Asymptotic fate of the finite-size flow

Determine whether the observed finite-size flow of the potential-energy-shell connectivity in the quantum random energy model eventually follows the predicted one-hop percolation asymptotics or crosses over to a different regime.

Background

The paper analyzes fragmentation of the QREM potential-energy shell using a one-hop site-percolation approximation on the Fock-space hypercube. At zero energy density, this approximation predicts a crossover scale proportional to 1/(L-1), while at nonzero energy density it predicts a competition between hypercube connectivity and the exponentially decreasing Gaussian density of states.

The numerically accessible system sizes remain pre-asymptotic, and the authors caution that the one-hop estimate is not yet a quantitative description of the observed data. In particular, finite-size behavior at nonzero energy density may either approach the predicted asymptotic form or exhibit a crossover to another scaling regime.