STML/98.B 
List Price:  $118.00 $88.50 
Softcover ISBN:  9781470467661 
Product Code:  STML/98 
List Price:  $59.00 
Individual Price:  $47.20 
eBook ISBN:  9781470472214 
Product Code:  STML/98.E 
List Price:  $59.00 
Individual Price:  $47.20 
Softcover ISBN:  9781470467661 
eBook ISBN:  9781470472214 
Product Code:  STML/98.B 
List Price:  $118.00 $88.50 

Book DetailsStudent Mathematical LibraryVolume: 98; 2022; 199 ppMSC: Primary 60
The title “Random Explorations” has two meanings. First, a few topics of advanced probability are deeply explored. Second, there is a recurring theme of analyzing a random object by exploring a random path.
This book is an outgrowth of lectures by the author in the University of Chicago Research Experiences for Undergraduate (REU) program in 2020. The idea of the course was to expose advanced undergraduates to ideas in probability research.
The book begins with Markov chains with an emphasis on transient or killed chains that have finite Green's function. This function, and its inverse called the Laplacian, is discussed next to relate two objects that arise in statistical physics, the looperased random walk (LERW) and the uniform spanning tree (UST). A modern approach is used including loop measures and soups. Understanding these approaches as the system size goes to infinity requires a deep understanding of the simple random walk so that is studied next, followed by a look at the infinite LERW and UST. Another model, the Gaussian free field (GFF), is introduced and related to loop measure. The emphasis in the book is on discrete models, but the final chapter gives an introduction to the continuous objects: Brownian motion, Brownian loop measures and soups, SchrammLoewner evolution (SLE), and the continuous Gaussian free field. A number of exercises scattered throughout the text will help a serious reader gain better understanding of the material.
ReadershipAdvanced undergraduate students, graduate students, and researchers interested in teaching and learning some aspects of random fields.

Table of Contents

Chapters

Markov chains

Looperased random walk

Loop soups

Random walk in $\mathbb {Z}$

LERW and spanning trees on $\mathbb {Z}^d$

Gaussian free field

Scaling limits

Some background and extra topics


Additional Material

RequestsReview Copy – for publishers of book reviewsAccessibility – to request an alternate format of an AMS title
 Book Details
 Table of Contents
 Additional Material
 Requests
The title “Random Explorations” has two meanings. First, a few topics of advanced probability are deeply explored. Second, there is a recurring theme of analyzing a random object by exploring a random path.
This book is an outgrowth of lectures by the author in the University of Chicago Research Experiences for Undergraduate (REU) program in 2020. The idea of the course was to expose advanced undergraduates to ideas in probability research.
The book begins with Markov chains with an emphasis on transient or killed chains that have finite Green's function. This function, and its inverse called the Laplacian, is discussed next to relate two objects that arise in statistical physics, the looperased random walk (LERW) and the uniform spanning tree (UST). A modern approach is used including loop measures and soups. Understanding these approaches as the system size goes to infinity requires a deep understanding of the simple random walk so that is studied next, followed by a look at the infinite LERW and UST. Another model, the Gaussian free field (GFF), is introduced and related to loop measure. The emphasis in the book is on discrete models, but the final chapter gives an introduction to the continuous objects: Brownian motion, Brownian loop measures and soups, SchrammLoewner evolution (SLE), and the continuous Gaussian free field. A number of exercises scattered throughout the text will help a serious reader gain better understanding of the material.
Advanced undergraduate students, graduate students, and researchers interested in teaching and learning some aspects of random fields.

Chapters

Markov chains

Looperased random walk

Loop soups

Random walk in $\mathbb {Z}$

LERW and spanning trees on $\mathbb {Z}^d$

Gaussian free field

Scaling limits

Some background and extra topics