Math 790 (Graduate minicourse): The Superconcentration Phenomenon, Spring 2026

Instructor: Nicholas Cook
WF 1:25--2:40, Feb 11--Mar 06, Physics 119
The concentration of measure phenomenon is a central concept in probability and high-dimensional geometry, with countless applications since it was first appreciated in the 70s. More recently it has been appreciated that certain important random processes concentrate at a scale significantly smaller than the (already surprisingly small) scale provided by "off-the-shelf" concentration inequalities. Examples include random variables associated to random matrix models, percolation models, spin glass models and random polymer models. A set of interrelated phenomena have been identified that help to understand when to expect such "superconcentration". Lectures will draw from Chatterjee's monograph and some more recent papers (see references below).

Credit will be based on attendance and completion of some exercises.

Notes

Topics to be covered:

• Overview of examples: First passage percolation, Random polymers, spin glass models, Tracy-Widom law for the norm of a random matrix.
• Markov semigroup method for Poincare inequalities, concentration of measure
• Connections between Superconcentration, 'Chaos', and 'Multiple Valleys' phenomena
• Superconcentration from semigroup hypercontractivity
• Superconcentration from spectral decomposition
• Noise sensitivity for the top eigenvector of random matrices
• (Time permitting) Disorder chaos for Brownian last-passage percolation

References:

• Superconcentration and Related Topics. Sourav Chatterjee. Available at Springerlink.
  
• Noise sensitivity of the top eigenvector of a Wigner matrix. C. Bordenave, G. Lugosi and N. Zhivotovskiy, Probab. Theory Related Fields, 177(3-4):1103--1135,2020. doi:10.1007/s00440-020-00970-1
  
• Stability and chaos in dynamical last passage percolation. S. Ganguly and A. Hammond, Commun. Am. Math. Soc. 4 (2024), 387--479. doi:10.1090/cams/35