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

TL;DR

This paper precisely characterizes the asymptotic return probability of a specific lamplighter walk on an infinite regular tree, demonstrating advanced probabilistic analysis.

Contribution

It provides the first sharp asymptotic formula for the return probability of the switch--walk--switch lamplighter walk on a regular tree.

Findings

Derived the exact asymptotic form of return probability

Confirmed the formula through automated proof generation

Demonstrated AI's capability in producing rigorous proofs

Abstract

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.