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Random Explorations (Paperback)
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Random Explorations (Paperback)
Series: Student Mathematical Library
Expected to ship within 12 - 17 working days
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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.
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