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One fundamental task for graphical models is to generate random samples from the associated distribution. The Markov chain Monte Carlo (MCMC) method is one of the simplest and most popular approaches to tackle such problems. Despite the popularity of graphical models and MCMC algorithms, theoretical guarantees of their performance are not known even for some simple models. I will describe a new tool called “spectral independence” to analyze MCMC algorithms and more importantly to reveal the underlying structure behind such models. I will also discuss how these structural properties can be applied to sampling when MCMC fails and to other statistical problems like parameter learning or model fitting.