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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.