大川 航平

We study the twisted Ihara zeta functions and the spectral theory of twisted non-backtracking operators on random d-regular graphs. For a uniformly random d-regular Ramanujan graph equipped with a random unitary local system (magnetic twist), we establish a uniform twisted spectral gap in probability. This spectral gap provides a uniform zero-free annulus for the associated twisted Ihara L-function, which holds with high probability as the number of vertices tends to infinity. As a primary consequence of this zero-free region, we prove a typical exponential cancellation phenomenon for twisted primitive cycles. Specifically, we show that the twisted prime cycle sum exhibits strong cancellation in the length parameter, far exceeding the bounds available for any fixed graph or fixed twist. Furthermore, we apply this exponential cancellation to establish magnetic quantum ergodicity. We prove that the quantum probability distribution of a magnetic quantum walk on a typical random regular graph converges weakly to the Haar measure on the unit circle. To the best of our knowledge, these are the first results establishing uniform zero-free regions and typical exponential cancellation for twisted prime cycles on random graphs.