Exit problem of particles interacting with their empirical law

The current article is devoted to the study of a mean-field system of particles. More precisely, we solve the exit-problem of the first particle (and from any particle) from a domain on $\bRb^d$. We establish a Kramers' type law with an exit-cost which converges to a given quantity as the number of particles is large. For doing so, we slightly modify the classical assumptions on the Freidlin-Wentzell theory and we use the recent result about the large deviations from Herrmann and us. The main improvement of the paper is that it is applied without assuming any global convexity of the external force nor on the interacting potential