Yan Fyodorov and Bertrand Lacroix-A-Chez-Toine – King’s College London
Wednesday, November 20th , 10:45AM EDT
Superposition of plane waves in high spatial dimensions: from landscape complexity to the ground state value.
We will discuss some statistical properties of a class of models of high-dimensional random landscapes defined in a Euclidean space of large dimension $Ngg 1$ via a superposition of $M$ plane waves whose amplitudes, directions of the wavevectors, and phases are taken to be random. Our main efforts are directed towards deriving, and then analysing for $Nto infty, Mto infty$ while keeping $alpha=M/N$ finite,
(i) the rates of asymptotic exponential growth with $N$ of the mean number of all critical points and of local minima known as the annealed complexities and
(ii) the expression for the mean (also expected to be typical) value of the deepest landscape minimum (the ground-state energy).
In particular, for the latter we derive the Parisi-like optimization functional and analyze conditions for the optimizer to reflect various phases: replica-symmetric, one-step and full replica symmetry broken, as well as criteria for the continuous, Gardner and random first order transitions between those phases.