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High-dimensional expanders (HDX) generalize expander graphs and have found various applications in math and TCS. One approach to proving that a simplicial complex is an HDX uses trickle-down theorems, where expansion of certain low-dimensional links implies expansion properties throughout the complex. In this talk, we will discuss some recent trickle-down theorems for partite HDX, with applications to approximate sampling. We will also show how these theorems arise from the theory of Lorentzian and log-concave polynomials. Joint work with Kasper Lindberg and Shayan Oveis Gharan.