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In this paper, we proceed as suggested in the final section ofarXiv:1812.03874v2 and prove a lower bound for the spectral gap of theconjugate Kac process with 3 interacting particles. This bound turns out to bearound $0.02$, which is already physically meaningful, and we perform MonteCarlo simulations to provide a better empirical estimate for this value viaentropy production inequalities. This finishes a complete quantitative estimateof the spectral gap of the Kac process.

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