We consider the critical Ising model on a randomly triangulated disk in the presence of a boundary magnetic field $h$, computing the partition function $Z$ and bulk magnetization $〈M〉$. These quantities depend only on an effective rescaled boundary length and bulk area, suggesting that the boundary field generates renormalization group flow towards $h=\infty$, just as in flat space. As $h>\mathrm{}0\mathrm{}$ increases, $〈M〉$ decreases, which we explain in terms of fluctuations of the geometry. We also derive an analytic expression for the boundary magnetization, which gives a finite nonzero boundary susceptibility at the critical temperature.

• Received 28 May 1996

DOI:https://doi.org/10.1103/PhysRevLett.77.3947