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Variational methods for a singular SPDE yielding the universality of the magnetisation ripple

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The magnetisation ripple is a microstructure formed by the magnetisation in thin ferromagnetic films, triggered by the random orientation of the grains in a polycrystalline material. It is described by the minimisers of a non-convex energy functional. The corresponding Euler—Lagrange equation is given by a singular elliptic SPDE in two dimensions driven by white noise. On large scales, the ripple exhibits a universal behaviour, in the sense that it does not depend on the statistical model of the grain distribution. In this talk, I will address the universality of the ripple based on a variational approach.

First, a suitable renormalisation of the random energy functional has to be preformed due to the roughness of white noise. Then, using the topology of $\Gamma$-convergence, one can give sense to the law of the renormalised energy functional, which is independent of the way white noise is approximated. This universality result holds in the class of (not necessarily Gaussian) approximations to white noise satisfying the spectral gap inequality, which allows us to obtain sharp stochastic estimates.

The talk is based on a joint work with Radu Ignat, Felix Otto, and Tobias Ried.

This talk is part of the Probability series.

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