Comments and questions are always welcome on these notes. If you have any to share, please email me at last_name + first_name at gmail dot com. Last updated Oct. 6 2024.
a. Replica calculations in the Sherrington-Kirkpatrick (SK) model (including an account of replica symmetry breaking). note
b. “Replica Method for the Machine Learning Theorist.” blog post with Blake Bordelon, Abdul Canatar, Boaz Barak, and Cengiz Pehlevan
a. Where does the Gibbs distribution (Boltzmann distribution) come from? A derivation from first principles (kind of). note
These calculations can be very frustrating. If you are feeling this way, remember that C.N. Yang felt the same:
I was thus led to a long calculation, the longest in my career. Full of local, tactical tricks, the calculation proceeded by twists and turns. There were many obstructions. But always, after a few days, a new trick was somehow found that pointed to a new path. The trouble was that I soon felt I was in a maze and was not sure whether in fact, after so many turns, I was anywhere nearer the goal than when I began. This kind of strategic overview was very depressing, and several times I almost gave up. But each time something drew me back, usually a new tactical trick that brightened the scene, even though only locally. Finally, after about six months of work off and on, all the pieces suddenly fitted together, producing miraculous cancellations, and I was staring at the amazingly simple final result … Since there were some limiting procedures in my calculation that were not rigorous, I did not feel quite secure …
Selected Papers, 1945-1980, with Commentary by C.N.Yang, 1982.
Here, Yang was describing his calculations of the spontaneous magnetization in the 2D Ising model.