Return Probability for the Switch--Walk--Switch Lamplighter Walk on a Regular Tree

We derive the sharp return-probability asymptotic for the switch--walk--switch lamplighter walk with lamp group $\mathbb Z_2$ over the infinite $d$-regular tree: $$p_{2n}(e,e) = \rho_d^{2n} \exp\left[ -\left(\pi^2(\log(d-1))^2+o(1)\right) \frac{n}{\log^2 n} \right].$$ The proofs were generated by QED, a multi-agent system co-developed by the authors, without human intervention beyond the specification of the problem. This provides a test case for the ability of AI systems to produce rigorous mathematical proofs.