For the switch–walk–switch lamplighter walk with Z₂ lamps over the infinite d-regular tree (d≥3), p₂ₙ(e,e)=ρ_d^{2n} exp[−(π²(log(d−1))²+o(1))n/(log n)²], where ρ_d=2√(d−1)/d.
For the switch–walk–switch lamplighter walk with Z₂ lamps over the infinite d-regular tree (d≥3), p₂ₙ(e,e)=ρ_d^{2n} exp[−(π²(log(d−1))²+o(1))n/(log n)²], where ρ_d=2√(d−1)/d.
For the switch–walk–switch lamplighter walk with Z₂ lamps over the infinite d-regular tree (d≥3), p₂ₙ(e,e)=ρ_d^{2n} exp[−(π²(log(d−1))²+o(1))n/(log n)²], where ρ_d=2√(d−1)/d.
If correct, gives a sharp asymptotic beyond the previously available spectral representation for this walk.
For the switch-walk-switch lamplighter walk on $\mathbb{Z}_2 \wr T_d$, prove the sharp asymptotic $p_{2n}(e,e) = \rho_d^{2n} \exp[-(\pi^2 (\log(d-1))^2 + o(1)) \frac{n}{\log^2 n}]$ with $\rho_d = \frac{2\sqrt{d-1}}{d}$.
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QED-reported sharp lamplighter return asymptotic on regular trees
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