On the hard sphere model and sphere packings in high dimensions

We prove a lower bound on the entropy of sphere packings of $\mathbb R^d$ of density $\Theta(d \cdot 2^{-d})$. The entropy measures how plentiful such packings are, and our result is significantly stronger than the trivial lower bound that can be obtained from the mere existence of a dense packing. Our method also provides a new, statistical-physics-based proof of the $\Omega(d \cdot 2^{-d})$ lower bound on the maximum sphere packing density by showing that the expected packing density of a random configuration from the hard sphere model is at least $(1+o_d(1)) \log(2/\sqrt{3}) d \cdot 2^{-d}$ when the ratio of the fugacity parameter to the volume covered by a single sphere is at least $3^{-d/2}$. Such a bound on the sphere packing density was first achieved by Rogers, with subsequent improvements to the leading constant by Davenport and Rogers, Ball, Vance, and Venkatesh

On kissing numbers and spherical codes in high dimensions

We prove a lower bound of $\Omega (d^{3/2} \cdot (2/\sqrt{3})^d)$ on the kissing number in dimension $d$. This improves the classical lower bound of Chabauty, Shannon, and Wyner by a linear factor in the dimension. We obtain a similar linear factor improvement to the best known lower bound on the maximal size of a spherical code of acute angle $\theta$ in high dimensions

Extremes of the internal energy of the Potts model on cubic graphs

We prove tight upper and lower bounds on the internal energy per particle (expected number of monochromatic edges per vertex) in the anti-ferromagnetic Potts model on cubic graphs at every temperature and for all $q \ge 2$. This immediately implies corresponding tight bounds on the anti-ferromagnetic Potts partition function. Taking the zero-temperature limit gives new results in extremal combinatorics: the number of $q$-colorings of a $3$-regular graph, for any $q \ge 2$, is maximized by a union of $K_{3,3}$'s. This proves the $d=3$ case of a conjecture of Galvin and Tetali

Exact Ramsey numbers of odd cycles via nonlinear optimisation

For a graph G, the k-colour Ramsey number R k(G) is the least integer N such that every k-colouring of the edges of the complete graph K N contains a monochromatic copy of G. Let C n denote the cycle on n vertices. We show that for fixed k≥2 and n odd and sufficiently large, R k(C n)=2 k−1(n−1)+1. This resolves a conjecture of Bondy and Erdős for large n. The proof is analytic in nature, the first step of which is to use the regularity method to relate this problem in Ramsey theory to one in nonlinear optimisation. This allows us to prove a stability-type generalisation of the above and establish a correspondence between extremal k-colourings for this problem and perfect matchings in the k-dimensional hypercube Q k