  Let $G$ be any $n$-vertex graph whose random walk matrix has its nontrivial
eigenvalues bounded in magnitude by $1/\sqrt{\Delta}$ (for example, a random
graph $G$ of average degree~$\Theta(\Delta)$ typically has this property). We
show that the $\exp\Big(c \frac{\log n}{\log \Delta}\Big)$-round Sherali--Adams
linear programming hierarchy certifies that the maximum cut in such a~$G$ is at
most $50.1\%$ (in fact, at most $\tfrac12 + 2^{-\Omega(c)}$). For example, in
random graphs with $n^{1.01}$ edges, $O(1)$ rounds suffice; in random graphs
with $n \cdot \text{polylog}(n)$ edges, $n^{O(1/\log \log n)} = n^{o(1)}$
rounds suffice.
  Our results stand in contrast to the conventional beliefs that linear
programming hierarchies perform poorly for \maxcut and other CSPs, and that
eigenvalue/SDP methods are needed for effective refutation. Indeed, our results
imply that constant-round Sherali--Adams can strongly refute random Boolean
$k$-CSP instances with $n^{\lceil k/2 \rceil + \delta}$ constraints; previously
this had only been done with spectral algorithms or the SOS SDP hierarchy.
