  We present a collection of simulations of the Edwards-Anderson lattice spin
glass at $T=0$ to elucidate the nature of low-energy excitations over a range
of dimensions that reach from physically realizable systems to the mean-field
limit. Using heuristic methods, we sample ground states of instances to
determine their energies while eliciting excitations through manipulating
boundary conditions. We exploit the universality of the phase diagram of
bond-diluted lattices to make such a study in higher dimensions computationally
feasible. As a result, we obtain a verity of accurate exponents for domain wall
stiffness and finite-size corrections that allow us to examine their
dimensional behavior and their connection with predictions from mean-field
theory. We also provide an experimentally testable prediction for the
thermal-to-percolative crossover exponent in dilute lattices Ising spin
glasses.
