Random Walk Resources

Question

I know several books in random walk like Spitzer book. As well we can find an overview from Feller book or Probability by Shiryaev. What I am looking for is a resource that even for a very specific type (like just simple random walk) it goes over many advanced results.

Answer 1

Révész, Pál. Random walk in random and non-random environments. World scientific, 2005.

Lawler, Gregory F. Intersections of random walks. Springer 2013.

Both of these references focus on simple random walk in one and higher dimensions, and prove quite advanced properties. I have used them extensively in my research.

See also Lawler, Gregory F., and Vlada Limic. Random walk: a modern introduction. Vol. 123. Cambridge University Press, 2010.

Answer 2

If you want something advanced, you may be interested in this book - https://books.google.ru/books/about/Quantum_Geometry.html?id=T6fWst-KdoYC&redir_esc=y. However, it is more oriented on the Physics applications - Statistical Physics, Quantum Field Theory, Gravity, and String theory.

Here the random walks are used to construct the propagator of a particle, and random surfaces - of a string,