The disordered ferromagnet is a disordered version of the ferromagnetic Ising model in which the coupling constants are quenched random, chosen independently from a distribution on the non-negative reals. A ground configuration is an infinite-volume configuration whose energy cannot be reduced by finite modifications. It is a long-standing challenge to ascertain whether the disordered ferromagnet on the Z^D lattice admits non-constant ground configurations. When D=2, the problem is equivalent to the existence of bigeodesics in first-passage percolation, so a negative answer is expected. We provide a positive answer in dimensions D>=4, when the distribution of the coupling constants is sufficiently concentrated.
Our result is proved by showing that the finite-volume interface formed by Dobrushin boundary conditions is localized, and converges to an infinite-volume interface. This may be expressed in purely combinatorial terms: A random environment is formed by endowing the lattice Z^D with independent, identically distributed edge capacities and we study the fluctuations of certain minimal cuts in this environment.
The talk will discuss the problem and its background, and present ideas from the proof. Based on joint work of the speakers with Shoni Gilboa.