Local and global expansion in random geometric graphs

Consider a random geometric 2-dimensional simplicial complex $X$ sampled as follows: first, sample $n$ vectors $\boldsymbol{u_1},\ldots,\boldsymbol{u_n}$ uniformly at random on $\mathbb{S}^{d-1}$; then, for each triple $i,j,k \in [n]$, add $\{i,j,k\}$ and all of its subsets to $X$ if and only if $\langle{\boldsymbol{u_i},\boldsymbol{u_j}}\rangle \ge τ, \langle{\boldsymbol{u_i},\boldsymbol{u_k}}\rangle \ge τ$, and $\langle \boldsymbol{u_j}, \boldsymbol{u_k}\rangle \ge τ$. We prove that for every $\varepsilon > 0$, there exists a choice of $d = Θ(\log n)$ and $τ= τ(\varepsilon,d)$ so that with high probability, $X$ is a high-dimensional expander of average degree $n^\varepsilon$ in which each $1$-link has spectral gap bounded away from $\frac{1}{2}$. To our knowledge, this is the first demonstration of a natural distribution over $2$-dimensional expanders of arbitrarily small polynomial average degree and spectral link expansion better than $\frac{1}{2}$. All previously known constructions are algebraic. This distribution also furnishes an example of simplicial complexes for which the trickle-down theorem is nearly tight. En route, we prove general bounds on the spectral expansion of random induced subgraphs of arbitrary vertex transitive graphs, which may be of independent interest. For example, one consequence is an almost-sharp bound on the second eigenvalue of random $n$-vertex geometric graphs on $\mathbb{S}^{d-1}$, which was previously unknown for most $n,d$ pairs.