A lower bound for the spectral gap of the conjugate Kac process with 3 interacting particles

In this paper, we proceed as suggested in the final section of arXiv:1812.03874v2 and prove a lower bound for the spectral gap of the conjugate Kac process with 3 interacting particles. This bound turns out to be around $0.02$, which is already physically meaningful, and we perform Monte Carlo simulations to provide a better empirical estimate for this value via entropy production inequalities. This finishes a complete quantitative estimate of the spectral gap of the Kac process.