Let $G_1$ and $G_2$ be compact simple Lie groups, and let $\mu_1$ and $\mu_2$ be symmetric probability measures on $G_1$ and $G_2$. Under mild conditions on $\mu_1$ and $\mu_2$, one knows that the distribution of the random walk on each group driven by the corresponding probability measure converges to the uniform distribution, and the speed of convergence is governed by the spectral gap.
A coupling of $\mu_1$ and $\mu_2$ is defined as a probability measure $\mu$ on $G_1 \times G_2$ with marginal distributions $\mu_1$ and $\mu_2$. The central question of our study is conditions under which $\mu$ will have a spectral gap depending on the spectralgaps of $\mu_1$ and $\mu_2$.
In this talk I will first review some of the old and new methods for establishing spectral gaps, with special focus on the pioneering works of Bourgain and Gamburd and then discuss the question stated above. The talk is based on a joint work with Amir Mohammadiand Alireza Salehi Golsefidy.