# Joint Dept. of Mathematics and CMSA Random Matrix and Probability Theory Seminar

RANDOM MATRIX

##### Speaker:

Antti Knowles *- University of Geneva*

*will speak on:*

**FIELD THEORY AS A LIMIT OF INTERACTING QUANTUM BOSE GASES**

We prove that the grand canonical Gibbs state of an interacting quantum Bose gas converges to the Gibbs measure of a nonlinear Schrödinger equation in the mean-field limit, where the density of the gas becomes large and the interaction strength is proportional to the inverse density. Our results hold in dimensions d = 1,2,3. For d > 1 the Gibbs measure is supported on distributions of negative regularity and we have to renormalize the interaction. The proof is based on a functional integral representation of the grand canonical Gibbs state, in which convergence to the mean-field limit follows formally from an infinite-dimensional stationary phase argument for ill-defined non-Gaussian measures. We make this argument rigorous by introducing a white-noise-type auxiliary field, through which the functional integral is expressed in terms of propagators of heat equations driven by time-dependent periodic random potentials. Joint work with Jürg Fröhlich, Benjamin Schlein, and Vedran Sohinger.

via Zoom: https://harvard.zoom.us/j/147308224