Insertion space in repulsive active matter

For equilibrium hard spheres, the statistical geometry of the insertion space, the room to accommodate another sphere, relates exactly to the equation of state. We begin to extend this idea to active matter, analyzing the insertion space for repulsive active particles in one and two dimensions using both on- and off-lattice models. In one dimension, we derive closed-form expressions for the mean insertion cavity size, cavity number, and total insertion volume, all in excellent agreement with simulations. Strikingly, activity increases the total insertion volume and tends to keep the insertion space more connected. We also find that insertion space metrics contain signatures of collective phase behaviors occurring at previously predicted packing fractions. Taken together, our work provides the first quantitative foundation for the statistical geometry of active matter.