Replicated liquid theory in 1 + <i>∞</i> dimensions

We develop a replicated liquid theory for structural glasses that exhibit spatial variation of physical quantities along one axis, say z-axis. The theory becomes exact with an infinite transverse dimension d - 1 → ∞. It provides an exact free-energy functional with a space-dependent glass order parameter Δab(z). As a first application of the scheme, we study diverging lengths associated with dynamic/static glass transitions of hard spheres with/without a confining cavity. The exponents agree with those obtained in previous studies on related mean-field models. Moreover, it predicts a non-trivial spatial profile of the glass order parameter Δab(z) within the cavity, which exhibits a scaling feature approaching the dynamical glass transition.