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. An online version will always remain free. You can order a hard copy of the book here. See also this flyer for the book.

Please let us know of any errors you find; send them to or

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 = {},
note = {Available at \url{}},

In the electronic version 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.

binary tree on 63 vertices

Downloads and Extras:

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
Glossary of Notation

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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