# Replicated liquid theory in $1+\infty$ dimensions

## TL;DR

This paper develops an exact replicated liquid theory for spatially varying glasses in high transverse dimensions, providing insights into diverging lengths and spatial profiles near glass transitions.

## Contribution

It introduces a novel space-dependent free-energy functional for glasses in infinite transverse dimensions, enabling analysis of spatial variations and transition behaviors.

## Key findings

- Exponents match previous mean-field models.
- Predicts non-trivial spatial profiles of the glass order parameter.
- Identifies diverging lengths near glass transitions.

## Abstract

We develop a replicated liquid theory for structural glasses which exhibit spatial variation of physical quantities along one axis, say $z$-axis. The theory becomes exact with infinite transverse dimension $d-1 \to \infty$. It provides an exact free-energy functional with space-dependent glass order parameter $\Delta_{ab}(z)$. As a first application of the scheme, we study diverging lengths associated with dynamic/static glass transitions of hardspheres with/without 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 $\Delta_{ab}(z)$ within the cavity which exhibits a scaling feature approaching the dynamical glass transition.