Probability on Trees and Networks

by Russell Lyons and Yuval Peres

This is close to the final version that was published by Cambridge University Press. The paperback version incorporates corrections, improvements, and updates; page numbers changed from the hardback, but not numbers of theorems, exercises, etc. An online version will always remain free. You can order a physical copy of the book here.

Please let us know of any errors you find; send them to rdlyons@indiana.edu or yperes@gmail.com.

You may wish to use the following if you refer to this book in a paper:


@book {MR3616205,
    AUTHOR = {Lyons, Russell and Peres, Yuval},
     TITLE = {Probability on Trees and Networks},
    SERIES = {Cambridge Series in Statistical and Probabilistic Mathematics},
    VOLUME = {42},
 PUBLISHER = {Cambridge University Press, New York},
      YEAR = {2016},
     PAGES = {xv+699},
      ISBN = {978-1-107-16015-6},
   MRCLASS = {60C05 (05C05 05C81 28A80 60J50 60J80 60K35 82B41)},
  MRNUMBER = {3616205},
       DOI = {10.1017/9781316672815},
       URL = {http://dx.doi.org/10.1017/9781316672815},
note = {Available at \url{https://rdlyons.pages.iu.edu/}},
}

or, especially if you cite something that is only in the paperback edition,


@book {MR3616205,
    AUTHOR = {Lyons, Russell and Peres, Yuval},
     TITLE = {Probability on Trees and Networks},
    SERIES = {Cambridge Series in Statistical and Probabilistic Mathematics},
    VOLUME = {42},
 PUBLISHER = {Cambridge University Press, New York},
      YEAR = {2021},
     PAGES = {xv+699},
      NOTE = {Paperback}, 
      ISBN = {978-1-108-73272-7},
   MRCLASS = {60C05 (05C05 05C81 28A80 60J50 60J80 60K35 82B41)},
note = {Available at \url{https://rdlyons.pages.iu.edu/}},
}

complete
binary tree on 63 vertices

In the electronic versions of this book, most symbols that are used with a fixed meaning are hyperlinked to their definitions, although the fact that such hyperlinks exist is not made visible (unless your mouse hovers over the symbol).

Because of current technology, color is used slightly differently in the electronic and printed versions.

Downloads and Extras:

Hardback version
PDF file of the hardback version of the book (xvi + 699 pages, 864 exercises, 768 references, 20.9 megabytes). Version of 8 Nov. 2016, essentially as published.
Errata for hardback version. Version of 6 Nov. 2020.
Almost all the above errata have been incorporated in the online-only corrected edition of the book. Nothing in the original has changed its page number. Corrections appear in dark red. Version of 8 Nov. 2020.
Paperback version
PDF file of the paperback version of the book (xvi + 699 pages, 864 exercises, 818 references, 21.0 megabytes). Version of 15 January 2024, essentially as published.
Tip: If you print this, use the page handling option (scaling) "fit to printable area" in order to maximize its size.
The paperback version incorporates all the above errata, as well minor updates and improvements. Some page numbers also changed. Here are additional errata: Errata for paperback version. Version of 15 January 2024.
A 3D UST in a 5 x 5 x 5 grid
A series of square tilings like the one below.

Chapter Titles:

Some Highlights
Random Walks and Electric Networks
Special Networks
Uniform Spanning Trees
Branching Processes, Second Moments, and Percolation
Isoperimetric Inequalities
Percolation on Transitive Graphs
The Mass-Transport Principle and Percolation
Infinite Electrical Networks and Dirichlet Functions
Uniform Spanning Forests
Minimal Spanning Forests
Limit Theorems for Galton–Watson Processes
Escape Rate of Random Walks and Embeddings
Random Walks on Groups and Poisson Boundaries
Hausdorff Dimension
Capacity and Stochastic Processes
Random Walks on Galton–Watson Trees
Comments on Exercises
Bibliography
Glossary of Notation
Index

free and wired uniform spanning forests for the (2, 3, 7)-triangle tessellation

square tiling generated by 10x10 electrical network

distances in the tree to the path connecting the corners in a uniform spanning tree of a 200x200 square grid

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