Probability and Computing

[2025 Summer] Statistical Physics and Computation

Location: 软件学院 1319

Schedule

[July 30] The SK model, RS solution

We study the SK model \(\mu(x)\propto \exp\tp{\frac{\beta}{2}x^\top J x}\) where \(J_{ij}\sim \+N(0,\frac{1}{n})\) and derive the critical temperature \(\beta_c=1\) using a replica symmetry argument.

One notably technique used in the calculation is the Hubbard-Stratonovich transformation, or equivalenty the formula for the Gaussian MGF.

Our calculations mainly follow those in [Nis21].

[July 23] The REM model, RS and 1-RSB solution for REM

We define the Random Energy Model (REM) and study its phase transition.

• The phase transition of its free energy can be rigorously derived via simple probabilistic argument. See [Tal03].
• We also demonstrate the prediction of the phase transition using the heuristic of the replica trick and 1-replica symmetric breaking. See [MM09] and [Kab].

References

• [Tal11] Mean Fields Models for Spin Glasses, Vol I. Michel Talagrand
• [Tal03] Spin Glasses: A Challenge for Mathematicians. Michel Talagrand (This is in fact the first version of the above. But I found something useful.)
• [MM09] Information, Physics, and Computation. Marc Mézard, Andrea Montanari
• [Kab] Introduction to the replica method. Yoshiyuki Kabashima
• [Nis21] Statistical Physics of Spin Glasses and Information Processing An Introduction. Hidetoshi Nishimori