  The interaction-driven Mott transition in the half-filled Hubbard model is a
first-order phase transition that terminates at a critical point
$(T_\mathrm{c},U_\mathrm{c})$ in the temperature-interaction plane $T-U$. A
number of crossovers occur along lines that extend for some range above
$(T_\mathrm{c},U_\mathrm{c})$. Asymptotically close to
$(T_\mathrm{c},U_\mathrm{c})$, these lines coalesce into the so-called Widom
line. The existence of $(T_\mathrm{c},U_\mathrm{c})$ and of the associated
crossovers becomes unclear when long-wavelength fluctuations or long-range
order occur above $(T_\mathrm{c},U_\mathrm{c})$. We study this problem using
continuous-time quantum Monte Carlo methods as impurity solvers for both
Dynamical Mean-Field Theory (DMFT) and Cellular Dynamical Mean-Field Theory
(CDMFT). We contrast the cases of the square lattice, where antiferromagnetic
fluctuations dominate in the vicinity of the Mott transition, and the
triangular lattice where they do not. The inflexion points and maxima found
near the Widom line for the square lattice can serve as proxy for the
triangular lattice case. But the only crossover observable in all cases at
sufficiently high temperature is that associated with the opening of the Mott
gap. The same physics also controls an analog crossover in the resistivity
called the "Quantum Widom line".
