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A striking example of dimensional reduction established by Brydges and Imbrie is an exact relation between the continuum hard sphere gas in $d$ dimensions and continuum branched polymers in $d+2$ dimensions. I will discuss a new proof of this result, which also proves dimensional reduction formulas for generalized models of continuum polymers associated to central hyperplane arrangements. The proof is essentially combinatorial, in contrast to the original proof which uses supersymmetry.