Spin Glasses
Last update: 23 Sep 2025 22:40First version: 6 May 2023
Yet Another Inadequate Placeholder
I am only being a little sarcastic when I say that a physicist interested in complex systems and trained in the 1990s (like me) was raised to look for spin glasses everywhere. There are times when I regard this as a childish thing I'm glad I put away, and times when it seems like a deep truth that the rest of the world would do well to appreciate...
Things I should already know the answer to: How does the Parisi et al.'s replica symmetry breaking relate to probabilists' ideas about graph limits? We've got this sequence of increasingly refined block-diagonal matrices, only every permutation of rows and columns is actually equally good, tending towards some sort of continuous limit function on the unit interval, so it seems like it ought to be a graphon, but maybe I'm missing something. For that matter, now that I start thinking about things in these terms, I am very puzzled about how overlap can be a physical order parameter... (Basically: any one sample from a mixture, no matter how large, can only tell you about that mixture component, not the distribution over mixtures.) --- I suspect that reading Talagrand would clear all this up for me, so I need to get around to doing so one of these decades.
- Recommended, big picture:
- K. H. Fischer and J. A. Hertz, Spin Glasses
- D. L. Stein, "Spin Glasses: Still Complex After All These Years?" cond-mat/0301104
- Daniel L. Stein and Charles M. Newman, Spin Glasses and Complexity
- Recommended, close-ups (very inadequate just now):
- Alia Abbara, Benjamin Aubin, Florent Krzakala, Lenka Zdeborová, "Rademacher complexity and spin glasses: A link between the replica and statistical theories of learning", arxiv:1912.02729
- Giulio Biroli, Marc Mézard, "Kernel Density Estimators in Large Dimensions", arxiv:2408.05807 [comments]
- Viktor Dotsenko, Introduction to the Theory of Spin Glasses and Neural Networks
- To read, over-views:
- Anton Bovier, Statistical Mechanics of Disordered Systems [enthusiastic review in J. Stat. Phys.]
- Tommaso Castellani and Andrea Cavagna, "Spin-Glass Theory for Pedestrians", cond-mat/0505032
- Andrea Montanari and Subhabrata Sen, "A Short Tutorial on Mean-Field Spin Glass Techniques for Non-Physicists", arxiv:2204.02909 ["short" = 108 pages!]
- Michel Talagrand, Mean Field Models for Spin Glasses
- To read, making sense of the replica trick (which I have used, but don't grasp):
- Christopher L. Baldwin and Brian Swingle, "Revisiting the Replica Trick: Competition Between Spin Glass and Conventional Order", Journal of Statistical Physics 190 (2023): 125
- Viktor Dotsenko, Introduction to the Replica Theory of Disordered Statistical Systems
- To read, applications to optimization, neural networks, and computational complexity:
- M. Baity-Jesi, L. Sagun, M. Geiger, S. Spigler, G. Ben Arous, C. Cammarota, Y. LeCun, M. Wyart, G. Biroli, "Comparing Dynamics: Deep Neural Networks versus Glassy Systems", arxiv:1803.06969
- Nicholas P Baskerville, Jonathan P Keating, Francesco Mezzadri, Joseph Najnudel, "A spin-glass model for the loss surfaces of generative adversarial networks", arxiv:2101.02524
- Sourav Chatterjee, "Chaos, concentration, and multiple valleys", arxiv:0810.4221
- David Gamarnik, Cristopher Moore, Lenka Zdeborová, "Disordered Systems Insights on Computational Hardness", arxiv:2210.08312
- Levent Sagun, V. Ugur Guney, Gerard Ben Arous, Yann LeCun, "Explorations on high dimensional landscapes", arxiv:1412.6615
- To read, not otherwise or not yet classified:
- Elena Agliari, Raffaella Burioni, Pierluigi Contucci, "A Diffusive Strategic Dynamics for Social Systems", arxiv:0812.1435
- Jorge Fernandez-de-Cossio-Diaz, Roberto Mulet, "Spin Glass Theory of Interacting Metabolic Networks", arxiv:1909.11070
- Matthew Garrod, Nick S. Jones, "Influencing dynamics on social networks without knowledge of network microstructure", arxiv:2011.05774
- Steffen Lauritzen, Caroline Uhler, Piotr Zwiernik, "Total positivity in exponential families with application to binary variables", Annals of Statistics 49 (2021): 1436--1459
- Andrew Lucas, Ching Hua Lee, "Multistable binary decision making on networks", Physical Review E 87 (2013): 032806, arxiv:1210.6044
- Stein (ed.), Spin Glasses and Biology
- Shubham Tripathi, David A. Kessler, Herbert Levine, "Biological Regulatory Networks are Minimally Frustrated", Physical Review Letters 125 (2020): 088101, arxiv:1911.10252
- David Wales, Energy Landscapes: Applications to Clusters, Biomolecules and Glasses