Probabilistic combinatorics at exponentially small scales

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.

💡 Research Summary

The paper surveys recent work on probabilistic combinatorics that goes beyond the classical “with high probability” regime and investigates phenomena that occur at exponentially small probability scales. The author, Julian Sahasrabudhe, uses his own research as a unifying thread, focusing on three main topics: flat Littlewood polynomials, high‑dimensional sphere packings (and spherical codes), and singularity probabilities for random symmetric ±1 matrices.

In the first part the author addresses a long‑standing conjecture of Littlewood: whether there exist degree‑n polynomials whose coefficients are all ±1 and whose modulus on the unit circle stays within Θ(√n). By combining tools from discrepancy theory with a refined construction based on Rudin–Shapiro polynomials, the author builds a cosine polynomial c(x) that is O(√n) everywhere and only small on a controlled collection of short intervals. Then, using Spencer’s “six‑standard‑deviations” theorem, a sign assignment α_I is chosen for each bad interval I, and a set of “progress” quantities Δ(k) is defined. Random signs ε̂_k are drawn with expectations proportional to Δ(k), producing a sine polynomial ŝ(x) whose expected value is large on each bad interval while remaining O(√n) globally. This yields a Littlewood polynomial P(z) with c₁√n ≤ |P(z)| ≤ c₂√n for all |z|=1, thus confirming the existence of flat Littlewood polynomials.

The second part turns to sphere packing in ℝ^d. By designing a biased random process that favours configurations achieving a particular packing density, the authors obtain the first asymptotic improvement over Rogers’ 1947 lower bound: θ(d) ≥ (1−o(1))·(d·log d)/2^{d+1}. The constructed packings are both dense and highly disordered, matching predictions from physics (Parisi–Zamponi). The same technique yields improved lower bounds for spherical codes: A(d,θ) ≥ (1−o(1))·(d·log d)/(2·s_d(θ)). The paper also discusses Klartag’s recent work on random lattices, which gives a comparable bound for lattice packings, and highlights the remaining exponential gaps between known upper and lower bounds as a major open problem.

The final section studies random symmetric matrices with entries in {−1,1}. The authors prove that the singularity probability satisfies P