## Random Close Packing is least random in three dimensions

## Random Close Packing is least random in three dimensions

Sam Wilken, Ashley Z. Guo, Dov Levine, Paul M. Chaikin

AbstractA simple dynamical model, Biased Random Organization, BRO, appears to produce the ensemble of configurations known as Random Close Packed (RCP) as its critical endpoint in dimension d=3. We conjecture that BRO likewise produces RCP in any dimension and come to the following conclusions: there is no RCP in d=1 or d=2 (where dynamics lead to crystalline order); in d=3, d=4, and d=5, we recover RCP behavior with previously estimated packing fractions 0.64, 0.45, and 0.30 respectively, and the systems are isostatic with average contact numbers 6, 8, and 10. BRO belongs to the Manna universality class of dynamical phase transitions, which has well-defined critical exponents and an upper critical dimension of 4. Exponents are mean field for $4 \le d \le \infty$, which we confirm in simulations. Further, a hyperscaling relation between the correlation function exponent and density fluctuations implies that when mean field exponents hold, density fluctuations are random and not hyperuniform. Hence, hyperuniformity in RCP is only observed in d=3.