Department of Mathematics - Seminar on Probability - Complexity of high dimensional Gaussian random fields with isotropic increments

Random fields with isotropic increments were introduced by Kolmogorov in the 1940s.  Gaussian random fields on N-dimensional Euclidean spaces with isotropic increments were classified as isotropic case and non-isotropic case by Yaglom in the 1950s.  Such models were used widely in statistical physics.  In particular, they were introduced to model a single particle in a random potential by Engel, Mezard and Parisi in 1990s.  A basic question is to count the number of critical points (or local minima, saddles, etc.) of the fields, which is commonly known as complexity.  In 2004, Fyodorov computed the large N limit of expected number of critical points for isotropic Gaussian random fields.  However, the non-isotropic case creates new difficulty.  In this talk, I will present some results on the large N behavior of complexity for this case. Connection to random matrices and large deviations will be explained.  This talk is based on joint work with Antonio Auffinger (Northwestern University).