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The bulk and boundary magnetizations are calculated for the critical Isingmodel on a randomly triangulated disk in the presence of a boundary magneticfield h. In the continuum limit this model corresponds to a c = 1/2 conformalfield theory coupled to 2D quantum gravity, with a boundary term breakingconformal invariance. It is found that as h increases, the averagemagnetization of a bulk spin decreases, an effect that is explained in terms offluctuations of the geometry. By introducing an $h$-dependent rescaling factor,the disk partition function and bulk magnetization can be expressed asfunctions of an effective boundary length and bulk area with no furtherdependence on h, except that the bulk magnetization is discontinuous andvanishes at h = 0. These results suggest that just as in flat space, theboundary field generates a renormalization group flow towards h = \infty. Anexact analytic expression for the boundary magnetization as a function of $h$is linear near h = 0, leading to a finite nonzero magnetic susceptibility atthe critical temperature.Comment: 9 pages inc. 2 figures, revtex, cosmetic change

Ising Model with a Boundary Magnetic Field on a Random Surface