ALT We prove Poisson local eigenvalue statistics for one-dimensional Schrödinger operators with nondegenerate Bernoulli potentials at any fixed nonzero coupling. After unfolding by the integrated density of states (IDS), the local eigenvalue process converges to a Poisson point process of unit intensity at every energy in the interior of the almost-sure spectrum. The theorem also holds for bounded nonconstant iid potentials, provided the IDS is continuous and strictly increasing near the reference energy. If the IDS also has a finite positive derivative at that energy, the usual linear normalization gives the same limit. The proof uses a phase coordinate with exact IDS drift to obtain a uniform exponential crossing law and joint Poisson counts in disjoint energy windows.
ALT We prove that below the multicritical (Nishimori) point the ferromagnetic boundary of a biased Ising spin glass, argued by Nishimori (1986) to be exactly vertical, can be vertical and can also be reentrant, so that cooling at fixed disorder destroys order. It is vertical in the Sherrington-Kirkpatrick (SK) model with a Curie-Weiss bias J₀, and reentrant near the triple point of fully connected p-spin glasses and, at one disorder strength, on a decorated planar ℤ²-periodic lattice. For every T<J the SK magnetization vanishes for J₀≤ J, continuously at J₀=J, and is macroscopic for J₀>J. For every integer p≥3, at each bias slightly larger than the triple-point bias, the p-spin magnetization is macroscopic at the Nishimori temperature and vanishes at an explicit lower temperature; this is Nishimori's recent replica prediction in two-temperature form. On the lattice (ℤ² with each bond replaced by a fixed series-parallel gadget; maximum degree four), with iid ± J couplings at ℙ(Jₑ=-1)=9/1000
ALT We prove Poisson local eigenvalue statistics for one-dimensional Schrödinger operators with nondegenerate Bernoulli potentials at any fixed nonzero coupling. After unfolding by the integrated density of states (IDS), the local eigenvalue process converges to a Poisson point process of unit intensity at every energy in the interior of the almost-sure spectrum. The theorem also holds for bounded nonconstant iid potentials, provided the IDS is continuous and strictly increasing near the reference energy. If the IDS also has a finite positive derivative at that energy, the usual linear normalization gives the same limit. The proof uses a phase coordinate with exact IDS drift to obtain a uniform exponential crossing law and joint Poisson counts in disjoint energy windows.
ALT Cover of the Proceedings of the American Mathematical Society journal featuring a list of contributing authors on a gray background.
