Dimensional Reduction for Generalized Continuum Polymers

Abstract

The Brydges-Imbrie dimensional reduction formula relates the pressure of a $d$-dimensional gas of hard spheres to a model of $(d+2)$-dimensional branched polymers. Brydges and Imbrie's proof was non-constructive and relied on a supersymmetric localization lemma. The main result of this article is a constructive proof of a more general dimensional reduction formula that contains the Brydges--Imbrie formula as a special case. Central to the proof are invariance lemmas, which were first introduced by Kenyon and Winkler for branched polymers. The new dimensional reduction formulas rely on invariance lemmas for central hyperplane arrangements that are due to M\'esz\'aros and Postnikov. Several applications are presented, notably dimensional reduction formulas for (i) non-spherical bodies and (ii) for corrections to the pressure due to symmetry effects.