Probabilistic combinatorics at exponentially small scales
Julian Sahasrabudhe
TL;DR
The paper investigates phenomena in probabilistic combinatorics and high-dimensional geometry that occur at exponentially small probabilities, connecting discrepancy theory, sphere packings and codes, and random matrix theory. It presents constructive discrepancy-based methods for flat Littlewood polynomials, algorithmic variants of Spencer’s theorem, and advances toward long-standing conjectures (Komlós, Beck–Fiala), while also introducing semi-random strategies and graph-nibble techniques to obtain sharp density bounds in sphere packing and spherical codes. In the random matrix realm, it surveys Littlewood–Offord theory, inverse-LLC results, and novel rank-splitting approaches that yield exponential bounds on singularity probabilities and refined least singular value control. Collectively, the work demonstrates how steering probabilistic spaces toward rare configurations can yield sharp lower bounds, algorithmic reconstructions, and deeper structural understanding in high-dimensional combinatorics and randomness.
Abstract
In many applications of the probabilistic method, one looks to study phenomena that occur ``with high probability''. More recently however, in an attempt to understand some of the most fundamental problems in combinatorics, researchers have been diving deeper into these probability spaces and understanding phenomena that occur at much smaller probability scales. Here I will survey a few of these ideas from the perspective of my own work in the area.