Finite-size Scaling for the 2d Ising Model with Fixed Boundary Conditions

We study the magnetization mL(h, β) for the Ising model on a large but finite lattice square under the minus boundary conditions. Using known large-deviation results evaluating the balance between the competing effects of the minus boundary conditions and the external magnetic field h, we describe the details of its dependence on h as exemplified by the finitesize rounding of the infinite-volume magnetization discontinuity and its shift with respect to the infinite-volume transition point.