Gaussian Free Field on the Tree Subject to a Hard Wall

This lecture explores the discrete Gaussian free field on the binary tree when all leaves are conditioned to be positive. Discover sharp asymptotics for the probability of this "hard-wall constraint" event and learn about the repulsion profile that enables this condition. Examine estimates for the mean, fluctuations, and covariances of the field under conditioning, demonstrating super-exponential tightness around the mean in the first log-many generations. Follow along as these results build toward a comprehensive, sharp asymptotic description of the field's law under conditioning, including both local statistics (the conditional law in a vertex neighborhood) and global statistics (conditional law of minimum, maximum, empirical population mean, and subcritical exponential martingales). The presentation resolves Velenik's 2006 open question by proving that even locally, the recentered repelled field is asymptotically not the unconditional field, though in the analogous tree case. This 57-minute talk by Oren Louidor represents joint work with Maximilian Fels (Technion) and Lisa Hartung (Mainz).