Seminar in probability theory: Anton Klimovsky (Duisburg-Essen)

High-dimensional Gaussian fields with isotropic increments seen through spin glasses

Finding the (space-height) distribution of the (local) extrema of high-dimensional strongly correlated random fields is a notorious hard problem with many applications. Following Fyodorov & Sommers (2007), we focus on the Gaussian fields with isotropic increments and take the viewpoint of statistical physics. By exploiting various probabilistic symmetries, we rigorously derive the Fyodorov-Sommers formula for the log-partition function in the high-dimensional limit. The formula suggests a rich picture for the distribution of the local extrema akin to the celebrated spherical Sherrington-Kirkpatrick model with mixed p-spin interactions.

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