Mixing times of Langevin dynamics for spiked matrix models

We investigate the Langevin dynamics for Wigner matrices with a spherical spike, in the regime where the signal-to-noise ratio $\theta$ is large, but order one. For large, order-$1$, signal-to-noise, the (worst-case) mixing time undergoes a sharp transition around the critical inverse temperature $\beta_c(\theta) = \frac{1}{\theta}$. Namely, if $\beta = \alpha/\theta$, and $\alpha<1$ then at large $\theta$ the mixing time is $O(\log N)$, and if $\alpha>1$ it is exponential in $N$. We show that initialized from the uniform-at-random spherical prior, however, the mixing time in the low-temperature $\alpha>1$ regime circumvents the exponential bottleneck and the mixing time is $O(\log N)$. In fact, this fast mixing holds for any initialization that is symmetric with respect to the top eigenvector of the spiked matrix. Using this, we are able to show a low-temperature metastability picture, pinning down the exact exponential rate of the (worst-case initialization) mixing time for low temperatures, showing it is given by the difference of the free energies of the spiked and null models.