### On a perceptron version of the Generalized Random Energy Model

Bolthausen, E *(Zürich)*

Monday 15 December 2008, 11:30-12:30

Seminar Room 1, Newton Institute

#### Abstract

We consider a macroscopically large family of cavity variables whose covariances are given in a hierarchical way, similar to Derrida's generalized random energy model. The model is similar to the perceptron where the cavity variables have a covariance structure given by the overlaps of two spin configurations. We therefore call the model a perceptron version of the GREM. In contrast to the GREM itself, the model has a rich asymptotic behavior. The free energy is given by a formula which closely resembles the Parisi formula for the SK model. The limiting ultrametric structure depends in a complicated way on the the temperature parameter. Also, in contrast to the GREM, the model has the chaos property (in temperature).

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