  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
$\langle{\boldsymbol{u_i},\boldsymbol{u_j}}\rangle \ge \tau,
\langle{\boldsymbol{u_i},\boldsymbol{u_k}}\rangle \ge \tau$, and $\langle
\boldsymbol{u_j}, \boldsymbol{u_k}\rangle \ge \tau$. We prove that for every
$\varepsilon > 0$, there exists a choice of $d = \Theta(\log n)$ and $\tau =
\tau(\varepsilon,d)$ 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.
