We design fast algorithms for repeatedly sampling from strongly Rayleigh distributions, which include random spanning tree distributions and determinantal point processes. For a graph $G=(V, E)$, we show how to approximately sample uniformly random spanning trees from $G$ in $\widetilde{O}(\lvert V\rvert)$ time per sample after an initial $\widetilde{O}(\lvert E\rvert)$ time preprocessing. For a determinantal point process on subsets of size $k$ of a ground set of $n$ elements, we show how to approximately sample in $\widetilde{O}(k^ω)$ time after an initial $\widetilde{O}(nk^{ω-1})$ time preprocessing, where $ω<2.372864$ is the matrix multiplication exponent. We even improve the state of the art for obtaining a single sample from determinantal point processes, from the prior runtime of $\widetilde{O}(\min\{nk^2, n^ω\})$ to $\widetilde{O}(nk^{ω-1})$. In our main technical result, we achieve the optimal limit on domain sparsification for strongly Rayleigh distributions. In domain sparsification, sampling from a distribution $μ$ on $\binom{[n]}{k}$ is reduced to sampling from related distributions on $\binom{[t]}{k}$ for $t\ll n$. We show that for strongly Rayleigh distributions, we can can achieve the optimal $t=\widetilde{O}(k)$. Our reduction involves sampling from $\widetilde{O}(1)$ domain-sparsified distributions, all of which can be produced efficiently assuming convenient access to approximate overestimates for marginals of $μ$. Having access to marginals is analogous to having access to the mean and covariance of a continuous distribution, or knowing "isotropy" for the distribution, the key assumption behind the Kannan-Lovász-Simonovits (KLS) conjecture and optimal samplers based on it. We view our result as a moral analog of the KLS conjecture and its consequences for sampling, for discrete strongly Rayleigh measures.