Math 541 - Reading materials
Ising model
Statistical Mechanics of Lattice Systems: a Concrete Mathematical Introduction
by Sacha Friedli and Yvan Velenik.
This is relatively new book by a couple of experts in the field.
A nice feature of the book is that the scope is limited.
They limit the topics they discuss; fortunately the Ising model is one of those topics - see chapter 3.
The Statistical Mechanics of Lattice Gases, Volume I
by Barry Simon. Available on-line through the UA library.
This is an exhaustive treatment of lattice models spread over two volumes.
However, the second volume has not appeared yet. (It is expected soon.)
The Ising model appears in volume I, but some important results about it,
especially the Peierls argument, will not appear until volume II.
Simon is famous for his many contributions to mathematical physics,
and he is an excellent expositor.
100 Years of the (Critical) Ising Model on the Hypercubic Lattice
by Hugo Duminil-Copin
Duminil-Copin was awarded the Fields medal in 2022 in large part for his work on the Ising model. This article is primarily a historical overview which covers a wide range of results on the model (with 100+ references for further reading!).
Percolation
Percolation
by Geoffrey Grimmett. This is a book available on-line through the UA library.
60 Years of Percolation
by Hugo Duminil-Copin
Fairly short (18 pages) and relatively recent (2017) review of percolation results and open questions.
There are no proofs, but it covers a wide range of results.
Lectures on two-dimensional percolation
by Wendelin Werner
There are lecture notes from a course he gave at IAS-Park City summer school
in 2007. The focus is on using SLE to study 2d percolation, so these notes get quite techinical after a brief introduction to percolation.
Renormalization Group for Ising-like Systems
Renormalization Theory for Ising-like Spin Systems
by Th. Niemeijer and J. M. J. van Leeuwen
This is an article from the well-known 20+ volumes series edited by
Domb, Green and Lebowitz. The link is to a scan of part of the chapter.
What is missing is the in-depth discussion on computational methods which is
very out dated.
Scaling and Renormalization in Statistical Physics
by John Cardy. This is a book available on-line through the UA library.
Chapter 3 is about the renormalization group idea.
The renormalization group: Critical phenomena and the Kondo problem
by Kenneth Wilson. Section VI is about RG for Ising type systems.
Lectures on Phase Transitions and the Renormalization Group by
Nigel Goldenfeld
Random walks that do not self-intersect
Random Walk: A Modern Introduction
by Gregory Lawler and Vlada Limic.
This book is almost entirely about the ordinary random walk, but there is a chapter on the LERW. Available on-line through the UA library.
Lectures on Self-Avoiding Walks by Bauerschmidt, Duminil-Copin, Goodman, and Slade. These lecture notes start with a good introduction to the SAW and go on to some deep stuff I did not cover, for example the lace expansion for high dimensions.
The self-avoiding walk
by Neal Madras and Gordon Slade. An entire book on the SAW. It was written before SLE, so there is no discusion of the conjectures relating SAW to SLE in two dimensions. Available on-line through the UA library.
Schramm-Loewner Evolution
Conformally Invariant Processes in the Plane by GF Lawler This is a book by Gregory Lawler available in the Library.
Schramm-Loewner Evolution by GF Lawler. These are notes by Lawler written after his book.
Random planar curves and Schramm-Loewner Evolutions - Wendenlin Werner
Lecture Notes by Werner.
Stochastic geometry of critical curves, Schramm-Loewner evolutions and conformal field theory
by Ilya A Gruzberg. J. Phys. A: Math. Gen. 39 12601
This is a review article written by a physicist
Systems with a continuous symmetry
Lectures on the Spin and Loop O(n) Models
Some lecture notes by Ron Peled and Yinon Spinka.
Quantum stuff
An Introduction to Quantum Spin Systems
Lecture notes by Bruno Nachtergaele and Robert Sims
Quantum Computation and Quantum Information
by Michael Nielsen and Isaac Chuang
A textbook on what the title indicates, available on-line from UA library.
I followed parts of this closely.