Consider a random polynomial H of degree p on the sphere in dimension N: when the coefficients are i.i.d. Gaussian, this is the spherical p-spin glass. This Hamiltonian H has exponentially many in N critical points of every index; moreover, its Langevin dynamics at low temperature are known to take exponentially long in N to equilibrate and are believed to exhibit aging on shorter timescales.
We prove an approximate phase diagram for (H(Xt),|∇ H(Xt)|2) on order-one time scales. We will discuss consequences of this phase diagram, e.g., uniformly over all starting states, Langevin dynamics at any temperature reaches and remains in a region of macroscopically negative energies, and is repelled by macroscopic neighborhoods of critical points.
This is a Joint work with G. Ben Arous and A. Jagannath.
Reza Gheissari is a 4th year Ph.D. student at NYU Courant co-advised by Charles Newman and Eyal Lubetzky. His research interests are in probability theory and mathematical physics, including dynamics of spin systems, their phase transitions and mixing times.
Seminar by the NYU-ECNU Institute of Mathematical Sciences at NYU Shanghai