Birthday Inequalities, Repulsion, and Hard Spheres

We study a birthday inequality in random geometric graphs: the probability of the empty graph is upper bounded by the product of the probabilities that each edge is absent. We show the birthday inequality holds at low densities, but does not hold in general. We give three different applications of the birthday inequality in statistical physics and combinatorics: we prove lower bounds on the free energy of the hard sphere model and upper bounds on the number of independent sets and matchings of a given size in d-regular graphs. The birthday inequality is implied by a repulsion inequality: the expected volume of the union of spheres of radius r around n randomly placed centers increases if we condition on the event that the centers are at pairwise distance greater than r. Surprisingly we show that the repulsion inequality is not true in general, and in particular that it fails in 24-dimensional Euclidean space: conditioning on the pairwise repulsion of centers of 24-dimensional spheres can decrease the expected volume of their union.