Spectral Gaps of Random Graphs and Applications

Abstract

We study the spectral gap of the Erdos-Rényi random graph through the connectivity threshold. In particular, we show that for any fixed δ > 0 if then the normalized graph Laplacian of an Erdos-Rényi graph has all of its nonzero eigenvalues tightly concentrated around 1. This is a strong expander property. We estimate both the decay rate of the spectral gap to 1 and the failure probability, up to a constant factor. We also show that the 1/2 in the above is optimal, and that if p = c log nn for c < 1/2, then there are eigenvalues of the Laplacian restricted to the giant component that are separated from 1. We then describe several applications of our spectral gap results to stochastic topology and geometric group theory. These all depend on Garland's method [24], a kind of spectral geometry for simplicial complexes. The following can all be considered to be higher-dimensional expander properties. First, we exhibit a sharp threshold for the fundamental group of the Bernoulli random 2-complex to have Kazhdan's property (T). We also obtain slightly more information and can describe the large-scale structure of the group just before the (T) threshold. In this regime, the random fundamental group is with high probability the free product of a (T) group with a free group, where the free group has one generator for every isolated edge. The (T) group plays a role analogous to that of a "giant component"in percolation theory. Next we give a new, short, self-contained proof of the Linial-Meshulam-Wallach theorem [35, 39], identifying the cohomology-vanishing threshold of Bernoulli random d-complexes. Since we use spectral techniques, it only holds for Q or R coefficients rather than finite field coefficients, as in [35] and [39]. However, it is sharp from a probabilistic point of view, providing for example, hitting time type results and limiting Poisson distributions inside the critical window. It is also a new method of proof, circumventing the combinatorial complications of cocycle counting. Similarly, results in an earlier preprint version of this article were already applied in [33] to obtain sharp cohomology-vanishing thresholds in every dimension for the random flag complex model.