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Random Explorations
 
Gregory F. Lawler University of Chicago, Chicago, IL
Softcover ISBN:  978-1-4704-6766-1
Product Code:  STML/98
List Price: $59.00
Individual Price: $47.20
eBook ISBN:  978-1-4704-7221-4
Product Code:  STML/98.E
List Price: $59.00
Individual Price: $47.20
Softcover ISBN:  978-1-4704-6766-1
eBook: ISBN:  978-1-4704-7221-4
Product Code:  STML/98.B
List Price: $118.00 $88.50
Click above image for expanded view
Random Explorations
Gregory F. Lawler University of Chicago, Chicago, IL
Softcover ISBN:  978-1-4704-6766-1
Product Code:  STML/98
List Price: $59.00
Individual Price: $47.20
eBook ISBN:  978-1-4704-7221-4
Product Code:  STML/98.E
List Price: $59.00
Individual Price: $47.20
Softcover ISBN:  978-1-4704-6766-1
eBook ISBN:  978-1-4704-7221-4
Product Code:  STML/98.B
List Price: $118.00 $88.50
  • Book Details
     
     
    Student Mathematical Library
    Volume: 982022; 199 pp
    MSC: 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 loop-erased 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, Schramm-Loewner 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.

    Readership

    Advanced undergraduate students, graduate students, and researchers interested in teaching and learning some aspects of random fields.

  • Table of Contents
     
     
    • Chapters
    • Markov chains
    • Loop-erased 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
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Accessibility – to request an alternate format of an AMS title
Volume: 982022; 199 pp
MSC: 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 loop-erased 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, Schramm-Loewner 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.

Readership

Advanced undergraduate students, graduate students, and researchers interested in teaching and learning some aspects of random fields.

  • Chapters
  • Markov chains
  • Loop-erased 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
Review Copy – for publishers of book reviews
Accessibility – to request an alternate format of an AMS title
Please select which format for which you are requesting permissions.