The occupation time of a random walk generated by the uniform random permutation--a GPT-6 Astra proof

Abstract

This note provide a proof of Conjecture 2.5 in Fang et al. (J. Appl. Probab., 58(4):851--867, 2021), with the help of \texttt{GPT-6 Astra}. We show that the number of edges lying above zero of a random walk generated from the uniform permutation has the discrete arcsine law. The proof is short but hinges on an unexpected representation. We also make several comments on solving the problem using large language models.

arXiv comment4 pages