How the Edwards-Anderson Model reaches its Mean-Field Limit; Simulations in d=3,...,7

Extensive computations of ground state energies of the Edwards-Anderson spin glass on bond-diluted, hypercubic lattices are conducted in dimensions $d=3,\ldots,7$. Results are presented for bond-densities exactly at the percolation threshold, $p=p_{c}$, and deep within the glassy regime, $p>p_{c}$, where finding ground-states becomes a hard combinatorial problem. The ``stiffness'' exponent $y$ that controls the formation of domain wall excitations at low temperatures is determined in all dimensions. Finite-size corrections of the form $1/N^{\omega}$ are shown to be consistent throughout with the prediction $\omega=1-y/d$. At $p=p_{c}$, an extrapolation for $d\to\infty$ appears to match our mean-field results for these corrections. In the glassy phase, $\omega$ does not approach the value of $2/3$ for large $d$ predicted from simulations of the Sherrington-Kirkpatrick spin glass. However, the value of $\omega$ reached at the upper critical dimension \emph{does} match certain mean-field spin glass models on sparse random networks of regular degree called Bethe lattices.\\[4pt] [1] S. Boettcher and S. Falkner, arXiv:1110.6242;\hfil\break [2] S. Boettcher and E. Marchetti, PRB77, 100405 (2008);\hfil\break [3] S. Boettcher, PRL95, 197205 (2005).