Spectral gap for a random walks on groups govern the rate of mixing. Suppose $G_1$ 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 any probability measure $\mu$ on $G_1 \times G_2$ whose marginal distributions are $\mu_1$ and $\mu_2$, respectively . A natural question is under what conditions a spectral gap for all couplings depending on spectral gaps of $\mu_1$ and $\mu_2$ can be established.
In this talk, I will present results in this direction which are based on joint work with Alireza S. Golsefidy and Amir Mohammadi.
